3.6.62 \(\int \frac {1}{\sqrt {d+e x} (a-c x^2)^3} \, dx\)

Optimal. Leaf size=315 \[ -\frac {3 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.63, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {741, 823, 827, 1166, 208} \begin {gather*} -\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac {3 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(a - c*x^2)^3),x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 - a*e^2)*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(c*d^2 - 7*a*e^2) - 6
*c*d*(c*d^2 - 2*a*e^2)*x))/(16*a^2*(c*d^2 - a*e^2)^2*(a - c*x^2)) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a
*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e
)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sq
rt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx &=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}+\frac {\int \frac {\frac {1}{2} \left (6 c d^2-7 a e^2\right )+\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a \left (c d^2-a e^2\right )}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\int \frac {-\frac {3}{4} c \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac {3}{2} c^2 d e \left (c d^2-2 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c \left (c d^2-a e^2\right )^2}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{2} c^2 d^2 e \left (c d^2-2 a e^2\right )-\frac {3}{4} c e \left (4 c^2 d^4-9 a c d^2 e^2+7 a^2 e^4\right )-\frac {3}{2} c^2 d e \left (c d^2-2 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c \left (c d^2-a e^2\right )^2}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\left (3 \sqrt {c} \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \left (\sqrt {c} d-\sqrt {a} e\right )^2}+\frac {\left (3 \sqrt {c} \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \left (\sqrt {c} d+\sqrt {a} e\right )^2}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.67, size = 335, normalized size = 1.06 \begin {gather*} \frac {\frac {2 \sqrt {d+e x} \left (7 a^2 e^3-a c d e (d+12 e x)+6 c^2 d^3 x\right )}{a-c x^2}+\frac {8 a \sqrt {d+e x} \left (c d^2-a e^2\right ) (c d x-a e)}{\left (a-c x^2\right )^2}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )-3 \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a} e} \sqrt {\sqrt {a} e+\sqrt {c} d}}}{32 a^2 \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)^3),x]

[Out]

((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*Sqrt[d + e*x])/(a - c*x^2)^2 + (2*Sqrt[d + e*x]*(7*a^2*e^3 + 6*c^2*d^3*
x - a*c*d*e*(d + 12*e*x)))/(a - c*x^2) + (-3*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e +
7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(4*c*
d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]
*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(32*a^2*(c*d^2 - a*e^2)^2)

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IntegrateAlgebraic [A]  time = 1.37, size = 496, normalized size = 1.57 \begin {gather*} \frac {3 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{32 a^{5/2} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {3 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\sqrt {d+e x} \left (11 a^3 e^7+4 a^2 c d^2 e^5-7 a^2 c e^5 (d+e x)^2-2 a^2 c d e^5 (d+e x)-21 a c^2 d^4 e^3+44 a c^2 d^3 e^3 (d+e x)-35 a c^2 d^2 e^3 (d+e x)^2+12 a c^2 d e^3 (d+e x)^3+6 c^3 d^6 e-18 c^3 d^5 e (d+e x)+18 c^3 d^4 e (d+e x)^2-6 c^3 d^3 e (d+e x)^3\right )}{16 a^2 \left (a e^2-c d^2\right )^2 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[d + e*x]*(a - c*x^2)^3),x]

[Out]

(Sqrt[d + e*x]*(6*c^3*d^6*e - 21*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 + 11*a^3*e^7 - 18*c^3*d^5*e*(d + e*x) + 44*a*
c^2*d^3*e^3*(d + e*x) - 2*a^2*c*d*e^5*(d + e*x) + 18*c^3*d^4*e*(d + e*x)^2 - 35*a*c^2*d^2*e^3*(d + e*x)^2 - 7*
a^2*c*e^5*(d + e*x)^2 - 6*c^3*d^3*e*(d + e*x)^3 + 12*a*c^2*d*e^3*(d + e*x)^3))/(16*a^2*(-(c*d^2) + a*e^2)^2*(-
(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)^2) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTan
[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(32*a^(5/2)*(Sqrt[c]*d + Sqrt[a]*e
)^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTan[(Sqrt[-
(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(32*a^(5/2)*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt
[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 1.79, size = 5776, normalized size = 18.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/64*(3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^
3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 21
0*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6
+ 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3
*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^
6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^
3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10
*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d
^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 5
15*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*
d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*
c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^
4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*
e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*
c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 +
105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8
- a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401
*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^
12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^
14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 +
5*a^9*c*d^2*e^8 - a^10*e^10))) - 3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^
2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^
2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d
^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974
*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^
16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^
12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e
^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*
c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e
^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*d^15 - 31*a^6
*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*
a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5
292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*
d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*
a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^
5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^
2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14
- 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c
^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 +
45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^
6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))) + 3*(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^
2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt((
16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 - 5*a^
6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10
- 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c
^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 21
0*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^
5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))*log(2
7*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x
+ d) + 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*a^6*c*d^2*e^12 + 343*a^7*e^1
4 + (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 -
169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12
+ 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7
*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 -
120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt((16*c^4*d^9 - 84*a*
c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10
*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e
^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*
a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^1
2 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6
*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10))) - 3*(a^4*c^2*d^4 - 2*a
^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*
e^2 + a^5*c*e^4)*x^2)*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*
d*e^8 - (a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e
^10)*sqrt((441*c^4*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^1
8)/(a^5*c^11*d^20 - 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 -
252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d
^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c
*d^2*e^8 - a^10*e^10))*log(27*(336*c^4*d^8*e^5 - 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 - 4802*a^3*c*d^2*e^
11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e^6 - 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 - 623*
a^6*c*d^2*e^12 + 343*a^7*e^14 + (4*a^5*c^8*d^15 - 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 - 205*a^8*c^5*d^9
*e^6 + 240*a^9*c^4*d^7*e^8 - 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 - 11*a^12*c*d*e^14)*sqrt((441*c^4*d^
8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 - 1
0*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10*e^
10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e^20
)))*sqrt((16*c^4*d^9 - 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 - 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d
^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^10)*sqrt((441*c^4
*d^8*e^10 - 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 - 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20
- 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 - 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 - 252*a^10*c^6*d^10
*e^10 + 210*a^11*c^5*d^8*e^12 - 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 - 10*a^14*c^2*d^2*e^18 + a^15*c*e
^20)))/(a^5*c^5*d^10 - 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 - 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 - a^10*e^
10))) - 4*(5*a^2*c*d^2*e - 11*a^3*e^3 + 6*(c^3*d^3 - 2*a*c^2*d*e^2)*x^3 - (a*c^2*d^2*e - 7*a^2*c*e^3)*x^2 - 2*
(5*a*c^2*d^3 - 8*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^4*c^2*d^4 - 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 - 2*a^
3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 - 2*(a^3*c^3*d^4 - 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)

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giac [B]  time = 0.78, size = 827, normalized size = 2.63 \begin {gather*} -\frac {3 \, {\left (4 \, c d^{2} + 10 \, \sqrt {a c} d e + 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} + \sqrt {{\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} - 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} - 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} - 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e + \sqrt {a c} a^{2} c d^{2} + \sqrt {a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} - \frac {3 \, {\left (4 \, c d^{2} - 10 \, \sqrt {a c} d e + 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} - \sqrt {{\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} - 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} - 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} - 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e - \sqrt {a c} a^{2} c d^{2} - \sqrt {a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e - 12 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} + 35 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} - 44 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} + 21 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 4 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} - 11 \, \sqrt {x e + d} a^{3} e^{7}}{16 \, {\left (a^{2} c^{2} d^{4} - 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-3/32*(4*c*d^2 + 10*sqrt(a*c)*d*e + 7*a*e^2)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^
2 + a^4*c*d*e^4 + sqrt((a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 - 3*a^3*c^2*d^4*e^2 +
3*a^4*c*d^2*e^4 - a^5*e^6)*(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 +
a^4*c*e^4)))/((2*a^3*c*d*e + sqrt(a*c)*a^2*c*d^2 + sqrt(a*c)*a^3*e^2)*sqrt(-c^2*d - sqrt(a*c)*c*e)) - 3/32*(4*
c*d^2 - 10*sqrt(a*c)*d*e + 7*a*e^2)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c
*d*e^4 - sqrt((a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 - 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d
^2*e^4 - a^5*e^6)*(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4
)))/((2*a^3*c*d*e - sqrt(a*c)*a^2*c*d^2 - sqrt(a*c)*a^3*e^2)*sqrt(-c^2*d + sqrt(a*c)*c*e)) - 1/16*(6*(x*e + d)
^(7/2)*c^3*d^3*e - 18*(x*e + d)^(5/2)*c^3*d^4*e + 18*(x*e + d)^(3/2)*c^3*d^5*e - 6*sqrt(x*e + d)*c^3*d^6*e - 1
2*(x*e + d)^(7/2)*a*c^2*d*e^3 + 35*(x*e + d)^(5/2)*a*c^2*d^2*e^3 - 44*(x*e + d)^(3/2)*a*c^2*d^3*e^3 + 21*sqrt(
x*e + d)*a*c^2*d^4*e^3 + 7*(x*e + d)^(5/2)*a^2*c*e^5 + 2*(x*e + d)^(3/2)*a^2*c*d*e^5 - 4*sqrt(x*e + d)*a^2*c*d
^2*e^5 - 11*sqrt(x*e + d)*a^3*e^7)/((a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 - a*e^2)^2)

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maple [B]  time = 0.60, size = 956, normalized size = 3.03 \begin {gather*} \frac {21 c \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \left (a \,e^{2}+c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {21 c \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \left (-a \,e^{2}-c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {3 c^{2} d^{2} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \left (a \,e^{2}+c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}-\frac {3 c^{2} d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \left (-a \,e^{2}-c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {15 c d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \left (a \,e^{2}+c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {15 c d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \left (-a \,e^{2}-c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} d e}{16 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (a \,e^{2}+c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) a^{2}}+\frac {3 \sqrt {e x +d}\, d e}{16 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (c d +\sqrt {a c \,e^{2}}\right ) a^{2}}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} d e}{16 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (a \,e^{2}+c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) a^{2}}+\frac {3 \sqrt {e x +d}\, d e}{16 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (c d -\sqrt {a c \,e^{2}}\right ) a^{2}}-\frac {9 \sqrt {a c \,e^{2}}\, \left (e x +d \right )^{\frac {3}{2}} e}{32 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (a \,e^{2}+c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) a^{2} c}+\frac {11 \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, e}{32 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (c d +\sqrt {a c \,e^{2}}\right ) a^{2} c}+\frac {9 \sqrt {a c \,e^{2}}\, \left (e x +d \right )^{\frac {3}{2}} e}{32 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (a \,e^{2}+c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) a^{2} c}-\frac {11 \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, e}{32 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} \left (c d -\sqrt {a c \,e^{2}}\right ) a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-c*x^2+a)^3/(e*x+d)^(1/2),x)

[Out]

-3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)*d-9/32*e/c*(a*c*e^2)^(1/
2)/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+3/16*e/a^2/(e*x-(a*c*e^2)^(1/
2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*d+11/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*d+(a*
c*e^2)^(1/2))*(e*x+d)^(1/2)+21/32*e^3*c/(a*c*e^2)^(1/2)/a/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1
/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)+3/8*e*c^2/(a*c*e^2)^(1/2)/a^2/(a*e^2+c*
d^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)
*c)*d^2+15/16*e*c/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/
((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d-3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e
*x+d)^(3/2)*d+9/32*e/c*(a*c*e^2)^(1/2)/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)
^(3/2)+3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*d-11/32*e/c*(a*c*e^2)^(1/2)/a^
2/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)-21/32*e^3*c/(a*c*e^2)^(1/2)/a/(-a*e^2-c*d^2+2*
(a*c*e^2)^(1/2)*d)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-3
/8*e*c^2/(a*c*e^2)^(1/2)/a^2/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d
)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2+15/16*e*c/a^2/(-a*e^2-c*d^2+2*(a*c*e^2)^(1/2)*d)/((-c*d+(a*c*e
^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (c x^{2} - a\right )}^{3} \sqrt {e x + d}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)^3*sqrt(e*x + d)), x)

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mupad [B]  time = 3.71, size = 8961, normalized size = 28.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - c*x^2)^3*(d + e*x)^(1/2)),x)

[Out]

- atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 3891
2*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4))
- ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4
- 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(40
96*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*
d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 -
16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)
))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d
^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10
- a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)
+ ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c
^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*
a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i - (((3*(
14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*
e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(
1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5
*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^
5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^
2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^1
0 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^
(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(
64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 -
49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a
^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c
^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i)/((((3*(14336*a^9*c^3*
e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a
^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*
a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c
^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64
*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49
*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^1
5*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5
*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*a^4
*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 +
a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^
15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) +
105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 1
0*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - (3*(144*c^6*d^7*e^3 - 684*a*c^5*d^5*e
^5 - 882*a^3*c^3*d*e^9 + 1233*a^2*c^4*d^3*e^7))/(1024*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^
6*e^2 + 6*a^8*c^2*d^4*e^4)) + (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*
a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2
+ 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2
+ 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^
7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3
*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 2
4576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^
2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d
^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)
))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^1
4*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c
^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^
2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210
*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^
15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^
8)))^(1/2)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210
*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^
15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^
8)))^(1/2)*2i - atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*
d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*
c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a
^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*
c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6
- 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7
*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^
6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 -
210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096
*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^
2*e^8)))^(1/2) + ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e
^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^
4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*1i - (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 3891
2*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4))
+ ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4
- 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(40
96*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*
d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 -
16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)
))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d
^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10
- a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)
- ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c
^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*
a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*
c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d
^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*1i)/((((3*(
14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*
e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(
1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2
*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^1
0 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/
2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5
*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^
5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^
2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^1
0 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) + ((d + e*x)^
(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(
64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 +
49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a
^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c
^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - (3*(144*c^6*d^7*e^3 - 6
84*a*c^5*d^5*e^5 - 882*a^3*c^3*d*e^9 + 1233*a^2*c^4*d^3*e^7))/(1024*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6
- 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^
6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^
7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a
^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 -
54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4
+ 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c
^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*
a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 +
189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*
(a^15*c)^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d
^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^
4 + 1089*a^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*
e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3
*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/
2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a
^14*c^2*d^2*e^8)))^(1/2)))*(-(9*(16*a^5*c^5*d^9 + 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3
*d^5*e^4 - 210*a^8*c^2*d^3*e^6 + 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(a^15*c)^(1/
2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a
^14*c^2*d^2*e^8)))^(1/2)*2i - ((e*(d + e*x)^(3/2)*(9*c^3*d^5 - 22*a*c^2*d^3*e^2 + a^2*c*d*e^4))/(8*a^2*(a*e^2
- c*d^2)^2) - (e*(d + e*x)^(1/2)*(11*a^2*e^4 - 6*c^2*d^4 + 15*a*c*d^2*e^2))/(16*a^2*(a*e^2 - c*d^2)) + (3*c*e*
(c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(7/2))/(8*a^2*(a*e^2 - c*d^2)^2) + (c*e*(d + e*x)^(5/2)*(7*a^2*e^4 - 18*c^2*
d^4 + 35*a*c*d^2*e^2))/(16*a^2*(a*e^2 - c*d^2)^2))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e
^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c*x**2+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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