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

Optimal. Leaf size=265 \[ -\frac {\sqrt [4]{c} \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\sqrt [4]{c} \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {a e-c d x}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {e \left (5 a e^2+c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^2} \]

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Rubi [A]  time = 0.48, antiderivative size = 265, 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, 829, 827, 1166, 208} \begin {gather*} -\frac {\sqrt [4]{c} \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\sqrt [4]{c} \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {a e-c d x}{2 a \left (a-c x^2\right ) \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {e \left (5 a e^2+c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(c*d^2 + 5*a*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x
]*(a - c*x^2)) - (c^(1/4)*(2*Sqrt[c]*d - 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]
*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + (c^(1/4)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*(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 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 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

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

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Mathematica [C]  time = 0.43, size = 331, normalized size = 1.25 \begin {gather*} \frac {\frac {3 c^{3/4} d \left (\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a}}+\frac {\left (5 a e^2+c d^2\right ) \left (\left (\sqrt {a} e+\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {a} e}\right )+\left (\sqrt {a} e-\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}\right )\right )}{2 \sqrt {a} \sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {a e-c d x}{\left (a-c x^2\right ) \sqrt {d+e x}}}{2 a \left (a e^2-c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e - c*d*x)/(Sqrt[d + e*x]*(a - c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sq
rt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt
[c]*d + Sqrt[a]*e]))/(2*Sqrt[a]) + ((c*d^2 + 5*a*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2,
(Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sq
rt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(2*Sqrt[a]*(c*d^2 - a*e^2)*Sqrt[d + e*x]))/(2*a*(-(c*d^2) + a*e^2)
)

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IntegrateAlgebraic [A]  time = 2.40, size = 379, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )} \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^3}-\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )} \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} \left (\sqrt {a} e-\sqrt {c} d\right )^3}+\frac {-4 a^2 e^5+4 a c d^2 e^3+5 a c e^3 (d+e x)^2-11 a c d e^3 (d+e x)-c^2 d^3 e (d+e x)+c^2 d^2 e (d+e x)^2}{2 a \sqrt {d+e x} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*c*d^2*e^3 - 4*a^2*e^5 - c^2*d^3*e*(d + e*x) - 11*a*c*d*e^3*(d + e*x) + c^2*d^2*e*(d + e*x)^2 + 5*a*c*e^3*
(d + e*x)^2)/(2*a*(-(c*d^2) + a*e^2)^2*Sqrt[d + e*x]*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)) - (
Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*
Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^3) - ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*
Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))]*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d -
Sqrt[a]*e)])/(4*a^(3/2)*(-(Sqrt[c]*d) + Sqrt[a]*e)^3)

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fricas [B]  time = 1.03, size = 5703, normalized size = 21.52

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d
^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 3
5*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^
4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*
c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*
e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^
3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a
^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 +
3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^
2*d^3*e^8 + 325*a^5*c*d*e^10 + (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^
6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*
a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d
^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*
c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^
7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d
^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966
*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*
d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^
10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 +
10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - (a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^
4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^
2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 1
05*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8
- a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 6
25*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6
*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a
^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*
a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*s
qrt(e*x + d) - (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 + 325*a^5*c*d*e^10 + (2*a^3*c^
7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e
^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 +
7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*
c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 4
5*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^
4 + 105*a^3*c*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^
2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^1
2 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^
7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 -
10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6
+ 5*a^7*c*d^2*e^8 - a^8*e^10))) + (a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e
^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a
^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c
^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10
780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c
^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*
a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d
^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*
d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c^4*d^7*e^4 - 609
*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 + 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^
5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*s
qrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)
/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 25
2*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18
+ a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*
a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6
- 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*
a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 +
210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*
c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - (a^2*
c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2
*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt((4*c^4*d^7 - 35*a*c^3*d^
5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6
*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^
10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 - 120*
a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6*e^14
+ 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6
*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log((140*c^4*d^6*e^3 - 1491*a*c^3*d^4*e^5 + 3750*a^2*
c^2*d^2*e^7 + 625*a^3*c*e^9)*sqrt(e*x + d) - (35*a^2*c^4*d^7*e^4 - 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8
+ 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 - 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 - 85*a^6*c^4*d^8*e^6 + 50*a^7
*c^3*d^6*e^8 + 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 + 5*a^10*e^14)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*
e^8 + 21966*a^2*c^3*d^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 +
45*a^5*c^8*d^16*e^4 - 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^
12 - 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt((4*c^4*d^7 - 35*a*c
^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 1
0*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((1225*c^5*d^8*e^6 - 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d
^4*e^10 + 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 - 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 -
120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 - 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 - 120*a^10*c^3*d^6
*e^14 + 45*a^11*c^2*d^4*e^16 - 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^
3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) - 4*(2*a*c*d^2*e + 4*a^2*e^3 - (c^2*d^2*e + 5*a
*c*e^3)*x^2 - (c^2*d^3 - a*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^2*d^5 - 2*a^3*c*d^3*e^2 + a^4*d*e^4 - (a*c^3*d^4*
e - 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 - (a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e -
2*a^3*c*d^2*e^3 + a^4*e^5)*x)

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giac [B]  time = 0.95, size = 1333, normalized size = 5.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e + 5*a*e^3)*abs(c) + (sqrt(a*c)*c^3*d^7*e - 15*sqrt(
a*c)*a*c^2*d^5*e^3 + 27*sqrt(a*c)*a^2*c*d^3*e^5 - 13*sqrt(a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 +
a^3*e^5)*abs(c) - 2*(a*c^6*d^12*e - 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 - 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*
d^4*e^9 - 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 +
sqrt((a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*
e^6)*(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8
*e - sqrt(a*c)*a*c^4*d^9 + 4*sqrt(a*c)*a^2*c^3*d^7*e^2 - 4*a^3*c^3*d^6*e^3 - 6*sqrt(a*c)*a^3*c^2*d^5*e^4 + 6*a
^4*c^2*d^4*e^5 + 4*sqrt(a*c)*a^4*c*d^3*e^6 - 4*a^5*c*d^2*e^7 - sqrt(a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^2*d - sq
rt(a*c)*c*e)*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)) - 1/4*((a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)^2*
(c*d^2*e + 5*a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^7*e - 15*sqrt(a*c)*a*c^2*d^5*e^3 + 27*sqrt(a*c)*a^2*c*d^3*e^5 -
13*sqrt(a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) - 2*(a*c^6*d^12*e - 8*a^2*c^5*d^10
*e^3 + 22*a^3*c^4*d^8*e^5 - 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 - 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(x*
e + d)/sqrt(-(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt((a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)
^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))
/(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8*e + sqrt(a*c)*a*c^4*d^9 - 4*sqrt(a*c)*a^2*c^3*d^7
*e^2 - 4*a^3*c^3*d^6*e^3 + 6*sqrt(a*c)*a^3*c^2*d^5*e^4 + 6*a^4*c^2*d^4*e^5 - 4*sqrt(a*c)*a^4*c*d^3*e^6 - 4*a^5
*c*d^2*e^7 + sqrt(a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a
^3*e^5)) - 1/2*((x*e + d)^2*c^2*d^2*e - (x*e + d)*c^2*d^3*e + 5*(x*e + d)^2*a*c*e^3 - 11*(x*e + d)*a*c*d*e^3 +
4*a*c*d^2*e^3 - 4*a^2*e^5)/((a*c^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e^4)*((x*e + d)^(5/2)*c - 2*(x*e + d)^(3/2)*c*
d + sqrt(x*e + d)*c*d^2 - sqrt(x*e + d)*a*e^2))

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maple [B]  time = 0.09, size = 783, normalized size = 2.95 \begin {gather*} \frac {c^{3} d^{3} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {c^{3} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {2 c^{2} d \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 c^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\sqrt {e x +d}\, c^{2} d^{3} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}+\frac {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 )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {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 )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {3 \sqrt {e x +d}\, c d \,e^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c \,e^{2} x^{2}-a \,e^{2}\right )}+\frac {5 c \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {5 c \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e x +d \right )^{\frac {3}{2}} c^{2} d^{2} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}-\frac {\left (e x +d \right )^{\frac {3}{2}} c \,e^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c \,e^{2} x^{2}-a \,e^{2}\right )}-\frac {2 e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(3/2)-1/2*e*c^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)/a*(e*x+
d)^(3/2)*d^2+3/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(1/2)+1/2*e*c^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-
a*e^2)*d^3/a*(e*x+d)^(1/2)-2*e^3*c^2/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((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+1/2*e*c^3/(a*e^2-c*d^2)^2/a/(a*c*e^2)^(1/2)/((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+5/4*e^3*c/(a*e^2-c*d^2)^2/((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)+1/4*e*c^2/(a*e^2-c*d^2)^2/a/
((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^2-2*e^3*c^2/(a*e^2-
c*d^2)^2/(a*c*e^2)^(1/2)/((-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+1/2*e*c^3/(a*e^2-c*d^2)^2/a/(a*c*e^2)^(1/2)/((-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^3-5/4*e^3*c/(a*e^2-c*d^2)^2/((-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)-1/4*e*c^2/(a*e^2-c*d^2)^2/a/((-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^2-2*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)

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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 )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.16, size = 8700, normalized size = 32.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*
(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*
e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 -
25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c
*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 -
10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c
^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c
^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*
d^3*e^20) - 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 - 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 -
107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 - 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 - 95
232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2
- 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008
*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^
7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^
6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^
2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i + ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^
4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^
(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*
d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^
2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a
^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^
16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d
^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*
d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 - 256*a^5*c^13*d^19*e^3 +
5376*a^6*c^12*d^17*e^5 - 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 - 204288*a^9*c^9*d^11*e^11 + 2472
96*a^10*c^8*d^9*e^13 - 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 - 26880*a^13*c^5*d^3*e^19) - (d + e*
x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848
*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6
*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5
*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10
- a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i)/(
((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c
)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 +
5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^
2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6
- 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^
8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c^13*d^
19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c^9*d^1
1*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*d^3*e^
20) - 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 - 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 - 107520
*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 - 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 - 95232*a^
12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 1760
*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008*a^8*c
^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25
*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*
e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10
*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5
*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(6
4*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6))
)^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4
+ 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^
10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*
e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 +
430080*a^10*c^10*d^13*e^10 - 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16
+ 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 - 256*a^5*c^13*d^19*e^3 + 5376*a^6
*c^12*d^17*e^5 - 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 - 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c
^8*d^9*e^13 - 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 - 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*
(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848*a^6*c^10
*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6*d^4*e^16
- 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*
e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5
*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) + 1000*a^10*c^
4*e^19 - 32*a^2*c^12*d^16*e^3 + 232*a^3*c^11*d^14*e^5 + 280*a^4*c^10*d^12*e^7 - 4760*a^5*c^9*d^10*e^9 + 13720*
a^6*c^8*d^8*e^11 - 19208*a^7*c^7*d^6*e^13 + 14728*a^8*c^6*d^4*e^15 - 5960*a^9*c^5*d^2*e^17))*(-(4*a^3*c^4*d^7
- 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*
c*d*e^6 - 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2
- 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*2i - ((2*e^3)/(a*e^2 - c*d^2) - (c*e*(5*a*e^2 + c*d^2)*(d +
e*x)^2)/(2*a*(a*e^2 - c*d^2)^2) + (c*d*e*(11*a*e^2 + c*d^2)*(d + e*x))/(2*a*(a*e^2 - c*d^2)^2))/((a*e^2 - c*d
^2)*(d + e*x)^(1/2) - c*(d + e*x)^(5/2) + 2*c*d*(d + e*x)^(3/2)) + atan((((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c
)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d
^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^
4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*
e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64
*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))
^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 24
5760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 2
45760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*d^3*e^20) - 3328*a^14*c^4*d*e^21 + 256*a^5*
c^13*d^19*e^3 - 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 - 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d
^11*e^11 - 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 - 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*
e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d
^14*e^6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14
- 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d
^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/
(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6
)))^(1/2)*1i + ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2
*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^
10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*
c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) +
105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*
d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20
480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 5160
96*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*
a^15*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 - 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 - 33792*a^7*c^11*d^
15*e^7 + 107520*a^8*c^10*d^13*e^9 - 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 - 193536*a^11*c^7*d^7*
e^15 + 95232*a^12*c^6*d^5*e^17 - 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*
d^18*e^2 - 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^1
0 - 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a
^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2)
+ 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c
^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i)/(((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2
) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5
*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10
*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 +
70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11
*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)
*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a
^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*
a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*d^3*e^20) - 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d
^19*e^3 - 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 - 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^
11 - 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 - 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19)
- (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2 - 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^
6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200
*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2
+ 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a
^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1
/2) - ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*
(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*
e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 +
25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c
*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 -
10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 - 20480*a^7*c
^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 - 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 - 516096*a^11*c
^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 - 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 - 20480*a^15*c^5*
d^3*e^20) + 3328*a^14*c^4*d*e^21 - 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 - 33792*a^7*c^11*d^15*e^7 +
107520*a^8*c^10*d^13*e^9 - 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 - 193536*a^11*c^7*d^7*e^15 + 95
232*a^12*c^6*d^5*e^17 - 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(800*a^12*c^4*e^20 + 128*a^3*c^13*d^18*e^2
- 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 - 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 - 51008
*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 - 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^
7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^
6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(64*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^
2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) + 1000*a^10*c^4*e^19 - 32*a^2*c^12*d^16*e^3 + 232*a^3*c^1
1*d^14*e^5 + 280*a^4*c^10*d^12*e^7 - 4760*a^5*c^9*d^10*e^9 + 13720*a^6*c^8*d^8*e^11 - 19208*a^7*c^7*d^6*e^13 +
14728*a^8*c^6*d^4*e^15 - 5960*a^9*c^5*d^2*e^17))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(a^9*c)^(1/2) - 35*a^4*c^3*d^5
*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c)^(1/2) + 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(a^9*c)^(1/2))/(6
4*(a^11*e^10 - a^6*c^5*d^10 - 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 - 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6))
)^(1/2)*2i

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

[Out]

Timed out

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