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

Optimal. Leaf size=281 \[ -\frac {\left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2} \]

[Out]

-1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(12*c*d^2+5*a*e^2-18*d*e*a^(1/2)*c^(1/2))/a^
(5/2)/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^(3/2)+1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(12
*c*d^2+5*a*e^2+18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(3/4)/(e*a^(1/2)+d*c^(1/2))^(3/2)+1/4*x*(e*x+d)^(1/2)/a/(-c*x
^2+a)^2-1/16*(a*d*e-(-5*a*e^2+6*c*d^2)*x)*(e*x+d)^(1/2)/a^2/(-a*e^2+c*d^2)/(-c*x^2+a)

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Rubi [A]  time = 0.49, antiderivative size = 281, 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 = {737, 823, 827, 1166, 208} \[ -\frac {\left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a - c*x^2)^3,x]

[Out]

(x*Sqrt[d + e*x])/(4*a*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*d*e - (6*c*d^2 - 5*a*e^2)*x))/(16*a^2*(c*d^2 - a*e^2
)*(a - c*x^2)) - ((12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
- Sqrt[a]*e]])/(32*a^(5/2)*c^(3/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + ((12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a
*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e
)^(3/2))

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 737

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*a*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x^2
)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1]
|| (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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 {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx &=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\int \frac {-3 d-\frac {5 e x}{2}}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a d e-\left (6 c d^2-5 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} c d \left (12 c d^2-13 a e^2\right )+\frac {1}{4} c e \left (6 c d^2-5 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 )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a d e-\left (6 c d^2-5 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} c d e \left (12 c d^2-13 a e^2\right )-\frac {1}{4} c d e \left (6 c d^2-5 a e^2\right )+\frac {1}{4} c e \left (6 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 )}{4 a^2 c \left (c d^2-a e^2\right )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a d e-\left (6 c d^2-5 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\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 )}+\frac {\left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\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 )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a d e-\left (6 c d^2-5 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 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} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 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} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.71, size = 368, normalized size = 1.31 \[ \frac {\frac {2 (d+e x)^{3/2} \left (5 a^2 e^3-a c d e (3 d+8 e x)+6 c^2 d^3 x\right )}{a-c x^2}+\frac {4 \sqrt {a} c^{3/4} d e \sqrt {d+e x} \left (3 c d^2-4 a e^2\right )+\sqrt {\sqrt {a} e+\sqrt {c} d} \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )-\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}}+\frac {8 a (d+e x)^{3/2} \left (c d^2-a e^2\right ) (c d x-a e)}{\left (a-c x^2\right )^2}}{32 a^2 \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*(d + e*x)^(3/2))/(a - c*x^2)^2 + (2*(d + e*x)^(3/2)*(5*a^2*e^3 + 6*c^2*
d^3*x - a*c*d*e*(3*d + 8*e*x)))/(a - c*x^2) + (4*Sqrt[a]*c^(3/4)*d*e*(3*c*d^2 - 4*a*e^2)*Sqrt[d + e*x] - Sqrt[
Sqrt[c]*d - Sqrt[a]*e]*(Sqrt[c]*d + Sqrt[a]*e)^2*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4
)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + (Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(12*c*d^
2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c
^(3/4)))/(32*a^2*(c*d^2 - a*e^2)^2)

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fricas [B]  time = 1.57, size = 3787, normalized size = 13.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*((a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d^2 - a^4*c*e^2)*x^2)*sqrt((144*c^3
*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^
2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e
^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^
5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))*log(-(3024*c^3*d^6*e^5 - 7884*a*c^2*d^4*e^7 +
5625*a^2*c*d^2*e^9 - 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 - 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^
10 + (12*a^5*c^7*d^10 - 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 - 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 - 5*
a^10*c^2*e^10)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 +
15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt((1
44*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*
c^2*d^2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*
d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)
))/(a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))) - (a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 -
a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d^2 - a^4*c*e^2)*x^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4
- 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 -
1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6
+ 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^
2*e^4 - a^8*c*e^6))*log(-(3024*c^3*d^6*e^5 - 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 - 625*a^3*e^11)*sqrt(e*x
+ d) - (126*a^3*c^3*d^5*e^6 - 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a^5*c^7*d^10 - 55*a^6*c^6*d^8*e^2 +
98*a^7*c^5*d^6*e^4 - 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 - 5*a^10*c^2*e^10)*sqrt((441*c^2*d^4*e^10 - 1050
*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 1
5*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d
^3*e^4 - 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e
^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d
^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*
c^2*d^2*e^4 - a^8*c*e^6))) + (a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d^2 - a^4*c*e
^2)*x^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - (a^5*c^4*d^6 - 3*a^6*c^3*
d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d
^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10
+ a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))*log(-(3024*c^3*d^6*e^5 -
7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 - 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 - 318*a^4*c^2*d^
3*e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10 - 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 - 84*a^8*c^4*d^4*e^6 + 3
4*a^9*c^3*d^2*e^8 - 5*a^10*c^2*e^10)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12
- 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^
11*c^3*e^12)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - (a^5*c^4*d^6 - 3*a^
6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5
*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2
*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))) - (a^4*c*d^2 - a^
5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d^2 - a^4*c*e^2)*x^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e
^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6)*sqr
t((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^
4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^6*c^3*
d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))*log(-(3024*c^3*d^6*e^5 - 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 - 6
25*a^3*e^11)*sqrt(e*x + d) - (126*a^3*c^3*d^5*e^6 - 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10
- 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 - 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 - 5*a^10*c^2*e^10)*sqrt((4
41*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 -
20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt((144*c^3*d^7 - 420*a*c^2
*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^
6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*
d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 - 3*a^
6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))) - 4*(a*c*d*e*x^2 - a^2*d*e - (6*c^2*d^2 - 5*a*c*e^2)*x^3 + (1
0*a*c*d^2 - 9*a^2*e^2)*x)*sqrt(e*x + d))/(a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d
^2 - a^4*c*e^2)*x^2)

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giac [B]  time = 0.88, size = 1074, normalized size = 3.82 \[ -\frac {{\left ({\left (a^{2} c d^{2} e - a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e - 5 \, a e^{3}\right )} {\left | c \right |} + 2 \, {\left (3 \, \sqrt {a c} a c^{2} d^{5} e - 7 \, \sqrt {a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {a c} a^{3} d e^{5}\right )} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} - {\left (12 \, a^{3} c^{4} d^{8} e - 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} - 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} - a^{3} c d e^{2} + \sqrt {{\left (a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} - 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} - a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e - \sqrt {a c} a^{3} c^{3} d^{5} + 2 \, \sqrt {a c} a^{4} c^{2} d^{3} e^{2} - 2 \, a^{5} c^{2} d^{2} e^{3} - \sqrt {a c} a^{5} c d e^{4} + a^{6} c e^{5}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {{\left ({\left (a^{2} c d^{2} e - a^{3} e^{3}\right )}^{2} {\left (6 \, \sqrt {a c} c d^{2} e - 5 \, \sqrt {a c} a e^{3}\right )} {\left | c \right |} - 2 \, {\left (3 \, a^{2} c^{3} d^{5} e - 7 \, a^{3} c^{2} d^{3} e^{3} + 4 \, a^{4} c d e^{5}\right )} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} - {\left (12 \, \sqrt {a c} a^{3} c^{4} d^{8} e - 37 \, \sqrt {a c} a^{4} c^{3} d^{6} e^{3} + 38 \, \sqrt {a c} a^{5} c^{2} d^{4} e^{5} - 13 \, \sqrt {a c} a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} - a^{3} c d e^{2} - \sqrt {{\left (a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} - 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} - a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{4} d^{5} + \sqrt {a c} a^{4} c^{3} d^{4} e - 2 \, a^{5} c^{3} d^{3} e^{2} - 2 \, \sqrt {a c} a^{5} c^{2} d^{2} e^{3} + a^{6} c^{2} d e^{4} + \sqrt {a c} a^{6} c e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {x e + d} c^{2} d^{5} e - 5 \, {\left (x e + d\right )}^{\frac {7}{2}} a c e^{3} + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c d e^{3} - 23 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} + 14 \, \sqrt {x e + d} a c d^{3} e^{3} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 8 \, \sqrt {x e + d} a^{2} d e^{5}}{16 \, {\left (a^{2} c d^{2} - a^{3} e^{2}\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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((a^2*c*d^2*e - a^3*e^3)^2*(6*c*d^2*e - 5*a*e^3)*abs(c) + 2*(3*sqrt(a*c)*a*c^2*d^5*e - 7*sqrt(a*c)*a^2*c
*d^3*e^3 + 4*sqrt(a*c)*a^3*d*e^5)*abs(a^2*c*d^2*e - a^3*e^3)*abs(c) - (12*a^3*c^4*d^8*e - 37*a^4*c^3*d^6*e^3 +
38*a^5*c^2*d^4*e^5 - 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d^3 - a^3*c*d*e^2 + sqrt((
a^2*c^2*d^3 - a^3*c*d*e^2)^2 - (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 - a^3*c*e^2)))/(a^2*c^2*
d^2 - a^3*c*e^2)))/((a^4*c^3*d^4*e - sqrt(a*c)*a^3*c^3*d^5 + 2*sqrt(a*c)*a^4*c^2*d^3*e^2 - 2*a^5*c^2*d^2*e^3 -
sqrt(a*c)*a^5*c*d*e^4 + a^6*c*e^5)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a^2*c*d^2*e - a^3*e^3)) - 1/32*((a^2*c*d^
2*e - a^3*e^3)^2*(6*sqrt(a*c)*c*d^2*e - 5*sqrt(a*c)*a*e^3)*abs(c) - 2*(3*a^2*c^3*d^5*e - 7*a^3*c^2*d^3*e^3 + 4
*a^4*c*d*e^5)*abs(a^2*c*d^2*e - a^3*e^3)*abs(c) - (12*sqrt(a*c)*a^3*c^4*d^8*e - 37*sqrt(a*c)*a^4*c^3*d^6*e^3 +
38*sqrt(a*c)*a^5*c^2*d^4*e^5 - 13*sqrt(a*c)*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d^3 -
a^3*c*d*e^2 - sqrt((a^2*c^2*d^3 - a^3*c*d*e^2)^2 - (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 - a^
3*c*e^2)))/(a^2*c^2*d^2 - a^3*c*e^2)))/((a^4*c^4*d^5 + sqrt(a*c)*a^4*c^3*d^4*e - 2*a^5*c^3*d^3*e^2 - 2*sqrt(a*
c)*a^5*c^2*d^2*e^3 + a^6*c^2*d*e^4 + sqrt(a*c)*a^6*c*e^5)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a^2*c*d^2*e - a^3*e
^3)) - 1/16*(6*(x*e + d)^(7/2)*c^2*d^2*e - 18*(x*e + d)^(5/2)*c^2*d^3*e + 18*(x*e + d)^(3/2)*c^2*d^4*e - 6*sqr
t(x*e + d)*c^2*d^5*e - 5*(x*e + d)^(7/2)*a*c*e^3 + 14*(x*e + d)^(5/2)*a*c*d*e^3 - 23*(x*e + d)^(3/2)*a*c*d^2*e
^3 + 14*sqrt(x*e + d)*a*c*d^3*e^3 + 9*(x*e + d)^(3/2)*a^2*e^5 - 8*sqrt(x*e + d)*a^2*d*e^5)/((a^2*c*d^2 - a^3*e
^2)*((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 = 0.64, size = 803, normalized size = 2.86 \[ \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 )}{32 \sqrt {a c \,e^{2}}\, \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\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 )}{32 \sqrt {a c \,e^{2}}\, \left (-c d +\sqrt {a c \,e^{2}}\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 (c d +\sqrt {a c \,e^{2}}\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 (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {9 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 (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {9 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 (-c d +\sqrt {a c \,e^{2}}\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 (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 (c d -\sqrt {a c \,e^{2}}\right ) a^{2}}+\frac {3 \sqrt {e x +d}\, d e}{16 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} a^{2} c}+\frac {3 \sqrt {e x +d}\, d e}{16 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} a^{2} c}-\frac {5 \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 (c d +\sqrt {a c \,e^{2}}\right ) a^{2} c}+\frac {5 \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 (c d -\sqrt {a c \,e^{2}}\right ) a^{2} c}+\frac {7 \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, e}{32 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} a^{2} c^{2}}-\frac {7 \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, e}{32 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)^(3/2)*d-5/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x-
(a*c*e^2)^(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(3/2)+3/16*e/c/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)*
d+7/32*e/c^2/a^2*(a*c*e^2)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)+5/32*e^3*c/a/(a*c*e^2)^(1/2)/(c*d+(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)+3/8*e*c
^2/a^2/(a*c*e^2)^(1/2)/(c*d+(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^2+9/16*e*c/a^2/(c*d+(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-3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)
^(3/2)*d+5/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(3/2)+3/16*e/c/a
^2/(e*x+(a*c*e^2)^(1/2)/c)^2*(e*x+d)^(1/2)*d-7/32*e/c^2/a^2*(a*c*e^2)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)^2*(e*x+d)^
(1/2)-5/32*e^3*c/a/(a*c*e^2)^(1/2)/(-c*d+(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)-3/8*e*c^2/a^2/(a*c*e^2)^(1/2)/(-c*d+(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^2+9/16*e*c/a^2/(-c*d+(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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.30, size = 6163, normalized size = 21.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(a - c*x^2)^3,x)

[Out]

- atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 -
2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144
*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*
c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1
/2))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c
^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^
13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6
*e^2 - 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5
*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^
5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i -
(((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*
c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^
5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e
^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(6
4*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^
6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*
e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 -
276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 -
25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15
*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i)/((((327
68*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e
^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^5*d^7 -
25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^1
5*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*
e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420
*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3
*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 - 276*a*c
^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^
7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(
1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (((32768*a^7*c^3
*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) + ((d
+ e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^5*d^7 - 25*a*e^7*
(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/
2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 + a^4*
c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d
^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*
d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 - 276*a*c^4*d^4*e^4
+ 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a*e^7*(a^15*c^
3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(409
6*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (125*a^3*c*e^9 - 864*c^4*d
^6*e^3 + 1944*a*c^3*d^4*e^5 - 1170*a^2*c^2*d^2*e^7)/(2048*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2))))*((144*a
^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*
d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2
)*2i - atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*
d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*
((144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4
- 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)
))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*
a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6
- a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^
5*d^6*e^2 - 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^
5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d
^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)
*1i - (((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2
*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a
^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*
d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2
))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2
*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13
*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e
^2 - 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d
^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*
(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i)/((
((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*
d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^5*
d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5
*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*
(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6
- 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^
6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 - 27
6*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25
*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c
^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (((32768*a^
7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2))
+ ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^5*d^7 + 25*a
*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3
)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 +
a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*
c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11
*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 - 276*a*c^4*d^
4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 + 25*a*e^7*(a^
15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^3)^(1/2))
/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (125*a^3*c*e^9 - 864*
c^4*d^6*e^3 + 1944*a*c^3*d^4*e^5 - 1170*a^2*c^2*d^2*e^7)/(2048*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2))))*((
144*a^5*c^5*d^7 + 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 -
21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))
^(1/2)*2i - (((4*a*d*e^3 - 3*c*d^3*e)*(d + e*x)^(1/2))/(8*a^2) - ((d + e*x)^(3/2)*(9*a^2*e^5 + 18*c^2*d^4*e -
23*a*c*d^2*e^3))/(16*a^2*(a*e^2 - c*d^2)) + (c*e*(5*a*e^2 - 6*c*d^2)*(d + e*x)^(7/2))/(16*a^2*(a*e^2 - c*d^2))
- (c*d*(7*a*e^3 - 9*c*d^2*e)*(d + e*x)^(5/2))/(8*a^2*(a*e^2 - c*d^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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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