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

Optimal. Leaf size=845 \[ \frac{e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt{d+e x} \left (c x^2+a\right )} \]

[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)*e*(c^(3/2)*d^3 + 13*a*Sqr
t[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sq
rt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2
+ a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e
^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*
d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^
2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a
*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2]
- Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*
(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e
^2]]) + (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*
d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*
d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(
5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 7.56634, antiderivative size = 845, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt{d+e x} \left (c x^2+a\right )} \]

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)*e*(c^(3/2)*d^3 + 13*a*Sqr
t[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sq
rt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2
+ a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e
^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*
d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^
2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a
*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2]
- Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*
(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e
^2]]) + (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*
d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*
d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(
5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.699454, size = 312, normalized size = 0.37 \[ \frac{\frac{2 \sqrt{a} \left (-4 a^2 e^3+a c e \left (2 d^2+d e x-5 e^2 x^2\right )+c^2 d^2 x (d+e x)\right )}{\left (a+c x^2\right ) \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{i \sqrt{c} \left (2 \sqrt{c} d-5 i \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d-i \sqrt{a} e\right )^2 \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i \sqrt{c} \left (2 \sqrt{c} d+5 i \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d+i \sqrt{a} e\right )^2 \sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{4 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.12, size = 10300, normalized size = 12.2 \[ \text{output too large to display} \]

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]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{2}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.833793, size = 7250, normalized size = 8.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*a*c*d^2*e - 16*a^2*e^3 + 4*(c^2*d^2*e - 5*a*c*e^3)*x^2 - (a^2*c^2*d^4 + 2
*a^3*c*d^2*e^2 + a^4*e^4 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt
(e*x + d)*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^1
4 + 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 + 2
1966*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^4 + 2*a^3
*c*d^2*e^2 + a^4*e^4 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x
+ d)*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 + 219
66*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 + 1
0*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^1
0))*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)*sq
rt(-(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^4 + 2*a^3*c*d
^2*e^2 + a^4*e^4 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d
)*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^1
1*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^1
6*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 + 21
0*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 + 1
0*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^4 + 2*a^3*c*d^2*e
^2 + a^4*e^4 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*sq
rt(-(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)*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 + 2
52*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(-(122
5*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*(c^2*d^3 + a*c*d*e^2)*x)/((a^2
*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^
4)*x^2)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{2} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out