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

Optimal. Leaf size=739 \[ \frac{\sqrt{d+e x} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

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

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Rubi [A]  time = 3.94962, antiderivative size = 739, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{\sqrt{d+e x} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(c*d^2 + 3*
a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[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^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])
- (e*(c*d^2 + 3*a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d
+ Sqrt[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^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt
[c*d^2 + a*e^2]]) - (e*(c*d^2 + 3*a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqr
t[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^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + 3*a*e^2 - Sqrt[c]*d*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^(1/4)*(c*d^2 + a*e^2)^(
3/2)*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/(c*x**2+a)**2/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.446945, size = 245, normalized size = 0.33 \[ \frac{-\frac{\left (\sqrt{a} \sqrt{c} d e+3 i a e^2+2 i c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i \left (i \sqrt{a} \sqrt{c} d e+3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}+\frac{2 \sqrt{a} \sqrt{d+e x} (a e+c d x)}{a+c x^2}}{4 a^{3/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.676, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{ex+d}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.334986, size = 4410, normalized size = 5.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a
*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 +
a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 +
6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^
8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*
c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*
x + d) + (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5*d^9 + 5
*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-
(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e
^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^
2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c
^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 +
90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^
8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^1
2)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^
2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 1
5*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt
(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10
*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*
d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^
6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) - (5*a^
2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e
^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^
6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^
5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c
*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4
*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8
+ 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6
*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*
d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) + (a^2*c*d^2 + a^3*e^2 +
(a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 -
(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*
e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*
c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9
*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((2
0*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) + (5*a^2*c^2*d^4*e^4
+ 24*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3
*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2
*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20
*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(
-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 +
3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10
)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 +
15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^
2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a
^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 +
3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d
^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 +
20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^
3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3
+ 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) - (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2
*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a
^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*
e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e
^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 +
15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e
^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^1
2 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^
4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*
a^5*c*d^2*e^4 + a^6*e^6))) + 4*(c*d*x + a*e)*sqrt(e*x + d))/(a^2*c*d^2 + a^3*e^2
+ (a*c^2*d^2 + a^2*c*e^2)*x^2)

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Sympy [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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError