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

Optimal. Leaf size=726 \[ -\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 3.46739, antiderivative size = 726, 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 \[ -\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 0.338844, size = 235, normalized size = 0.32 \[ \frac{-\frac{\left (\sqrt{a} \sqrt{c} d e+i a e^2+2 i c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i \left (i \sqrt{a} \sqrt{c} d e+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}+\frac{2 \sqrt{a} \sqrt{d+e x} (c d x-a e)}{a+c x^2}}{4 a^{3/2} c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/(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
]*Sqrt[c]*d*e + I*a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sq
rt[c]*e]])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e] + (I*(2*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e
+ a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]])/Sqrt
[c*d + I*Sqrt[a]*Sqrt[c]*e])/(4*a^(3/2)*c)

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Maple [B]  time = 0.084, size = 4421, normalized size = 6.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e/(c*e^2*x^2+a*e^2)*d/a*(e*x+d)^(3/2)-1/2*e^3/(c*e^2*x^2+a*e^2)/c*(e*x+d)^(1
/2)-1/2*e/(c*e^2*x^2+a*e^2)/a*(e*x+d)^(1/2)*d^2-1/16/e/a^2/(a*c^2*e^2)^(1/2)*ln(
-(a*c^2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(
1/2)*(e*x+d)^(1/2)-(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^
4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*d^2+1/16/e^3/a^3/c^2*ln(-(a*c^2*e^2)^(1/2)*(e*x+d)
+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(e*x+d)^(1/2)-(a*c*e^
2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1
/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*d+1/16/e^3/a^3/c^2/(a*c^2*e^2)^(1/2)*ln(-(a*
c^2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)
*(e*x+d)^(1/2)-(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(
1/2)+2*a*c^2*e^2*d)^(1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)*d-1/16/e^5/a^4/c^4
*ln(-(a*c^2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*
d)^(1/2)*(e*x+d)^(1/2)-(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^
2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*(a^3*c^3*e^6+a
^2*c^4*d^2*e^4)^(1/2)-1/2*e/a/c/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/
2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2)*arctan((-2*(a*c^2*e^
2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)
)/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2
))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)+1/8/e/a^2/(4*(a*c
^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-
2*a*c^2*e^2*d)^(1/2)*arctan((-2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*
(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+
c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*
c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)/(a*c^2*e^2)^(1/2)*(2*(a^3*c^3*
e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*d^2-1/8/e^3/a^3/c^2/(4*(a*c^2*e^
2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c
^2*e^2*d)^(1/2)*arctan((-2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^
2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2
))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*c^3*e
^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1
/2)+2*a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*d-1/8/e^3/a^3/c^2/(4*(a
*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2
)-2*a*c^2*e^2*d)^(1/2)*arctan((-2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^
4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^
2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^
2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)/(a*c^2*e^2)^(1/2)*(2*(a^3*c^
3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^
(1/2)*d+1/8/e^5/a^4/c^4/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^
2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2)*arctan((-2*(a*c^2*e^2)^(1/2)
*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*
c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)
-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)
*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^
2*e^2)^(1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+1/16/e/a^2/(a*c^2*e^2)^(1/2)*ln
((a*c^2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(
1/2)*(e*x+d)^(1/2)+(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^
4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*d^2-1/16/e^3/a^3/c^2*ln((a*c^2*e^2)^(1/2)*(e*x+d)+
(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(e*x+d)^(1/2)+(a*c*e^2
*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/
2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*d-1/16/e^3/a^3/c^2/(a*c^2*e^2)^(1/2)*ln((a*c^
2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(
e*x+d)^(1/2)+(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/
2)+2*a*c^2*e^2*d)^(1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)*d+1/16/e^5/a^4/c^4*l
n((a*c^2*e^2)^(1/2)*(e*x+d)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^
(1/2)*(e*x+d)^(1/2)+(a*c*e^2*(a*e^2+c*d^2))^(1/2))*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e
^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*(a^3*c^3*e^6+a^2*
c^4*d^2*e^4)^(1/2)+1/2*e/a/c/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-
2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2)*arctan((2*(a*c^2*e^2)^(
1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4
*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(
1/2)-2*a*c^2*e^2*d)^(1/2))*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)-1/8/e/a^2/(4*(a*c^2*e
^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*
c^2*e^2*d)^(1/2)*arctan((2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^
2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2
))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*c^3*e
^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)/(a*c^2*e^2)^(1/2)*(2*(a^3*c^3*e^6+a
^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*d^2+1/8/e^3/a^3/c^2/(4*(a*c^2*e^2)^(1
/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^
2*d)^(1/2)*arctan((2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^2+c*d^
2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/
2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*c^3*e^4*(a*
e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(2*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*
a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(1/2)*d+1/8/e^3/a^3/c^2/(4*(a*c^2*e
^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*
c^2*e^2*d)^(1/2)*arctan((2*(a*c^2*e^2)^(1/2)*(e*x+d)^(1/2)+(2*(a^2*c^3*e^4*(a*e^
2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2
))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2))*(2*(a^2*c^3*e
^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)/(a*c^2*e^2)^(1/2)*(2*(a^3*c^3*e^6+a
^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)*d
-1/8/e^5/a^4/c^4/(4*(a*c^2*e^2)^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e
^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2*e^2*d)^(1/2)*arctan((2*(a*c^2*e^2)^(1/2)*(e*x+d)
^(1/2)+(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2))/(4*(a*c^2*e^2)
^(1/2)*(a*c*e^2*(a*e^2+c*d^2))^(1/2)-2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)-2*a*c^2
*e^2*d)^(1/2))*(2*(a^2*c^3*e^4*(a*e^2+c*d^2))^(1/2)+2*a*c^2*e^2*d)^(1/2)*(2*(a^3
*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)+2*a*c^2*e^2*d)^(1/2)*(a^2*c*e^4+a*c^2*d^2*e^2)^(
1/2)*(a^3*c^3*e^6+a^2*c^4*d^2*e^4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234394, size = 917, normalized size = 1.26 \[ \frac{{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) +{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \,{\left (c d x - a e\right )} \sqrt{e x + d}}{8 \,{\left (a c^{2} x^{2} + a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e
^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(-e^6/
(a^3*c^5)) + a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e^
2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d
^3 + 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (2*a^3*c^4*
d*sqrt(-e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^
3 + 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^
5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) +
(2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)
) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))) + (a*c^2*x^2 + a^2*c)*sqrt((a^3*c^2*sqrt(-e
^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e
*x + d) - (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6
/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))) + 4*(c*d*x - a*e)*sqrt(e*x + d))/
(a*c^2*x^2 + a^2*c)

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

[Out]

Exception raised: TypeError