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

Optimal. Leaf size=769 \[ -\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{\sqrt{d+e x} (a e+6 c d x)}{16 a^2 c \left (a+c x^2\right )}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*a*c*(a + c*x^2)^2) + ((a*e + 6*c*d*x)*Sqrt[d +
e*x])/(16*a^2*c*(a + c*x^2)) + (3*e*(2*c*d^2 + a*e^2 + 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]]])/(32*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*
d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c*d^2 + a*e^2 + 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]]])/(32*Sqrt[
2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e
*(2*c*d^2 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - S
qrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d
+ e*x)])/(64*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2
+ a*e^2]]) + (3*e*(2*c*d^2 + a*e^2 - 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)])/(64*Sqrt[2]*a^2*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 = 5.04796, antiderivative size = 769, 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 \[ -\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{\sqrt{d+e x} (a e+6 c d x)}{16 a^2 c \left (a+c x^2\right )}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*a*c*(a + c*x^2)^2) + ((a*e + 6*c*d*x)*Sqrt[d +
e*x])/(16*a^2*c*(a + c*x^2)) + (3*e*(2*c*d^2 + a*e^2 + 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]]])/(32*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*
d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c*d^2 + a*e^2 + 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]]])/(32*Sqrt[
2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e
*(2*c*d^2 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - S
qrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d
+ e*x)])/(64*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2
+ a*e^2]]) + (3*e*(2*c*d^2 + a*e^2 - 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)])/(64*Sqrt[2]*a^2*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)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.4167, size = 253, normalized size = 0.33 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (-3 a^2 e+a c x (10 d+e x)+6 c^2 d x^3\right )}{\left (a+c x^2\right )^2}-\frac{3 i \left (-2 i \sqrt{a} \sqrt{c} d e+a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{3 i \left (2 i \sqrt{a} \sqrt{c} d e+a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{32 a^{5/2} c} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(-3*a^2*e + 6*c^2*d*x^3 + a*c*x*(10*d + e*x)))/(a + c*
x^2)^2 - ((3*I)*(4*c*d^2 - (2*I)*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTanh[(Sqrt[c]*S
qrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]
+ ((3*I)*(4*c*d^2 + (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])/(32*a^
(5/2)*c)

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Maple [B]  time = 0.117, size = 9288, normalized size = 12.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270346, size = 2365, normalized size = 3.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2
+ 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6
*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 +
12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (4
*a^5*c^6*d^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*
a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^
4 + (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^
7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^
4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^6*c^2*e
^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 +
a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) -
27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (4*a^5*c^6*d^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*
c^4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(1
6*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^1
0/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))
) + 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 +
5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d
^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 + 12
*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (4*a
^5*c^6*d^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^
6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4
- (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*
c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*
c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2
)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^
6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 2
7*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (4*a^5*c^6*d^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*c^
4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*
c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/
(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2)))
+ 4*(6*c^2*d*x^3 + a*c*e*x^2 + 10*a*c*d*x - 3*a^2*e)*sqrt(e*x + d))/(a^2*c^3*x^4
+ 2*a^3*c^2*x^2 + a^4*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)**3,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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError