3.551 \(\int \frac{(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ \frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]

[Out]

-((2*A*b - 3*a*B)*e^2*(e*x)^(3/2))/(3*b^2*Sqrt[a + b*x^3]) + (B*(e*x)^(9/2))/(3*
b*e*Sqrt[a + b*x^3]) + ((2*A*b - 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e
^(3/2)*Sqrt[a + b*x^3])])/(3*b^(5/2))

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Rubi [A]  time = 0.261849, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]

[Out]

-((2*A*b - 3*a*B)*e^2*(e*x)^(3/2))/(3*b^2*Sqrt[a + b*x^3]) + (B*(e*x)^(9/2))/(3*
b*e*Sqrt[a + b*x^3]) + ((2*A*b - 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e
^(3/2)*Sqrt[a + b*x^3])])/(3*b^(5/2))

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Rubi in Sympy [A]  time = 24.4732, size = 112, normalized size = 0.93 \[ \frac{B \left (e x\right )^{\frac{9}{2}}}{3 b e \sqrt{a + b x^{3}}} - \frac{2 e^{2} \left (e x\right )^{\frac{3}{2}} \left (A b - \frac{3 B a}{2}\right )}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 e^{\frac{7}{2}} \left (A b - \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(3/2),x)

[Out]

B*(e*x)**(9/2)/(3*b*e*sqrt(a + b*x**3)) - 2*e**2*(e*x)**(3/2)*(A*b - 3*B*a/2)/(3
*b**2*sqrt(a + b*x**3)) + 2*e**(7/2)*(A*b - 3*B*a/2)*atanh(sqrt(b)*(e*x)**(3/2)/
(e**(3/2)*sqrt(a + b*x**3)))/(3*b**(5/2))

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Mathematica [A]  time = 0.228545, size = 91, normalized size = 0.76 \[ \frac{e^2 (e x)^{3/2} \left (\sqrt{b} \left (3 a B-2 A b+b B x^3\right )+\sqrt{\frac{a}{x^3}+b} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{3 b^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]

[Out]

(e^2*(e*x)^(3/2)*(Sqrt[b]*(-2*A*b + 3*a*B + b*B*x^3) + (2*A*b - 3*a*B)*Sqrt[b +
a/x^3]*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(3*b^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.079, size = 7016, normalized size = 58.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(3/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.652507, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{12 \,{\left (b^{3} x^{3} + a b^{2}\right )}}, -\frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

[-1/12*(((3*B*a*b - 2*A*b^2)*e^3*x^3 + (3*B*a^2 - 2*A*a*b)*e^3)*sqrt(e/b)*log(-8
*b^2*e*x^6 - 8*a*b*e*x^3 - a^2*e - 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(e*
x)*sqrt(e/b)) - 4*(B*b*e^3*x^4 + (3*B*a - 2*A*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x
))/(b^3*x^3 + a*b^2), -1/6*(((3*B*a*b - 2*A*b^2)*e^3*x^3 + (3*B*a^2 - 2*A*a*b)*e
^3)*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*x/((2*b*x^3 + a)*sqrt(-e/b)))
- 2*(B*b*e^3*x^4 + (3*B*a - 2*A*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/(b^3*x^3 +
a*b^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((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(3/2),x)

[Out]

Timed out

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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((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

Timed out