3.232 \(\int \frac{A+B x^3}{x^4 \left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}-\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.214431, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}-\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.0876, size = 78, normalized size = 0.91 \[ - \frac{A}{3 a x^{3} \sqrt{a + b x^{3}}} - \frac{2 \left (\frac{3 A b}{2} - B a\right )}{3 a^{2} \sqrt{a + b x^{3}}} + \frac{2 \left (\frac{3 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**4/(b*x**3+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.424707, size = 73, normalized size = 0.85 \[ \frac{\sqrt{\frac{b x^3}{a}+1} (3 A b-2 a B) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )-\frac{a A}{x^3}+2 a B-3 A b}{3 a^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

(-3*A*b + 2*a*B - (a*A)/x^3 + (3*A*b - 2*a*B)*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1
+ (b*x^3)/a]])/(3*a^2*Sqrt[a + b*x^3])

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Maple [A]  time = 0.012, size = 100, normalized size = 1.2 \[ A \left ( -{\frac{1}{3\,{a}^{2}{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{2\,b}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}+{b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) +B \left ({\frac{2}{3\,a}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^4/(b*x^3+a)^(3/2),x)

[Out]

A*(-1/3/a^2*(b*x^3+a)^(1/2)/x^3-2/3*b/a^2/((x^3+a/b)*b)^(1/2)+b/a^(5/2)*arctanh(
(b*x^3+a)^(1/2)/a^(1/2)))+B*(2/3/a/((x^3+a/b)*b)^(1/2)-2/3/a^(3/2)*arctanh((b*x^
3+a)^(1/2)/a^(1/2)))

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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)/((b*x^3 + a)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255995, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{b x^{3} + a}{\left (2 \, B a - 3 \, A b\right )} x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left ({\left (2 \, B a - 3 \, A b\right )} x^{3} - A a\right )} \sqrt{a}}{6 \, \sqrt{b x^{3} + a} a^{\frac{5}{2}} x^{3}}, \frac{\sqrt{b x^{3} + a}{\left (2 \, B a - 3 \, A b\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left ({\left (2 \, B a - 3 \, A b\right )} x^{3} - A a\right )} \sqrt{-a}}{3 \, \sqrt{b x^{3} + a} \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[-1/6*(sqrt(b*x^3 + a)*(2*B*a - 3*A*b)*x^3*log(((b*x^3 + 2*a)*sqrt(a) + 2*sqrt(b
*x^3 + a)*a)/x^3) - 2*((2*B*a - 3*A*b)*x^3 - A*a)*sqrt(a))/(sqrt(b*x^3 + a)*a^(5
/2)*x^3), 1/3*(sqrt(b*x^3 + a)*(2*B*a - 3*A*b)*x^3*arctan(a/(sqrt(b*x^3 + a)*sqr
t(-a))) + ((2*B*a - 3*A*b)*x^3 - A*a)*sqrt(-a))/(sqrt(b*x^3 + a)*sqrt(-a)*a^2*x^
3)]

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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((B*x**3+A)/x**4/(b*x**3+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220098, size = 134, normalized size = 1.56 \[ \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{2}} + \frac{2 \,{\left (b x^{3} + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x^{3} + a\right )} A b + 2 \, A a b}{3 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{3} + a} a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

1/3*(2*B*a - 3*A*b)*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^2) + 1/3*(2*(b*
x^3 + a)*B*a - 2*B*a^2 - 3*(b*x^3 + a)*A*b + 2*A*a*b)/(((b*x^3 + a)^(3/2) - sqrt
(b*x^3 + a)*a)*a^2)