3.248 \(\int \frac{A+B x^3}{x^4 \left (a+b x^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{7/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.274155, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{7/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^4*(a + b*x^3)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.4813, size = 104, normalized size = 0.92 \[ - \frac{A}{3 a x^{3} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{5 A b}{2} - B a\right )}{9 a^{2} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{5 A b}{2} - B a\right )}{3 a^{3} \sqrt{a + b x^{3}}} + \frac{2 \left (\frac{5 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**4/(b*x**3+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.309363, size = 108, normalized size = 0.96 \[ \frac{\sqrt{a+b x^3} \left (\frac{2 a^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac{6 a (a B-2 A b)}{a+b x^3}+\frac{3 (5 A b-2 a B) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{3 a A}{x^3}\right )}{9 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(5/2)),x]

[Out]

(Sqrt[a + b*x^3]*((-3*a*A)/x^3 + (2*a^2*(-(A*b) + a*B))/(a + b*x^3)^2 + (6*a*(-2
*A*b + a*B))/(a + b*x^3) + (3*(5*A*b - 2*a*B)*ArcTanh[Sqrt[1 + (b*x^3)/a]])/Sqrt
[1 + (b*x^3)/a]))/(9*a^4)

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Maple [A]  time = 0.047, size = 157, normalized size = 1.4 \[ A \left ( -{\frac{1}{3\,{a}^{3}{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{2}{9\,{a}^{2}b}\sqrt{b{x}^{3}+a} \left ({x}^{3}+{\frac{a}{b}} \right ) ^{-2}}-{\frac{4\,b}{3\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}+{\frac{5\,b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{7}{2}}}} \right ) +B \left ({\frac{2}{9\,a{b}^{2}}\sqrt{b{x}^{3}+a} \left ({x}^{3}+{\frac{a}{b}} \right ) ^{-2}}+{\frac{2}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^4/(b*x^3+a)^(5/2),x)

[Out]

A*(-1/3/a^3*(b*x^3+a)^(1/2)/x^3-2/9/a^2/b*(b*x^3+a)^(1/2)/(x^3+a/b)^2-4/3*b/a^3/
((x^3+a/b)*b)^(1/2)+5/3*b/a^(7/2)*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+B*(2/9/a/b^2
*(b*x^3+a)^(1/2)/(x^3+a/b)^2+2/3/a^2/((x^3+a/b)*b)^(1/2)-2/3/a^(5/2)*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)^(5/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296907, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{6} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt{b x^{3} + a} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{6} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{3} - 3 \, A a^{2}\right )} \sqrt{a}}{18 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{a}}, \frac{3 \,{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{6} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt{b x^{3} + a} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{6} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{3} - 3 \, A a^{2}\right )} \sqrt{-a}}{9 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^4),x, algorithm="fricas")

[Out]

[-1/18*(3*((2*B*a*b - 5*A*b^2)*x^6 + (2*B*a^2 - 5*A*a*b)*x^3)*sqrt(b*x^3 + a)*lo
g(((b*x^3 + 2*a)*sqrt(a) + 2*sqrt(b*x^3 + a)*a)/x^3) - 2*(3*(2*B*a*b - 5*A*b^2)*
x^6 + 4*(2*B*a^2 - 5*A*a*b)*x^3 - 3*A*a^2)*sqrt(a))/((a^3*b*x^6 + a^4*x^3)*sqrt(
b*x^3 + a)*sqrt(a)), 1/9*(3*((2*B*a*b - 5*A*b^2)*x^6 + (2*B*a^2 - 5*A*a*b)*x^3)*
sqrt(b*x^3 + a)*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (3*(2*B*a*b - 5*A*b^2)*x^
6 + 4*(2*B*a^2 - 5*A*a*b)*x^3 - 3*A*a^2)*sqrt(-a))/((a^3*b*x^6 + a^4*x^3)*sqrt(b
*x^3 + a)*sqrt(-a))]

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222165, size = 136, normalized size = 1.2 \[ \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{3}} + \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )} B a + B a^{2} - 6 \,{\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}} - \frac{\sqrt{b x^{3} + a} A}{3 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^4),x, algorithm="giac")

[Out]

1/3*(2*B*a - 5*A*b)*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^3) + 2/9*(3*(b*
x^3 + a)*B*a + B*a^2 - 6*(b*x^3 + a)*A*b - A*a*b)/((b*x^3 + a)^(3/2)*a^3) - 1/3*
sqrt(b*x^3 + a)*A/(a^3*x^3)