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

Optimal. Leaf size=77 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}+\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.178925, antiderivative size = 77, 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 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}+\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4026, size = 70, normalized size = 0.91 \[ \frac{2 A}{3 a^{2} \sqrt{a + b x^{3}}} - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{5}{2}}} + \frac{2 \left (A b - B a\right )}{9 a b \left (a + b x^{3}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.223342, size = 83, normalized size = 1.08 \[ \frac{2 \left (a \left (-\frac{a^2 B}{b}+4 a A+3 A b x^3\right )-\frac{3 A \left (a+b x^3\right )^2 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right )}{9 a^3 \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.042, size = 85, normalized size = 1.1 \[ A \left ({\frac{2}{9\,a{b}^{2}}\sqrt{b{x}^{3}+a} \left ({x}^{3}+{\frac{a}{b}} \right ) ^{-2}}+{\frac{2}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) -{\frac{2\,B}{9\,b} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(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)))-2/9*B/b/(b*x^3+a)^(3/2)

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27424, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (A b^{2} x^{3} + A a b\right )} \sqrt{b x^{3} + a} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (3 \, A b^{2} x^{3} - B a^{2} + 4 \, A a b\right )} \sqrt{a}}{9 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{b x^{3} + a} \sqrt{a}}, \frac{2 \,{\left (3 \,{\left (A b^{2} x^{3} + A a b\right )} \sqrt{b x^{3} + a} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \, A b^{2} x^{3} - B a^{2} + 4 \, A a b\right )} \sqrt{-a}\right )}}{9 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{b x^{3} + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/9*(3*(A*b^2*x^3 + A*a*b)*sqrt(b*x^3 + a)*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(
b*x^3 + a)*a)/x^3) + 2*(3*A*b^2*x^3 - B*a^2 + 4*A*a*b)*sqrt(a))/((a^2*b^2*x^3 +
a^3*b)*sqrt(b*x^3 + a)*sqrt(a)), 2/9*(3*(A*b^2*x^3 + A*a*b)*sqrt(b*x^3 + a)*arct
an(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (3*A*b^2*x^3 - B*a^2 + 4*A*a*b)*sqrt(-a))/((a
^2*b^2*x^3 + a^3*b)*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/(b*x**3+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218687, size = 90, normalized size = 1.17 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{2}} - \frac{2 \,{\left (B a^{2} - 3 \,{\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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