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

Optimal. Leaf size=58 \[ \frac{2 (A b-a B)}{3 a b \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.143756, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (A b-a B)}{3 a b \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5889, size = 51, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} + \frac{2 \left (A b - B a\right )}{3 a b \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.242802, size = 61, normalized size = 1.05 \[ -\frac{2 \left (A b \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a B-A b\right )}{3 a b \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 57, normalized size = 1. \[ A \left ({\frac{2}{3\,a}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) -{\frac{2\,B}{3\,b}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(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)))-2/3*B
/b/(b*x^3+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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265079, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x^{3} + a} A b \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left (B a - A b\right )} \sqrt{a}}{3 \, \sqrt{b x^{3} + a} a^{\frac{3}{2}} b}, \frac{2 \,{\left (\sqrt{b x^{3} + a} A b \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) -{\left (B a - A b\right )} \sqrt{-a}\right )}}{3 \, \sqrt{b x^{3} + a} \sqrt{-a} a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(sqrt(b*x^3 + a)*A*b*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(b*x^3 + a)*a)/x^3)
- 2*(B*a - A*b)*sqrt(a))/(sqrt(b*x^3 + a)*a^(3/2)*b), 2/3*(sqrt(b*x^3 + a)*A*b*
arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) - (B*a - A*b)*sqrt(-a))/(sqrt(b*x^3 + a)*sq
rt(-a)*a*b)]

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Sympy [A]  time = 57.5709, size = 214, normalized size = 3.69 \[ A \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{3}}{a}}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{2} b x^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{2} b x^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}}\right ) + B \left (\begin{cases} - \frac{2}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(2*a**3*sqrt(1 + b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) + a**3*log(b*x**3/
a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) - 2*a**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(
9/2) + 3*a**(7/2)*b*x**3) + a**2*b*x**3*log(b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b
*x**3) - 2*a**2*b*x**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2) + 3*a**(7/2)*b*x*
*3)) + B*Piecewise((-2/(3*b*sqrt(a + b*x**3)), Ne(b, 0)), (x**3/(3*a**(3/2)), Tr
ue))

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GIAC/XCAS [A]  time = 0.218203, size = 72, normalized size = 1.24 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a} - \frac{2 \,{\left (B a - A b\right )}}{3 \, \sqrt{b x^{3} + a} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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