3.217 \(\int \frac{A+B x^3}{x^4 \sqrt{a+b x^3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.158416, 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 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{A \sqrt{a+b x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.3931, size = 51, normalized size = 0.88 \[ - \frac{A \sqrt{a + b x^{3}}}{3 a x^{3}} + \frac{2 \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**4/(b*x**3+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.240244, size = 64, normalized size = 1.1 \[ \frac{\sqrt{a+b x^3} \left (\frac{(A b-2 a B) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a A}{x^3}\right )}{3 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 62, normalized size = 1.1 \[ A \left ( -{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) -{\frac{2\,B}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^4/(b*x^3+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25743, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \, \sqrt{b x^{3} + a} A \sqrt{a}}{6 \, a^{\frac{3}{2}} x^{3}}, \frac{{\left (2 \, B a - A b\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) - \sqrt{b x^{3} + a} A \sqrt{-a}}{3 \, \sqrt{-a} a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 23.8201, size = 80, normalized size = 1.38 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} - \frac{2 B \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**4/(b*x**3+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219299, size = 84, normalized size = 1.45 \[ \frac{\frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{b x^{3} + a} A b}{a x^{3}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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