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

Optimal. Leaf size=71 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{3/2} b^{3/2}}+\frac{x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.12672, antiderivative size = 71, 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 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{3/2} b^{3/2}}+\frac{x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8777, size = 58, normalized size = 0.82 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(3/2)*(A*b - B*a)/(3*a*b*(a + b*x**3)) + (A*b + B*a)*atan(sqrt(b)*x**(3/2)/sq
rt(a))/(3*a**(3/2)*b**(3/2))

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Mathematica [A]  time = 0.20188, size = 139, normalized size = 1.96 \[ \frac{\frac{\sqrt{a} \sqrt{b} x^{3/2} (A b-a B)}{a+b x^3}-(a B+A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+(a B+A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )-(a B+A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{3/2} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*Sqrt[b]*(A*b - a*B)*x^(3/2))/(a + b*x^3) - (A*b + a*B)*ArcTan[Sqrt[3]
- (2*b^(1/6)*Sqrt[x])/a^(1/6)] + (A*b + a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x]
)/a^(1/6)] - (A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(3*a^(3/2)*b^(3/2))

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Maple [A]  time = 0.014, size = 74, normalized size = 1. \[{\frac{Ab-Ba}{3\,ab \left ( b{x}^{3}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{A}{3\,a}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249534, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (B a - A b\right )} \sqrt{-a b} x^{\frac{3}{2}} -{\left ({\left (B a b + A b^{2}\right )} x^{3} + B a^{2} + A a b\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{6 \,{\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt{-a b}}, -\frac{{\left (B a - A b\right )} \sqrt{a b} x^{\frac{3}{2}} -{\left ({\left (B a b + A b^{2}\right )} x^{3} + B a^{2} + A a b\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )}{3 \,{\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**(1/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221267, size = 85, normalized size = 1.2 \[ \frac{{\left (B a + A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a b} - \frac{B a x^{\frac{3}{2}} - A b x^{\frac{3}{2}}}{3 \,{\left (b x^{3} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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