3.163 \(\int \frac{x^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=95 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{5/2}}-\frac{x^{3/2} (A b-3 a B)}{3 a b^2}+\frac{x^{9/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.169996, antiderivative size = 95, 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 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{5/2}}-\frac{x^{3/2} (A b-3 a B)}{3 a b^2}+\frac{x^{9/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.9803, size = 80, normalized size = 0.84 \[ \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x^{\frac{3}{2}} \left (A b - 3 B a\right )}{3 a b^{2}} + \frac{\left (A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.255638, size = 160, normalized size = 1.68 \[ \frac{\frac{\sqrt{b} x^{3/2} (a B-A b)}{a+b x^3}+\frac{(3 a B-A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{a}}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt{a}}+\frac{(3 a B-A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{a}}+2 \sqrt{b} B x^{3/2}}{3 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(2*Sqrt[b]*B*x^(3/2) + (Sqrt[b]*(-(A*b) + a*B)*x^(3/2))/(a + b*x^3) + ((-(A*b) +
3*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/Sqrt[a] + ((A*b - 3*a*B)*
ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/Sqrt[a] + ((-(A*b) + 3*a*B)*ArcTa
n[(b^(1/6)*Sqrt[x])/a^(1/6)])/Sqrt[a])/(3*b^(5/2))

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Maple [A]  time = 0.024, size = 93, normalized size = 1. \[{\frac{2\,B}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{A}{3\,b \left ( b{x}^{3}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{Ba}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{A}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{Ba}{{b}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

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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)*x^(7/2)/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242472, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x^{4} +{\left (3 \, B a - A b\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left ({\left (3 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )} \sqrt{-a b}}, \frac{{\left (2 \, B b x^{4} +{\left (3 \, B a - A b\right )} x\right )} \sqrt{a b} \sqrt{x} -{\left ({\left (3 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

[1/6*(2*(2*B*b*x^4 + (3*B*a - A*b)*x)*sqrt(-a*b)*sqrt(x) - ((3*B*a*b - A*b^2)*x^
3 + 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)))/
((b^3*x^3 + a*b^2)*sqrt(-a*b)), 1/3*((2*B*b*x^4 + (3*B*a - A*b)*x)*sqrt(a*b)*sqr
t(x) - ((3*B*a*b - A*b^2)*x^3 + 3*B*a^2 - A*a*b)*arctan(sqrt(a*b)*x^(3/2)/a))/((
b^3*x^3 + a*b^2)*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(x**(7/2)*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216011, size = 92, normalized size = 0.97 \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, b^{2}} - \frac{{\left (3 \, B a - A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b^{2}} + \frac{B a x^{\frac{3}{2}} - A b x^{\frac{3}{2}}}{3 \,{\left (b x^{3} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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