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

Optimal. Leaf size=104 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

((A*b - a*B)*x^(9/2))/(6*a*b*(a + b*x^3)^2) - ((A*b + 3*a*B)*x^(3/2))/(12*a*b^2*
(a + b*x^3)) + ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(3/2)*b^(
5/2))

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Rubi [A]  time = 0.173112, antiderivative size = 104, 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 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(9/2))/(6*a*b*(a + b*x^3)^2) - ((A*b + 3*a*B)*x^(3/2))/(12*a*b^2*
(a + b*x^3)) + ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(3/2)*b^(
5/2))

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Rubi in Sympy [A]  time = 19.3873, size = 88, normalized size = 0.85 \[ \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{x^{\frac{3}{2}} \left (A b + 3 B a\right )}{12 a b^{2} \left (a + b x^{3}\right )} + \frac{\left (A b + 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(9/2)*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) - x**(3/2)*(A*b + 3*B*a)/(12*a*b**2
*(a + b*x**3)) + (A*b + 3*B*a)*atan(sqrt(b)*x**(3/2)/sqrt(a))/(12*a**(3/2)*b**(5
/2))

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Mathematica [A]  time = 0.33952, size = 179, normalized size = 1.72 \[ \frac{-\frac{(3 a B+A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{3/2}}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{3/2}}-\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} x^{3/2} (A b-5 a B)}{a \left (a+b x^3\right )}-\frac{2 \sqrt{b} x^{3/2} (A b-a B)}{\left (a+b x^3\right )^2}}{12 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[b]*(A*b - a*B)*x^(3/2))/(a + b*x^3)^2 + (Sqrt[b]*(A*b - 5*a*B)*x^(3/2)
)/(a*(a + b*x^3)) - ((A*b + 3*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)]
)/a^(3/2) + ((A*b + 3*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/a^(3/2
) - ((A*b + 3*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/a^(3/2))/(12*b^(5/2))

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Maple [A]  time = 0.025, size = 96, normalized size = 0.9 \[{\frac{2}{3\, \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{Ab-5\,Ba}{8\,ab}{x}^{{\frac{9}{2}}}}-{\frac{Ab+3\,Ba}{8\,{b}^{2}}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{A}{12\,ab}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{4\,{b}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(1/8*(A*b-5*B*a)/a/b*x^(9/2)-1/8*(A*b+3*B*a)/b^2*x^(3/2))/(b*x^3+a)^2+1/12/b
/a/(a*b)^(1/2)*arctan(x^(3/2)*b/(a*b)^(1/2))*A+1/4/b^2/(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)*x^(7/2)/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254935, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left ({\left (5 \, B a b - A b^{2}\right )} x^{4} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{24 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left ({\left (5 \, B a b - A b^{2}\right )} x^{4} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{a b} \sqrt{x} -{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )}{12 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(2*((5*B*a*b - A*b^2)*x^4 + (3*B*a^2 + A*a*b)*x)*sqrt(-a*b)*sqrt(x) - ((3
*B*a*b^2 + A*b^3)*x^6 + 3*B*a^3 + A*a^2*b + 2*(3*B*a^2*b + A*a*b^2)*x^3)*log((2*
a*b*x^(3/2) + (b*x^3 - a)*sqrt(-a*b))/(b*x^3 + a)))/((a*b^4*x^6 + 2*a^2*b^3*x^3
+ a^3*b^2)*sqrt(-a*b)), -1/12*(((5*B*a*b - A*b^2)*x^4 + (3*B*a^2 + A*a*b)*x)*sqr
t(a*b)*sqrt(x) - ((3*B*a*b^2 + A*b^3)*x^6 + 3*B*a^3 + A*a^2*b + 2*(3*B*a^2*b + A
*a*b^2)*x^3)*arctan(sqrt(a*b)*x^(3/2)/a))/((a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222604, size = 113, normalized size = 1.09 \[ \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{12 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{\frac{9}{2}} - A b^{2} x^{\frac{9}{2}} + 3 \, B a^{2} x^{\frac{3}{2}} + A a b x^{\frac{3}{2}}}{12 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*B*a + A*b)*arctan(b*x^(3/2)/sqrt(a*b))/(sqrt(a*b)*a*b^2) - 1/12*(5*B*a*b
*x^(9/2) - A*b^2*x^(9/2) + 3*B*a^2*x^(3/2) + A*a*b*x^(3/2))/((b*x^3 + a)^2*a*b^2
)