Optimal. Leaf size=104 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.0571408, 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, Rules used = {457, 288, 329, 275, 205} \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 288
Rule 329
Rule 275
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^{9/2}}{6 a b \left (a+b x^3\right )^2}+\frac{\left (\frac{3 A b}{2}+\frac{9 a B}{2}\right ) \int \frac{x^{7/2}}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^{9/2}}{6 a b \left (a+b x^3\right )^2}-\frac{(A b+3 a B) x^{3/2}}{12 a b^2 \left (a+b x^3\right )}+\frac{(A b+3 a B) \int \frac{\sqrt{x}}{a+b x^3} \, dx}{8 a b^2}\\ &=\frac{(A b-a B) x^{9/2}}{6 a b \left (a+b x^3\right )^2}-\frac{(A b+3 a B) x^{3/2}}{12 a b^2 \left (a+b x^3\right )}+\frac{(A b+3 a B) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^6} \, dx,x,\sqrt{x}\right )}{4 a b^2}\\ &=\frac{(A b-a B) x^{9/2}}{6 a b \left (a+b x^3\right )^2}-\frac{(A b+3 a B) x^{3/2}}{12 a b^2 \left (a+b x^3\right )}+\frac{(A b+3 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^{3/2}\right )}{12 a b^2}\\ &=\frac{(A b-a B) x^{9/2}}{6 a b \left (a+b x^3\right )^2}-\frac{(A b+3 a B) x^{3/2}}{12 a b^2 \left (a+b x^3\right )}+\frac{(A b+3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.141612, size = 93, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \sqrt{b} x^{3/2} \left (-3 a^2 B-a b \left (A+5 B x^3\right )+A b^2 x^3\right )}{\left (a+b x^3\right )^2}+(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 96, normalized size = 0.9 \begin{align*}{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74468, size = 664, normalized size = 6.38 \begin{align*} \left [-\frac{{\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 )} \sqrt{-a b} \log \left (\frac{b x^{3} - 2 \, \sqrt{-a b} x^{\frac{3}{2}} - a}{b x^{3} + a}\right ) + 2 \,{\left ({\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} +{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{24 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, \frac{{\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 )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right ) -{\left ({\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} +{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{12 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11753, size = 113, normalized size = 1.09 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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