Optimal. Leaf size=74 \[ \frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{315 a^3}-\frac{8 b x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{63 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{5/2}}{9 a} \]
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Rubi [A] time = 0.0226424, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{315 a^3}-\frac{8 b x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{63 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{5/2}}{9 a} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} x^{7/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{9 a}-\frac{(4 b) \int \left (a+\frac{b}{x}\right )^{3/2} x^{5/2} \, dx}{9 a}\\ &=-\frac{8 b \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{63 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{9 a}+\frac{\left (8 b^2\right ) \int \left (a+\frac{b}{x}\right )^{3/2} x^{3/2} \, dx}{63 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}}{315 a^3}-\frac{8 b \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{63 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{9 a}\\ \end{align*}
Mathematica [A] time = 0.015161, size = 49, normalized size = 0.66 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (35 a^2 x^2-20 a b x+8 b^2\right )}{315 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 35\,{a}^{2}{x}^{2}-20\,xab+8\,{b}^{2} \right ) }{315\,{a}^{3}}{x}^{{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976777, size = 70, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (35 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} x^{\frac{9}{2}} - 90 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b x^{\frac{7}{2}} + 63 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{2} x^{\frac{5}{2}}\right )}}{315 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52703, size = 136, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (35 \, a^{4} x^{4} + 50 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{315 \, a^{3}} \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 [B] time = 1.22338, size = 154, normalized size = 2.08 \begin{align*} -\frac{2}{105} \, b{\left (\frac{8 \, b^{\frac{7}{2}}}{a^{3}} - \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}}{a^{3}}\right )} \mathrm{sgn}\left (x\right ) + \frac{2}{315} \, a{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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