Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]
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Rubi [A] time = 0.0142352, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} (A+B x)}{x^{11/2}} \, dx &=-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac{\left (2 \left (-A b+\frac{9 a B}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac{2 (2 A b-9 a B) (a+b x)^{7/2}}{63 a^2 x^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0203764, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{7/2} (7 a A+9 a B x-2 A b x)}{63 a^2 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.6 \begin{align*} -{\frac{-4\,Abx+18\,Bax+14\,Aa}{63\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{9}{2}}}} \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 [B] time = 2.6027, size = 223, normalized size = 4.21 \begin{align*} -\frac{2 \,{\left (7 \, A a^{4} +{\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} +{\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{3} + 3 \,{\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{63 \, a^{2} x^{\frac{9}{2}}} \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 = 1.3019, size = 116, normalized size = 2.19 \begin{align*} \frac{{\left (b x + a\right )}^{\frac{7}{2}} b{\left (\frac{{\left (9 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}}{64512 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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