Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{5/2} (2 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{5/2}}{7 a x^{7/2}} \]
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Rubi [A] time = 0.0146028, 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} \[ \frac{2 (a+b x)^{5/2} (2 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{5/2}}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx &=-\frac{2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac{\left (2 \left (-A b+\frac{7 a B}{2}\right )\right ) \int \frac{(a+b x)^{3/2}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac{2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac{2 (2 A b-7 a B) (a+b x)^{5/2}}{35 a^2 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0157798, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{5/2} (5 a A+7 a B x-2 A b x)}{35 a^2 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.6 \begin{align*} -{\frac{-4\,Abx+14\,Bax+10\,Aa}{35\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{7}{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 [A] time = 2.59215, size = 173, normalized size = 3.26 \begin{align*} -\frac{2 \,{\left (5 \, A a^{3} +{\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} +{\left (14 \, B a^{2} b + A a b^{2}\right )} x^{2} +{\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} x\right )} \sqrt{b x + a}}{35 \, a^{2} x^{\frac{7}{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 = 1.3, size = 113, normalized size = 2.13 \begin{align*} \frac{{\left (b x + a\right )}^{\frac{5}{2}} b{\left (\frac{{\left (7 \, B a^{2} b^{6} - 2 \, A a b^{7}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{7 \,{\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{4} b^{12}}\right )}}{26880 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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