Optimal. Leaf size=67 \[ \frac{2 (a+b x)^{7/2} (A b-2 a B)}{7 b^3}-\frac{2 a (a+b x)^{5/2} (A b-a B)}{5 b^3}+\frac{2 B (a+b x)^{9/2}}{9 b^3} \]
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Rubi [A] time = 0.0246166, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{2 (a+b x)^{7/2} (A b-2 a B)}{7 b^3}-\frac{2 a (a+b x)^{5/2} (A b-a B)}{5 b^3}+\frac{2 B (a+b x)^{9/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int x (a+b x)^{3/2} (A+B x) \, dx &=\int \left (\frac{a (-A b+a B) (a+b x)^{3/2}}{b^2}+\frac{(A b-2 a B) (a+b x)^{5/2}}{b^2}+\frac{B (a+b x)^{7/2}}{b^2}\right ) \, dx\\ &=-\frac{2 a (A b-a B) (a+b x)^{5/2}}{5 b^3}+\frac{2 (A b-2 a B) (a+b x)^{7/2}}{7 b^3}+\frac{2 B (a+b x)^{9/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0336395, size = 49, normalized size = 0.73 \[ \frac{2 (a+b x)^{5/2} \left (8 a^2 B-2 a b (9 A+10 B x)+5 b^2 x (9 A+7 B x)\right )}{315 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 47, normalized size = 0.7 \begin{align*} -{\frac{-70\,{b}^{2}B{x}^{2}-90\,{b}^{2}Ax+40\,abBx+36\,Aab-16\,B{a}^{2}}{315\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14123, size = 73, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 63 \,{\left (B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{315 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4864, size = 212, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (35 \, B b^{4} x^{4} + 8 \, B a^{4} - 18 \, A a^{3} b + 5 \,{\left (10 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + 24 \, A a b^{3}\right )} x^{2} -{\left (4 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.04559, size = 178, normalized size = 2.66 \begin{align*} \frac{2 A a \left (- \frac{a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}} + \frac{2 A \left (\frac{a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{2}} + \frac{2 B a \left (\frac{a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}} + \frac{2 B \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33004, size = 215, normalized size = 3.21 \begin{align*} \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a}{b} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} B a}{b^{2}} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} A}{b} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b^{2}}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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