Optimal. Leaf size=83 \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]
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Rubi [A] time = 0.0373372, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {76} \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{\sqrt{x}} \, dx &=\int \left (\frac{a^3 A}{\sqrt{x}}+a^2 (3 A b+a B) \sqrt{x}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{5/2}+b^3 B x^{7/2}\right ) \, dx\\ &=2 a^3 A \sqrt{x}+\frac{2}{3} a^2 (3 A b+a B) x^{3/2}+\frac{6}{5} a b (A b+a B) x^{5/2}+\frac{2}{7} b^2 (A b+3 a B) x^{7/2}+\frac{2}{9} b^3 B x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0246229, size = 70, normalized size = 0.84 \[ \frac{2}{315} \sqrt{x} \left (63 a^2 b x (5 A+3 B x)+105 a^3 (3 A+B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{70\,B{b}^{3}{x}^{4}+90\,A{b}^{3}{x}^{3}+270\,B{x}^{3}a{b}^{2}+378\,aA{b}^{2}{x}^{2}+378\,B{x}^{2}{a}^{2}b+630\,{a}^{2}Abx+210\,{a}^{3}Bx+630\,{a}^{3}A}{315}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09674, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27004, size = 174, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18233, size = 110, normalized size = 1.33 \begin{align*} 2 A a^{3} \sqrt{x} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15491, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} + 2 \, A a^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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