Optimal. Leaf size=81 \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]
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Rubi [A] time = 0.035759, antiderivative size = 81, 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} \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac{a^3 A}{x^{5/2}}+\frac{a^2 (3 A b+a B)}{x^{3/2}}+\frac{3 a b (A b+a B)}{\sqrt{x}}+b^2 (A b+3 a B) \sqrt{x}+b^3 B x^{3/2}\right ) \, dx\\ &=-\frac{2 a^3 A}{3 x^{3/2}}-\frac{2 a^2 (3 A b+a B)}{\sqrt{x}}+6 a b (A b+a B) \sqrt{x}+\frac{2}{3} b^2 (A b+3 a B) x^{3/2}+\frac{2}{5} b^3 B x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0207379, size = 66, normalized size = 0.81 \[ \frac{2 \left (45 a^2 b x (B x-A)-5 a^3 (A+3 B x)+15 a b^2 x^2 (3 A+B x)+b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 76, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{b}^{3}{x}^{4}-10\,A{b}^{3}{x}^{3}-30\,B{x}^{3}a{b}^{2}-90\,aA{b}^{2}{x}^{2}-90\,B{x}^{2}{a}^{2}b+90\,{a}^{2}Abx+30\,{a}^{3}Bx+10\,{a}^{3}A}{15}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08934, size = 99, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{3} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28619, size = 165, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69225, size = 105, normalized size = 1.3 \begin{align*} - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A a^{2} b}{\sqrt{x}} + 6 A a b^{2} \sqrt{x} + \frac{2 A b^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{\sqrt{x}} + 6 B a^{2} b \sqrt{x} + 2 B a b^{2} x^{\frac{3}{2}} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34927, size = 101, normalized size = 1.25 \begin{align*} \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + 2 \, B a b^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + 6 \, B a^{2} b \sqrt{x} + 6 \, A a b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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