Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}-\frac{2 b (2 a B+A b)}{\sqrt{x}}+2 b^2 B \sqrt{x} \]
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Rubi [A] time = 0.0250903, antiderivative size = 59, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}-\frac{2 b (2 a B+A b)}{\sqrt{x}}+2 b^2 B \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{x^{7/2}} \, dx &=\int \left (\frac{a^2 A}{x^{7/2}}+\frac{a (2 A b+a B)}{x^{5/2}}+\frac{b (A b+2 a B)}{x^{3/2}}+\frac{b^2 B}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (2 A b+a B)}{3 x^{3/2}}-\frac{2 b (A b+2 a B)}{\sqrt{x}}+2 b^2 B \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.0151995, size = 47, normalized size = 0.8 \[ -\frac{2 \left (a^2 (3 A+5 B x)+10 a b x (A+3 B x)+15 b^2 x^2 (A-B x)\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 52, normalized size = 0.9 \begin{align*} -{\frac{-30\,B{b}^{2}{x}^{3}+30\,A{b}^{2}{x}^{2}+60\,B{x}^{2}ab+20\,aAbx+10\,{a}^{2}Bx+6\,{a}^{2}A}{15}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0908, size = 70, normalized size = 1.19 \begin{align*} 2 \, B b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36147, size = 122, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.1037, size = 75, normalized size = 1.27 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{3 x^{\frac{3}{2}}} - \frac{2 A b^{2}}{\sqrt{x}} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 B a b}{\sqrt{x}} + 2 B b^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18931, size = 70, normalized size = 1.19 \begin{align*} 2 \, B b^{2} \sqrt{x} - \frac{2 \,{\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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