Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{7 x^{7/2}}-\frac{2 a (a B+2 A b)}{5 x^{5/2}}-\frac{2 b (2 a B+A b)}{3 x^{3/2}}-\frac{2 b^2 B}{\sqrt{x}} \]
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Rubi [A] time = 0.0272062, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{2 a^2 A}{7 x^{7/2}}-\frac{2 a (a B+2 A b)}{5 x^{5/2}}-\frac{2 b (2 a B+A b)}{3 x^{3/2}}-\frac{2 b^2 B}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{9/2}} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^{9/2}} \, dx\\ &=\int \left (\frac{a^2 A}{x^{9/2}}+\frac{a (2 A b+a B)}{x^{7/2}}+\frac{b (A b+2 a B)}{x^{5/2}}+\frac{b^2 B}{x^{3/2}}\right ) \, dx\\ &=-\frac{2 a^2 A}{7 x^{7/2}}-\frac{2 a (2 A b+a B)}{5 x^{5/2}}-\frac{2 b (A b+2 a B)}{3 x^{3/2}}-\frac{2 b^2 B}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0163576, size = 50, normalized size = 0.82 \[ -\frac{2 \left (3 a^2 (5 A+7 B x)+14 a b x (3 A+5 B x)+35 b^2 x^2 (A+3 B x)\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 52, normalized size = 0.9 \begin{align*} -{\frac{210\,{b}^{2}B{x}^{3}+70\,A{b}^{2}{x}^{2}+140\,B{x}^{2}ab+84\,aAbx+42\,{a}^{2}Bx+30\,A{a}^{2}}{105}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06024, size = 69, normalized size = 1.13 \begin{align*} -\frac{2 \,{\left (105 \, B b^{2} x^{3} + 15 \, A a^{2} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54523, size = 128, normalized size = 2.1 \begin{align*} -\frac{2 \,{\left (105 \, B b^{2} x^{3} + 15 \, A a^{2} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.51759, size = 80, normalized size = 1.31 \begin{align*} - \frac{2 A a^{2}}{7 x^{\frac{7}{2}}} - \frac{4 A a b}{5 x^{\frac{5}{2}}} - \frac{2 A b^{2}}{3 x^{\frac{3}{2}}} - \frac{2 B a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 B a b}{3 x^{\frac{3}{2}}} - \frac{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 = 1.28703, size = 69, normalized size = 1.13 \begin{align*} -\frac{2 \,{\left (105 \, B b^{2} x^{3} + 70 \, B a b x^{2} + 35 \, A b^{2} x^{2} + 21 \, B a^{2} x + 42 \, A a b x + 15 \, A a^{2}\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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