Optimal. Leaf size=120 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.0427632, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{9/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right ) (A+B x)}{x^{9/2}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{x^{9/2}}+\frac{b (A b+a B)}{x^{7/2}}+\frac{b^2 B}{x^{5/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0284987, size = 51, normalized size = 0.42 \[ -\frac{2 \sqrt{(a+b x)^2} (3 a (5 A+7 B x)+7 b x (3 A+5 B x))}{105 x^{7/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 0.4 \begin{align*} -{\frac{70\,Bb{x}^{2}+42\,Abx+42\,aBx+30\,aA}{105\,bx+105\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05445, size = 47, normalized size = 0.39 \begin{align*} -\frac{2 \,{\left (5 \, b x^{2} + 3 \, a x\right )} B}{15 \, x^{\frac{7}{2}}} - \frac{2 \,{\left (7 \, b x^{2} + 5 \, a x\right )} A}{35 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49881, size = 77, normalized size = 0.64 \begin{align*} -\frac{2 \,{\left (35 \, B b x^{2} + 15 \, A a + 21 \,{\left (B 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13674, size = 69, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (35 \, B b x^{2} \mathrm{sgn}\left (b x + a\right ) + 21 \, B a x \mathrm{sgn}\left (b x + a\right ) + 21 \, A b x \mathrm{sgn}\left (b x + a\right ) + 15 \, A a \mathrm{sgn}\left (b x + a\right )\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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