Optimal. Leaf size=108 \[ -\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]
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Rubi [A] time = 0.0463346, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 50, 63, 205} \[ -\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{(a+b x)^2} \, dx &=\frac{(A b-a B) x^{5/2}}{a b (a+b x)}-\frac{\left (\frac{3 A b}{2}-\frac{5 a B}{2}\right ) \int \frac{x^{3/2}}{a+b x} \, dx}{a b}\\ &=-\frac{(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac{(A b-a B) x^{5/2}}{a b (a+b x)}+\frac{(3 A b-5 a B) \int \frac{\sqrt{x}}{a+b x} \, dx}{2 b^2}\\ &=\frac{(3 A b-5 a B) \sqrt{x}}{b^3}-\frac{(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac{(A b-a B) x^{5/2}}{a b (a+b x)}-\frac{(a (3 A b-5 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 b^3}\\ &=\frac{(3 A b-5 a B) \sqrt{x}}{b^3}-\frac{(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac{(A b-a B) x^{5/2}}{a b (a+b x)}-\frac{(a (3 A b-5 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{(3 A b-5 a B) \sqrt{x}}{b^3}-\frac{(3 A b-5 a B) x^{3/2}}{3 a b^2}+\frac{(A b-a B) x^{5/2}}{a b (a+b x)}-\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0592108, size = 88, normalized size = 0.81 \[ \frac{\sqrt{x} \left (-15 a^2 B+a b (9 A-10 B x)+2 b^2 x (3 A+B x)\right )}{3 b^3 (a+b x)}+\frac{\sqrt{a} (5 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 113, normalized size = 1.1 \begin{align*}{\frac{2\,B}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{2}}}-4\,{\frac{Ba\sqrt{x}}{{b}^{3}}}+{\frac{Aa}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}-{\frac{B{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{Aa}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+5\,{\frac{B{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49922, size = 524, normalized size = 4.85 \begin{align*} \left [-\frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.9693, size = 932, normalized size = 8.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2631, size = 128, normalized size = 1.19 \begin{align*} \frac{{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{B a^{2} \sqrt{x} - A a b \sqrt{x}}{{\left (b x + a\right )} b^{3}} + \frac{2 \,{\left (B b^{4} x^{\frac{3}{2}} - 6 \, B a b^{3} \sqrt{x} + 3 \, A b^{4} \sqrt{x}\right )}}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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