Optimal. Leaf size=117 \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]
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Rubi [A] time = 0.0509159, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^{5/2}} \, dx &=-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac{\left (2 \left (A b+\frac{3 a B}{2}\right )\right ) \int \frac{(a+b x)^{3/2}}{x^{3/2}} \, dx}{3 a}\\ &=-\frac{2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac{(b (2 A b+3 a B)) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx}{a}\\ &=\frac{b (2 A b+3 a B) \sqrt{x} \sqrt{a+b x}}{a}-\frac{2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac{1}{2} (b (2 A b+3 a B)) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=\frac{b (2 A b+3 a B) \sqrt{x} \sqrt{a+b x}}{a}-\frac{2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+(b (2 A b+3 a B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{b (2 A b+3 a B) \sqrt{x} \sqrt{a+b x}}{a}-\frac{2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+(b (2 A b+3 a B)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )\\ &=\frac{b (2 A b+3 a B) \sqrt{x} \sqrt{a+b x}}{a}-\frac{2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\sqrt{b} (2 A b+3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0740862, size = 73, normalized size = 0.62 \[ \frac{2 \sqrt{a+b x} \left (-\frac{x (3 a B+2 A b) \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{\frac{b x}{a}+1}}-\frac{A (a+b x)^2}{a}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 162, normalized size = 1.4 \begin{align*}{\frac{1}{6}\sqrt{bx+a} \left ( 6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{b}^{2}+9\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}ab+6\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{x}^{2}-16\,Ax{b}^{3/2}\sqrt{x \left ( bx+a \right ) }-12\,Bxa\sqrt{x \left ( bx+a \right ) }\sqrt{b}-4\,Aa\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \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 = 2.7326, size = 424, normalized size = 3.62 \begin{align*} \left [\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{6 \, x^{2}}, -\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 77.8419, size = 168, normalized size = 1.44 \begin{align*} A \left (- \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )}\right ) + B \left (- \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 + \frac{b x}{a}}} + 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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