Optimal. Leaf size=133 \[ \frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.0521983, antiderivative size = 133, 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} \[ \frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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{x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}-\frac{\left (2 \left (A b-\frac{5 a B}{2}\right )\right ) \int \frac{x^{3/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{a b^2}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{2 b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.053889, size = 80, normalized size = 0.6 \[ \frac{2 x^{5/2} \left ((a+b x) \sqrt{\frac{b x}{a}+1} (5 a B-2 A b) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b x}{a}\right )+5 a (A b-a B)\right )}{15 a^2 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 315, normalized size = 2.4 \begin{align*}{\frac{1}{6} \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}^{3}-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{2}+6\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+12\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xa{b}^{2}-16\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x-30\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b+40\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}xa+6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b-12\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}a-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}+30\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{2} \right ) \sqrt{x}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{b}^{-{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \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 = 2.83938, size = 729, normalized size = 5.48 \begin{align*} \left [-\frac{3 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b +{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{6 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{3 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b +{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \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 [B] time = 85.9072, size = 417, normalized size = 3.14 \begin{align*} \frac{\sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} B{\left | b \right |}}{b^{5}} + \frac{{\left (5 \, B a \sqrt{b}{\left | b \right |} - 2 \, A b^{\frac{3}{2}}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{5}} + \frac{4 \,{\left (9 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 6 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 7 \, B a^{4} b^{\frac{5}{2}}{\left | b \right |} - 6 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 4 \, A a^{3} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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