Optimal. Leaf size=92 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}} \]
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Rubi [A] time = 0.0302455, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx &=\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}}+\frac{a \int \frac{\sqrt{a+b x}}{x (c+d x)^{3/2}} \, dx}{c}\\ &=\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{a^2 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c^2}\\ &=\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{c^2}\\ &=\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0824594, size = 82, normalized size = 0.89 \[ \frac{2 \sqrt{a+b x} (4 a c+3 a d x+b c x)}{3 c^2 (c+d x)^{3/2}}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 248, normalized size = 2.7 \begin{align*} -{\frac{1}{3\,{c}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}cd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xad-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xbc-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ac\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \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 [B] time = 5.23116, size = 755, normalized size = 8.21 \begin{align*} \left [\frac{3 \,{\left (a d^{2} x^{2} + 2 \, a c d x + a c^{2}\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (4 \, a c +{\left (b c + 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (c^{2} d^{2} x^{2} + 2 \, c^{3} d x + c^{4}\right )}}, \frac{3 \,{\left (a d^{2} x^{2} + 2 \, a c d x + a c^{2}\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{a}{c}}}{2 \,{\left (a b d x^{2} + a^{2} c +{\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \,{\left (4 \, a c +{\left (b c + 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (c^{2} d^{2} x^{2} + 2 \, c^{3} d x + c^{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 = 2.39226, size = 379, normalized size = 4.12 \begin{align*} -\frac{2 \, \sqrt{b d} a^{2} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} - \frac{\sqrt{b x + a}{\left (\frac{{\left (b^{5} c^{4} d{\left | b \right |} + 2 \, a b^{4} c^{3} d^{2}{\left | b \right |} - 3 \, a^{2} b^{3} c^{2} d^{3}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (a b^{5} c^{4} d{\left | b \right |} - 2 \, a^{2} b^{4} c^{3} d^{2}{\left | b \right |} + a^{3} b^{3} c^{2} d^{3}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{48 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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