Optimal. Leaf size=162 \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{c d} \]
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Rubi [A] time = 0.154712, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {98, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{c d} \]
Antiderivative was successfully verified.
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Rule 98
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x^2 \sqrt{c+d x}} \, dx &=-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{\int \frac{\sqrt{a+b x} \left (-\frac{1}{2} a (5 b c-a d)-b (b c+a d) x\right )}{x \sqrt{c+d x}} \, dx}{c}\\ &=\frac{b (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{c d}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{\int \frac{-\frac{1}{2} a^2 d (5 b c-a d)+\frac{1}{2} b^2 c (b c-5 a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c d}\\ &=\frac{b (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{c d}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{\left (b^2 (b c-5 a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 d}+\frac{\left (a^2 (5 b c-a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 c}\\ &=\frac{b (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{c d}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{(b (b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{d}+\frac{\left (a^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{c}\\ &=\frac{b (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{c d}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{(b (b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{b (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{c d}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}-\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.820756, size = 176, normalized size = 1.09 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (b^2 c x-a^2 d\right )}{c d x}+\frac{a^{3/2} (a d-5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{(b c-5 a d) (b c-a d)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{d^{3/2} (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 320, normalized size = 2. \begin{align*}{\frac{1}{2\,cdx}\sqrt{bx+a}\sqrt{dx+c} \left ( 5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{3}{c}^{2}\sqrt{ac}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{3}{d}^{2}\sqrt{bd}-5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}bcd\sqrt{bd}+2\,x{b}^{2}c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,{a}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \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 = 14.1628, size = 2188, normalized size = 13.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{5}{2}}}{x^{2} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.93411, size = 710, normalized size = 4.38 \begin{align*} \frac{b{\left (\frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a} b}{d} + \frac{{\left (\sqrt{b d} b^{2} c - 5 \, \sqrt{b d} a b d\right )} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2}} - \frac{2 \,{\left (5 \, \sqrt{b d} a^{2} b^{2} c - \sqrt{b d} a^{3} b d\right )} \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} b c} - \frac{4 \,{\left (\sqrt{b d} a^{2} b^{4} c^{2} - 2 \, \sqrt{b d} a^{3} b^{3} c d + \sqrt{b d} a^{4} b^{2} d^{2} - \sqrt{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} a^{2} b^{2} c - \sqrt{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} a^{3} b d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}\right )} c}\right )}}{2 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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