Optimal. Leaf size=160 \[ \frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{a b} \]
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Rubi [A] time = 0.157803, antiderivative size = 160, 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} \[ \frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{a b} \]
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{(c+d x)^{5/2}}{x^2 \sqrt{a+b x}} \, dx &=-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}-\frac{\int \frac{\sqrt{c+d x} \left (\frac{1}{2} c (b c-5 a d)-d (b c+a d) x\right )}{x \sqrt{a+b x}} \, dx}{a}\\ &=\frac{d (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{a b}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}-\frac{\int \frac{\frac{1}{2} b c^2 (b c-5 a d)-\frac{1}{2} a d^2 (5 b c-a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a b}\\ &=\frac{d (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{a b}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}-\frac{\left (c^2 (b c-5 a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a}+\frac{\left (d^2 (5 b c-a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 b}\\ &=\frac{d (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{a b}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}-\frac{\left (c^2 (b c-5 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 )}{a}+\frac{\left (d^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b^2}\\ &=\frac{d (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{a b}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{\left (d^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b^2}\\ &=\frac{d (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{a b}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.95816, size = 195, normalized size = 1.22 \[ \frac{a^{3/2} d^{3/2} x \sqrt{b c-a d} (5 b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )+b \left (\sqrt{a} \sqrt{a+b x} (c+d x) \left (a d^2 x-b c^2\right )+b c^{3/2} x \sqrt{c+d x} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )}{a^{3/2} b^2 x \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 320, normalized size = 2. \begin{align*} -{\frac{1}{2\,axb}\sqrt{bx+a}\sqrt{dx+c} \left ( \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{a}^{2}{d}^{3}\sqrt{ac}-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 ) xabc{d}^{2}\sqrt{ac}+5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}d\sqrt{bd}-\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{b}^{2}{c}^{3}\sqrt{bd}-2\,xa{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+2\,b{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \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 = 17.3902, size = 2192, normalized size = 13.7 \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 (c + d x\right )^{\frac{5}{2}}}{x^{2} \sqrt{a + b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.01747, size = 738, normalized size = 4.61 \begin{align*} \frac{\frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a} d^{2}{\left | b \right |}}{b^{2}} - \frac{{\left (5 \, \sqrt{b d} b c d{\left | b \right |} - \sqrt{b d} a d^{2}{\left | b \right |}\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 )}{b^{2}} + \frac{2 \,{\left (\sqrt{b d} b^{2} c^{3}{\left | b \right |} - 5 \, \sqrt{b d} a b c^{2} d{\left | b \right |}\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} a b} - \frac{4 \,{\left (\sqrt{b d} b^{4} c^{4}{\left | b \right |} - 2 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} + \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} - \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} b^{2} c^{3}{\left | b \right |} - \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 b c^{2} d{\left | b \right |}\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 )} a}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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