Optimal. Leaf size=119 \[ \frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]
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Rubi [A] time = 0.0785604, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {98, 157, 63, 217, 206, 93, 208} \[ \frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
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Rule 98
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{x^2 \sqrt{a+b x}} \, dx &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x}-\frac{\int \frac{\frac{1}{2} c (b c-3 a d)-a d^2 x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x}+d^2 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx-\frac{(c (b c-3 a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x}+\frac{\left (2 d^2\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}-\frac{(c (b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{a}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x}+\frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{\left (2 d^2\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}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x}+\frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 1.01433, size = 154, normalized size = 1.29 \[ \frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \sqrt{c+d x} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 223, normalized size = 1.9 \begin{align*}{\frac{1}{2\,ax}\sqrt{bx+a}\sqrt{dx+c} \left ( 2\,\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{d}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xacd\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 ) xb{c}^{2}\sqrt{bd}-2\,c\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 [B] time = 6.04008, size = 1955, normalized size = 16.43 \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{3}{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 = 2.55445, size = 653, normalized size = 5.49 \begin{align*} -\frac{\frac{\sqrt{b d} d{\left | b \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} - \frac{{\left (\sqrt{b d} b^{2} c^{2}{\left | b \right |} - 3 \, \sqrt{b d} a b c 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{2 \,{\left (\sqrt{b d} b^{4} c^{3}{\left | b \right |} - 2 \, \sqrt{b d} a b^{3} c^{2} d{\left | b \right |} + \sqrt{b d} a^{2} b^{2} c 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^{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} a b c 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}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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