Optimal. Leaf size=116 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]
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Rubi [A] time = 0.0843423, antiderivative size = 116, 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 = {102, 157, 63, 217, 206, 93, 208} \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 102
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x \sqrt{c+d x}} \, dx &=\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d}+\frac{\int \frac{a^2 d-\frac{1}{2} b (b c-3 a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{d}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d}+a^2 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx-\frac{(b (b c-3 a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 d}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d}+\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 )-\frac{(b c-3 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{b \sqrt{a+b x} \sqrt{c+d x}}{d}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{(b c-3 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 \sqrt{a+b x} \sqrt{c+d x}}{d}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 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 = 0.642659, size = 176, normalized size = 1.52 \[ \frac{\sqrt{d} \left (b \sqrt{a+b x} (c+d x)-\frac{2 a^{3/2} d \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}\right )-\frac{\left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}}{d^{3/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 220, normalized size = 1.9 \begin{align*}{\frac{1}{2\,d}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) abd\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 ){b}^{2}c\sqrt{ac}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}d\sqrt{bd}+2\,b\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 [B] time = 8.09439, size = 1909, normalized size = 16.46 \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{3}{2}}}{x \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31976, size = 267, normalized size = 2.3 \begin{align*} -\frac{{\left (\frac{4 \, \sqrt{b d} a^{2} \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} - \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 c - 3 \, \sqrt{b d} a 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 )}{b d^{2}}\right )} b^{2}}{2 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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