Optimal. Leaf size=119 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}+\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x} (b c-a d)}{c d \sqrt{c+d x}} \]
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Rubi [A] time = 0.0680015, 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} \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}+\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x} (b c-a d)}{c d \sqrt{c+d x}} \]
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{(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx &=-\frac{2 (b c-a d) \sqrt{a+b x}}{c d \sqrt{c+d x}}+\frac{2 \int \frac{\frac{a^2 d}{2}+\frac{1}{2} b^2 c x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c d}\\ &=-\frac{2 (b c-a d) \sqrt{a+b x}}{c d \sqrt{c+d x}}+\frac{a^2 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c}+\frac{b^2 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{d}\\ &=-\frac{2 (b c-a d) \sqrt{a+b x}}{c d \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}+\frac{(2 b) \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{2 (b c-a d) \sqrt{a+b x}}{c d \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^{3/2}}+\frac{(2 b) \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{2 (b c-a d) \sqrt{a+b x}}{c d \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^{3/2}}+\frac{2 b^{3/2} \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 = 0.794246, size = 162, normalized size = 1.36 \[ \frac{2 \left (\frac{\sqrt{d} \left (\sqrt{c} \sqrt{a+b x} (a d-b c)-a^{3/2} d \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )}{c^{3/2}}+b \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{d^{3/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 306, normalized size = 2.6 \begin{align*}{\frac{1}{dc}\sqrt{bx+a} \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{b}^{2}cd\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}^{2}{d}^{2}\sqrt{bd}+\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 ) \sqrt{ac}{b}^{2}{c}^{2}-\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 ){a}^{2}cd+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}ad-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}bc \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{dx+c}}}} \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 = 6.69364, size = 2140, normalized size = 17.98 \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 \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.34973, size = 277, normalized size = 2.33 \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{\left | b \right |}} - \frac{\sqrt{b d} b^{2} \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}{\left | b \right |}} - \frac{2 \,{\left (b^{3} c{\left | b \right |} - a b^{2} d{\left | b \right |}\right )} \sqrt{b x + a}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} b^{2} c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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