Optimal. Leaf size=125 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d} \]
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Rubi [A] time = 0.0638791, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {80, 50, 63, 217, 206} \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x \sqrt{a+b x}}{\sqrt{c+d x}} \, dx &=\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d}-\frac{(3 b c+a d) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{4 b d}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d}+\frac{((b c-a d) (3 b c+a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 b d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d}+\frac{((b c-a d) (3 b c+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 )}{4 b^2 d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d}+\frac{((b c-a d) (3 b c+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 )}{4 b^2 d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d}+\frac{(b c-a d) (3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.286898, size = 126, normalized size = 1.01 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) (a d-3 b c+2 b d x)+(a d+3 b c) (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{4 b^2 d^{5/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 250, normalized size = 2. \begin{align*} -{\frac{1}{8\,{d}^{2}b}\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 ){a}^{2}{d}^{2}+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 ) abcd-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 ){b}^{2}{c}^{2}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xbd-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}ad+6\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bc \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \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 = 1.98944, size = 713, normalized size = 5.7 \begin{align*} \left [-\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, b^{2} d^{3}}, -\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, b^{2} d^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.96473, size = 193, normalized size = 1.54 \begin{align*} \frac{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b d} - \frac{3 \, b^{2} c d + a b d^{2}}{b^{2} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\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 |}\right )}{\sqrt{b d} d^{2}}}{4 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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