Optimal. Leaf size=125 \[ -\frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d} \]
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Rubi [A] time = 0.0638365, 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} \[ -\frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d} \]
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{c+d x}}{\sqrt{a+b x}} \, dx &=\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d}-\frac{(b c+3 a d) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{4 b d}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d}-\frac{((b c-a d) (b c+3 a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 b^2 d}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d}-\frac{((b c-a d) (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 )}{4 b^3 d}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d}-\frac{((b c-a d) (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 )}{4 b^3 d}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d}-\frac{(b c-a d) (b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.277263, size = 121, normalized size = 0.97 \[ \frac{\sqrt{c+d x} \left (\sqrt{d} \sqrt{a+b x} (b (c+2 d x)-3 a d)-\frac{\sqrt{b c-a d} (3 a d+b c) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{4 b^2 d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 251, normalized size = 2. \begin{align*}{\frac{1}{8\,{b}^{2}d}\sqrt{bx+a}\sqrt{dx+c} \left ( 4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xbd+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 ){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-\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}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}ad+2\,\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 = 2.16683, size = 711, normalized size = 5.69 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, 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 + b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, b^{3} d^{2}}, \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, 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 + b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, b^{3} d^{2}}\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 = 2.6957, size = 190, normalized size = 1.52 \begin{align*} \frac{{\left (\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^{4} d^{2}} + \frac{b c d - 5 \, a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, 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} b^{3} d^{3}}\right )}{\left | b \right |}}{48 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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