Optimal. Leaf size=163 \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]
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Rubi [A] time = 0.0905002, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x \sqrt{a+b x} \sqrt{c+d x} \, dx &=\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}-\frac{(b c+a d) \int \sqrt{a+b x} \sqrt{c+d x} \, dx}{2 b d}\\ &=-\frac{(b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}-\frac{\left (c^2-\frac{a^2 d^2}{b^2}\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d}\\ &=-\frac{\left (c^2-\frac{a^2 d^2}{b^2}\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}-\frac{(b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac{\left ((b c-a d)^2 (b c+a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b^2 d^2}\\ &=-\frac{\left (c^2-\frac{a^2 d^2}{b^2}\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}-\frac{(b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac{\left ((b c-a d)^2 (b c+a d)\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 )}{8 b^3 d^2}\\ &=-\frac{\left (c^2-\frac{a^2 d^2}{b^2}\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}-\frac{(b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac{\left ((b c-a d)^2 (b c+a d)\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 )}{8 b^3 d^2}\\ &=-\frac{\left (c^2-\frac{a^2 d^2}{b^2}\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}-\frac{(b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d}+\frac{(b c-a d)^2 (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.471125, size = 156, normalized size = 0.96 \[ \frac{3 (b c-a d)^{5/2} (a d+b c) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )-b \sqrt{d} \sqrt{a+b x} (c+d x) \left (3 a^2 d^2-2 a b d (c+d x)+b^2 \left (3 c^2-2 c d x-8 d^2 x^2\right )\right )}{24 b^3 d^{5/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 472, normalized size = 2.9 \begin{align*}{\frac{1}{48\,{b}^{2}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}xab{d}^{2}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}x{b}^{2}cd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}{d}^{2}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}abcd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\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 = 2.92442, size = 902, normalized size = 5.53 \begin{align*} \left [\frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, b^{3} d^{3}}, -\frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, b^{3} d^{3}}\right ] \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 x \sqrt{a + b x} \sqrt{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23676, size = 259, normalized size = 1.59 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\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^{5} d^{4}}\right )}{\left | b \right |}}{1920 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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