Optimal. Leaf size=221 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)}{96 b^2 d^2}-\frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{7/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+5 b c)}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d} \]
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Rubi [A] time = 0.125251, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)}{96 b^2 d^2}-\frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{7/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+5 b c)}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x (a+b x)^{3/2} \sqrt{c+d x} \, dx &=\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}+\frac{\left (-\frac{5 b c}{2}-\frac{3 a d}{2}\right ) \int (a+b x)^{3/2} \sqrt{c+d x} \, dx}{4 b d}\\ &=-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}-\frac{((b c-a d) (5 b c+3 a d)) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{48 b^2 d}\\ &=-\frac{(b c-a d) (5 b c+3 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^2}-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}+\frac{\left ((b c-a d)^2 (5 b c+3 a d)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{64 b^2 d^2}\\ &=\frac{(b c-a d)^2 (5 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^3}-\frac{(b c-a d) (5 b c+3 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^2}-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}-\frac{\left ((b c-a d)^3 (5 b c+3 a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^2 d^3}\\ &=\frac{(b c-a d)^2 (5 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^3}-\frac{(b c-a d) (5 b c+3 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^2}-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}-\frac{\left ((b c-a d)^3 (5 b c+3 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 )}{64 b^3 d^3}\\ &=\frac{(b c-a d)^2 (5 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^3}-\frac{(b c-a d) (5 b c+3 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^2}-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}-\frac{\left ((b c-a d)^3 (5 b c+3 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 )}{64 b^3 d^3}\\ &=\frac{(b c-a d)^2 (5 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^3}-\frac{(b c-a d) (5 b c+3 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^2}-\frac{(5 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d}-\frac{(b c-a d)^3 (5 b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 1.36968, size = 299, normalized size = 1.35 \[ \frac{(a+b x)^{5/2} (c+d x)^{3/2} \left (5-\frac{5 (3 a d+5 b c) \left (8 b^3 d^3 (a+b x)^3 \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}-b (b c-a d) \left (-2 b^2 d^2 (a+b x)^2 \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}+3 b^2 d (a+b x) (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-3 b^2 \sqrt{d} \sqrt{a+b x} (b c-a d)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )\right )}{48 b^3 d^3 (a+b x)^3 (b c-a d)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2}}\right )}{20 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 686, normalized size = 3.1 \begin{align*}{\frac{1}{384\,{b}^{2}{d}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+144\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+16\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-12\,\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}bc{d}^{3}-18\,\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}{b}^{2}{c}^{2}{d}^{2}+36\,\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}^{3}{c}^{3}d-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}{c}^{4}+12\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{d}^{3}+40\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{2}c{d}^{2}-20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{3}{c}^{2}d-18\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{d}^{3}+18\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}bc{d}^{2}-62\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{2}{c}^{2}d+30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}{c}^{3} \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 = 2.38243, size = 1204, normalized size = 5.45 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 31 \, a b^{3} c^{2} d^{2} + 9 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \,{\left (b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} - 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{3} d^{4}}, \frac{3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 31 \, a b^{3} c^{2} d^{2} + 9 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \,{\left (b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} - 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b^{3} d^{4}}\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 [B] time = 1.35229, size = 653, normalized size = 2.95 \begin{align*} \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\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 d^{3}}\right )}{\left | b \right |}}{b} + \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 )} a{\left | b \right |}}{b^{3}}}{1920 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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