Optimal. Leaf size=258 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{7/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]
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Rubi [A] time = 0.154667, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{7/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]
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} (c+d x)^{3/2} \, dx &=\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{(b c+a d) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{2 b d}\\ &=-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{\left (3 \left (c^2-\frac{a^2 d^2}{b^2}\right )\right ) \int (a+b x)^{3/2} \sqrt{c+d x} \, dx}{16 d}\\ &=-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{\left ((b c-a d)^2 (b c+a d)\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{32 b^3 d}\\ &=-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^3 d^2}-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}+\frac{\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b^3 d^2}\\ &=\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^3 d^3}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^3 d^2}-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^3 d^3}\\ &=\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^3 d^3}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^3 d^2}-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{\left (3 (b c-a d)^4 (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 )}{128 b^4 d^3}\\ &=\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^3 d^3}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^3 d^2}-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{\left (3 (b c-a d)^4 (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 )}{128 b^4 d^3}\\ &=\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^3 d^3}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^3 d^2}-\frac{(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{16 b^3 d}-\frac{(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac{3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.683078, size = 250, normalized size = 0.97 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) \left (2 a^2 b^2 d^2 \left (9 c^2+13 c d x+4 d^2 x^2\right )-10 a^3 b d^3 (4 c+d x)+15 a^4 d^4+2 a b^3 d \left (13 c^2 d x-20 c^3+136 c d^2 x^2+88 d^3 x^3\right )+b^4 \left (8 c^2 d^2 x^2-10 c^3 d x+15 c^4+176 c d^3 x^3+128 d^4 x^4\right )\right )-15 (b c-a d)^{9/2} (a d+b c) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{640 b^4 d^{7/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 942, normalized size = 3.7 \begin{align*} -{\frac{1}{1280\,{b}^{3}{d}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( -256\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-352\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-352\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-16\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-544\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-16\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-45\,\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}bc{d}^{4}+30\,\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}{b}^{2}{c}^{2}{d}^{3}+30\,\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}^{3}{c}^{3}{d}^{2}-45\,\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}^{4}{c}^{4}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}^{5}{c}^{5}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-52\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}-52\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-36\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \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 = 2.73758, size = 1513, normalized size = 5.86 \begin{align*} \left [\frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \,{\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, b^{4} d^{4}}, \frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \,{\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, b^{4} d^{4}}\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 \left (a + b x\right )^{\frac{3}{2}} \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 = 1.50312, size = 1536, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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