Optimal. Leaf size=237 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)}{64 b^3 d^3}-\frac{5 (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{24 b^2 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (4 a b c d-5 (a d+b c)^2\right )}{32 b^3 d^2}+\frac{\left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{7/2}}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d} \]
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Rubi [A] time = 0.195934, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)}{64 b^3 d^3}-\frac{5 (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{24 b^2 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (4 a b c d-5 (a d+b c)^2\right )}{32 b^3 d^2}+\frac{\left (4 a b c d-5 (a d+b c)^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{7/2}}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b x} \sqrt{c+d x} \, dx &=\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac{\int \sqrt{a+b x} \sqrt{c+d x} \left (-a c-\frac{5}{2} (b c+a d) x\right ) \, dx}{4 b d}\\ &=-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) \int \sqrt{a+b x} \sqrt{c+d x} \, dx}{16 b^2 d^2}\\ &=-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3 d^2}-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}-\frac{\left ((b c-a d) \left (4 a b c d-5 (b c+a d)^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{64 b^3 d^2}\\ &=-\frac{(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d^3}-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3 d^2}-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^3 d^3}\\ &=-\frac{(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d^3}-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3 d^2}-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right )\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^4 d^3}\\ &=-\frac{(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d^3}-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3 d^2}-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right )\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^4 d^3}\\ &=-\frac{(b c-a d) \left (4 a b c d-5 (b c+a d)^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d^3}-\frac{\left (4 a b c d-5 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3 d^2}-\frac{5 (b c+a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{3/2}}{4 b d}+\frac{(b c-a d)^2 \left (4 a b c d-5 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.494562, size = 214, normalized size = 0.9 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) \left (-a^2 b d^2 (7 c+10 d x)+15 a^3 d^3+a b^2 d \left (-7 c^2+4 c d x+8 d^2 x^2\right )+b^3 \left (-10 c^2 d x+15 c^3+8 c d^2 x^2+48 d^3 x^3\right )\right )-3 (b c-a d)^{5/2} \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{192 b^4 d^{7/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 686, normalized size = 2.9 \begin{align*} -{\frac{1}{384\,{b}^{3}{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}-16\,{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}+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}^{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}-6\,\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}-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{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}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{d}^{3}-8\,\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-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{d}^{3}+14\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}bc{d}^{2}+14\,\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 = 3.04908, size = 1192, normalized size = 5.03 \begin{align*} \left [\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 )} \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 - 7 \, a b^{3} c^{2} d^{2} - 7 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{4} d^{4}}, \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 )} \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 - 7 \, a b^{3} c^{2} d^{2} - 7 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, 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^{2} \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 = 1.35884, size = 386, normalized size = 1.63 \begin{align*} \frac{{\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 |}}{192 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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