Optimal. Leaf size=213 \[ -\frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 d}{b}+8 a c-\frac{b c^2}{d}\right )-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]
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Rubi [A] time = 0.197545, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 d}{b}+8 a c-\frac{b c^2}{d}\right )-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx &=\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac{1}{3} \int \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a c-\frac{3}{2} (b c+a d) x\right )}{x} \, dx\\ &=\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac{\int \frac{\sqrt{c+d x} \left (-6 a^2 c d+\frac{3}{4} \left (b^2 c^2-8 a b c d-a^2 d^2\right ) x\right )}{x \sqrt{a+b x}} \, dx}{6 d}\\ &=\frac{1}{8} \left (8 a c-\frac{b c^2}{d}+\frac{a^2 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}+\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac{\int \frac{-6 a^2 b c^2 d+\frac{3}{8} (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 b d}\\ &=\frac{1}{8} \left (8 a c-\frac{b c^2}{d}+\frac{a^2 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}+\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (a^2 c^2\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx-\frac{\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b d}\\ &=\frac{1}{8} \left (8 a c-\frac{b c^2}{d}+\frac{a^2 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}+\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )-\frac{\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 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 )}{8 b^2 d}\\ &=\frac{1}{8} \left (8 a c-\frac{b c^2}{d}+\frac{a^2 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}+\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 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 )}{8 b^2 d}\\ &=\frac{1}{8} \left (8 a c-\frac{b c^2}{d}+\frac{a^2 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}+\frac{(b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{4 d}+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.753446, size = 243, normalized size = 1.14 \[ \frac{\sqrt{d} \left (\sqrt{a+b x} (c+d x) \left (3 a^2 d^2+2 a b d (19 c+7 d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )-48 a^{3/2} b c^{3/2} d \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )-\frac{3 \sqrt{b c-a d} \left (-9 a^2 b c d^2+a^3 d^3-9 a b^2 c^2 d+b^3 c^3\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{24 b d^{3/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 587, normalized size = 2.8 \begin{align*} -{\frac{1}{48\,bd}\sqrt{bx+a}\sqrt{dx+c} \left ( -16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{3}{d}^{3}-27\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{2}bc{d}^{2}-27\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}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 ) \sqrt{ac}{b}^{3}{c}^{3}+48\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){a}^{2}b{c}^{2}d-28\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}xab{d}^{2}-28\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}x{b}^{2}cd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{a}^{2}{d}^{2}-76\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}abcd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \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 = 51.023, size = 2722, normalized size = 12.78 \begin{align*} \text{result too large to display} \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 \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.66533, size = 447, normalized size = 2.1 \begin{align*} -\frac{2 \, \sqrt{b d} a^{2} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{24} \, \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 )} d{\left | b \right |}}{b^{3}} + \frac{7 \, b^{5} c d^{4}{\left | b \right |} - a b^{4} d^{5}{\left | b \right |}}{b^{7} d^{4}}\right )} + \frac{3 \,{\left (b^{6} c^{2} d^{3}{\left | b \right |} + 8 \, a b^{5} c d^{4}{\left | b \right |} - a^{2} b^{4} d^{5}{\left | b \right |}\right )}}{b^{7} d^{4}}\right )} + \frac{{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} - 9 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} - 9 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + \sqrt{b d} a^{3} d^{3}{\left | b \right |}\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 )}^{2}\right )}{16 \, b^{3} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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