Optimal. Leaf size=144 \[ -\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}} \]
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Rubi [A] time = 0.112782, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 97
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} \sqrt{c+d x}}{x^2} \, dx &=-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+\int \frac{\sqrt{a+b x} \left (\frac{1}{2} (3 b c+a d)+2 b d x\right )}{x \sqrt{c+d x}} \, dx\\ &=2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+\frac{\int \frac{\frac{1}{2} a d (3 b c+a d)+\frac{1}{2} b d (b c+3 a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{d}\\ &=2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+\frac{1}{2} (a (3 b c+a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{1}{2} (b (b c+3 a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx\\ &=2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+(a (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )+(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )\\ &=2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}-\frac{\sqrt{a} (3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )\\ &=2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}-\frac{\sqrt{a} (3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.83051, size = 164, normalized size = 1.14 \[ \frac{\sqrt{a+b x} (b x-a) \sqrt{c+d x}}{x}+\frac{\sqrt{b c-a d} (3 a d+b c) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{d} \sqrt{c+d x}}-\frac{\sqrt{a} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 347, normalized size = 2.4 \begin{align*}{\frac{1}{2\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabd\sqrt{ac}+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{2}c\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) x{a}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xabc\sqrt{bd}+2\,xb\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-2\,a\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \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 = 7.50872, size = 2026, normalized size = 14.07 \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}} \sqrt{c + d x}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.40549, size = 709, normalized size = 4.92 \begin{align*} \frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left | b \right |} - \frac{{\left (\sqrt{b d} b c{\left | b \right |} + 3 \, \sqrt{b d} a d{\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 )}{d} - \frac{2 \,{\left (3 \, \sqrt{b d} a b^{2} c{\left | b \right |} + \sqrt{b d} a^{2} b d{\left | b \right |}\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{4 \,{\left (\sqrt{b d} a b^{4} c^{2}{\left | b \right |} - 2 \, \sqrt{b d} a^{2} b^{3} c d{\left | b \right |} + \sqrt{b d} a^{3} b^{2} d^{2}{\left | b \right |} - \sqrt{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} a b^{2} c{\left | b \right |} - \sqrt{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} a^{2} b d{\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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