Optimal. Leaf size=254 \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d} \]
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Rubi [A] time = 0.230748, antiderivative size = 254, 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} \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d} \]
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 \frac{x^2 (c+d x)^{3/2}}{\sqrt{a+b x}} \, dx &=\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\int \frac{(c+d x)^{3/2} \left (-a c-\frac{1}{2} (3 b c+7 a d) x\right )}{\sqrt{a+b x}} \, dx}{4 b d}\\ &=-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^2 d^2}\\ &=\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\left ((b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{64 b^3 d^2}\\ &=\frac{(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d^2}+\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^4 d^2}\\ &=\frac{(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d^2}+\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 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 )}{64 b^5 d^2}\\ &=\frac{(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d^2}+\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 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 )}{64 b^5 d^2}\\ &=\frac{(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d^2}+\frac{\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac{(3 b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d}+\frac{(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.66582, size = 209, normalized size = 0.82 \[ \frac{\sqrt{c+d x} \left (\sqrt{d} \sqrt{a+b x} \left (5 a^2 b d^2 (29 c+14 d x)-105 a^3 d^3-a b^2 d \left (15 c^2+92 c d x+56 d^2 x^2\right )+b^3 \left (6 c^2 d x-9 c^3+72 c d^2 x^2+48 d^3 x^3\right )\right )+\frac{3 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{192 b^4 d^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 574, normalized size = 2.3 \begin{align*}{\frac{1}{384\,{b}^{4}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-180\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+54\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\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{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}{c}^{3}d+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}{c}^{4}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}-184\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}+12\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-210\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}+290\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d-18\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}{c}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\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 = 2.32311, size = 1226, normalized size = 4.83 \begin{align*} \left [\frac{3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, 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} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \,{\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{5} d^{3}}, -\frac{3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, 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} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \,{\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b^{5} d^{3}}\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 = 1.39028, size = 676, normalized size = 2.66 \begin{align*} \frac{\frac{8 \,{\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^{2}} + \frac{b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, 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 d^{2}}\right )} c{\left | b \right |}}{b^{2}} + \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^{3}} + \frac{b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac{5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, 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^{2} d^{3}}\right )} d{\left | b \right |}}{b^{2}}}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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