Optimal. Leaf size=254 \[ \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d} \]
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Rubi [A] time = 0.229475, 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} \[ \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d} \]
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 (a+b x)^{3/2}}{\sqrt{c+d x}} \, dx &=\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{\int \frac{(a+b x)^{3/2} \left (-a c-\frac{1}{2} (7 b c+3 a d) x\right )}{\sqrt{c+d x}} \, dx}{4 b d}\\ &=-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{48 b^2 d^2}\\ &=\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^3}-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}-\frac{\left ((b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{64 b^2 d^3}\\ &=-\frac{(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^4}+\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^3}-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^2 d^4}\\ &=-\frac{(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^4}+\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^3}-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 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^3 d^4}\\ &=-\frac{(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^4}+\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^3}-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 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^3 d^4}\\ &=-\frac{(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^2 d^4}+\frac{\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{96 b^2 d^3}-\frac{(7 b c+3 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d}+\frac{(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.549862, size = 215, normalized size = 0.85 \[ \frac{3 (b c-a d)^{5/2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )-b \sqrt{d} \sqrt{a+b x} (c+d x) \left (3 a^2 b d^2 (5 c-2 d x)+9 a^3 d^3+a b^2 d \left (-145 c^2+92 c d x-72 d^2 x^2\right )+b^3 \left (-70 c^2 d x+105 c^3+56 c d^2 x^2-48 d^3 x^3\right )\right )}{192 b^3 d^{9/2} \sqrt{c+d x}} \]
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}^{2}{d}^{4}}\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}+144\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+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 ){a}^{4}{d}^{4}+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}^{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}-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{b}^{3}{c}^{3}d+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 ){b}^{4}{c}^{4}+12\,\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}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-18\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}+290\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d-210\,\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 = 3.31748, size = 1226, normalized size = 4.83 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, 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} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{3} d^{5}}, -\frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, 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} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b^{3} d^{5}}\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 [A] time = 1.32684, size = 393, normalized size = 1.55 \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^{3} d} - \frac{7 \, b^{7} c d^{5} + 9 \, a b^{6} d^{6}}{b^{9} d^{7}}\right )} + \frac{35 \, b^{8} c^{2} d^{4} + 10 \, a b^{7} c d^{5} + 3 \, a^{2} b^{6} d^{6}}{b^{9} d^{7}}\right )} - \frac{3 \,{\left (35 \, b^{9} c^{3} d^{3} - 25 \, a b^{8} c^{2} d^{4} - 7 \, a^{2} b^{7} c d^{5} - 3 \, a^{3} b^{6} d^{6}\right )}}{b^{9} d^{7}}\right )} \sqrt{b x + a} - \frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, 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^{4}}\right )} b}{192 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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