Optimal. Leaf size=126 \[ \frac{2 c^2 \sqrt{a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c \sqrt{a+b x} (2 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}} \]
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Rubi [A] time = 0.0888877, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {89, 78, 63, 217, 206} \[ \frac{2 c^2 \sqrt{a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c \sqrt{a+b x} (2 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx &=\frac{2 c^2 \sqrt{a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{2 \int \frac{\frac{1}{2} c (b c-3 a d)-\frac{3}{2} d (b c-a d) x}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac{2 c^2 \sqrt{a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (2 b c-3 a d) \sqrt{a+b x}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{d^2}\\ &=\frac{2 c^2 \sqrt{a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (2 b c-3 a d) \sqrt{a+b x}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b d^2}\\ &=\frac{2 c^2 \sqrt{a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (2 b c-3 a d) \sqrt{a+b x}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b d^2}\\ &=\frac{2 c^2 \sqrt{a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (2 b c-3 a d) \sqrt{a+b x}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.309907, size = 148, normalized size = 1.17 \[ \frac{2 \left (-\frac{3 (b c-a d)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b^2 \sqrt{d}}-c^2 \sqrt{a+b x}+\frac{2 c \sqrt{a+b x} (c+d x) (2 b c-3 a d)}{b c-a d}\right )}{3 d^2 (c+d x)^{3/2} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 604, normalized size = 4.8 \begin{align*}{\frac{1}{3\, \left ( ad-bc \right ) ^{2}{d}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{4}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}{c}^{2}{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}c{d}^{3}-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 ) xab{c}^{2}{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{3}d+3\,\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}{c}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{3}d+3\,\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}^{2}{c}^{4}+12\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xac{d}^{2}-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xb{c}^{2}d+10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}a{c}^{2}d-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}b{c}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \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 [B] time = 4.81249, size = 1423, normalized size = 11.29 \begin{align*} \left [\frac{3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\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 (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \,{\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} +{\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}, -\frac{3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\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 (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \,{\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} +{\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}\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 \frac{x^{2}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49788, size = 288, normalized size = 2.29 \begin{align*} \frac{\sqrt{b x + a}{\left (\frac{2 \,{\left (2 \, b^{6} c^{2} d^{2} - 3 \, a b^{5} c d^{3}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 2 \, a^{2} b^{5} c d^{3}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{\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 )}{4 \, \sqrt{b d} b^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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