Optimal. Leaf size=152 \[ -\frac{\sqrt{d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{c+d x}}-\frac{c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}+\frac{\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^5}+\frac{b x^3}{2 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]
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Rubi [A] time = 0.11468, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {460, 89, 12, 78, 63, 217, 206} \[ -\frac{\sqrt{d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{c+d x}}-\frac{c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}+\frac{\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^5}+\frac{b x^3}{2 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 460
Rule 89
Rule 12
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{1}{2} \left (-2 a-\frac{3 b c^2}{d^2}\right ) \int \frac{x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (-2 a-\frac{3 b c^2}{d^2}\right ) \int \frac{c d^2 x}{\sqrt{-c+d x} (c+d x)^{3/2}} \, dx}{2 c d^3}\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{\left (3 b c^2+2 a d^2\right ) \int \frac{x}{\sqrt{-c+d x} (c+d x)^{3/2}} \, dx}{2 d^3}\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (3 b c^2+2 a d^2\right ) \sqrt{-c+d x}}{2 d^5 \sqrt{c+d x}}+\frac{\left (3 b c^2+2 a d^2\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{2 d^4}\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (3 b c^2+2 a d^2\right ) \sqrt{-c+d x}}{2 d^5 \sqrt{c+d x}}+\frac{\left (3 b c^2+2 a d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )}{d^5}\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (3 b c^2+2 a d^2\right ) \sqrt{-c+d x}}{2 d^5 \sqrt{c+d x}}+\frac{\left (3 b c^2+2 a d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{d^5}\\ &=-\frac{c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^3}{2 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (3 b c^2+2 a d^2\right ) \sqrt{-c+d x}}{2 d^5 \sqrt{c+d x}}+\frac{\left (3 b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{d^5}\\ \end{align*}
Mathematica [A] time = 0.107926, size = 90, normalized size = 0.59 \[ \frac{c \sqrt{1-\frac{d^2 x^2}{c^2}} \left (2 a d^2+3 b c^2\right ) \sin ^{-1}\left (\frac{d x}{c}\right )-2 a d^3 x-3 b c^2 d x+b d^3 x^3}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 254, normalized size = 1.7 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{5}} \left ({\it csgn} \left ( d \right ){x}^{3}b{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+2\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}a{d}^{4}+3\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}b{c}^{2}{d}^{2}-2\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xa-3\,{\it csgn} \left ( d \right ) d\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xb{c}^{2}-2\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) a{c}^{2}{d}^{2}-3\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) b{c}^{4} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955846, size = 211, normalized size = 1.39 \begin{align*} \frac{b x^{3}}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{3 \, b c^{2} x}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} - \frac{a x}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{3 \, b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{\sqrt{d^{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54815, size = 325, normalized size = 2.14 \begin{align*} \frac{2 \, b c^{4} + 2 \, a c^{2} d^{2} - 2 \,{\left (b c^{2} d^{2} + a d^{4}\right )} x^{2} +{\left (b d^{3} x^{3} -{\left (3 \, b c^{2} d + 2 \, a d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} +{\left (3 \, b c^{4} + 2 \, a c^{2} d^{2} -{\left (3 \, b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{2 \,{\left (d^{7} x^{2} - c^{2} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 66.7475, size = 212, normalized size = 1.39 \begin{align*} a \left (\frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, \frac{1}{2}, 1, 1 \\- \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 1 & \\- \frac{3}{4}, - \frac{1}{4} & - \frac{3}{2}, -1, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}}\right ) + b \left (\frac{c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, - \frac{1}{2}, 0, 1 \\- \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}} + \frac{i c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, 1 & \\- \frac{7}{4}, - \frac{5}{4} & - \frac{5}{2}, -2, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28983, size = 208, normalized size = 1.37 \begin{align*} -\frac{{\left ({\left (3 \, b d^{15} - \frac{{\left (d x + c\right )} b d^{15}}{c}\right )}{\left (d x + c\right )} - \frac{b c^{2} d^{15} - a d^{17}}{c}\right )} \sqrt{d x + c}}{768 \, \sqrt{d x - c}} - \frac{{\left (3 \, b c^{2} d^{15} + 2 \, a d^{17}\right )} \log \left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}{768 \, c} - \frac{2 \,{\left (b c^{3} + a c d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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