Optimal. Leaf size=161 \[ -\frac{x^3 \left (4 a d^2+5 b c^2\right )}{4 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{3 x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+5 b c^2\right )}{8 d^6}+\frac{3 c^2 \left (4 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^7}+\frac{b x^5}{4 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]
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Rubi [A] time = 0.122936, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {460, 98, 21, 90, 12, 63, 217, 206} \[ -\frac{x^3 \left (4 a d^2+5 b c^2\right )}{4 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{3 x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+5 b c^2\right )}{8 d^6}+\frac{3 c^2 \left (4 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^7}+\frac{b x^5}{4 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 460
Rule 98
Rule 21
Rule 90
Rule 12
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{1}{4} \left (-4 a-\frac{5 b c^2}{d^2}\right ) \int \frac{x^4}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (4 a+\frac{5 b c^2}{d^2}\right ) \int \frac{x^2 \left (-3 c^2-3 c d x\right )}{\sqrt{-c+d x} (c+d x)^{3/2}} \, dx}{4 c d^2}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{\left (3 \left (5 b c^2+4 a d^2\right )\right ) \int \frac{x^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{4 d^4}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{3 \left (5 b c^2+4 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{8 d^6}+\frac{\left (3 \left (5 b c^2+4 a d^2\right )\right ) \int \frac{c^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{8 d^6}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{3 \left (5 b c^2+4 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{8 d^6}+\frac{\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{8 d^6}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{3 \left (5 b c^2+4 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{8 d^6}+\frac{\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )}{4 d^7}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{3 \left (5 b c^2+4 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{8 d^6}+\frac{\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{4 d^7}\\ &=-\frac{\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{-c+d x} \sqrt{c+d x}}+\frac{3 \left (5 b c^2+4 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{8 d^6}+\frac{3 c^2 \left (5 b c^2+4 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{4 d^7}\\ \end{align*}
Mathematica [A] time = 0.141607, size = 119, normalized size = 0.74 \[ \frac{3 c^3 \sqrt{1-\frac{d^2 x^2}{c^2}} \left (4 a d^2+5 b c^2\right ) \sin ^{-1}\left (\frac{d x}{c}\right )+4 a d^3 x \left (d^2 x^2-3 c^2\right )+b d x \left (5 c^2 d^2 x^2-15 c^4+2 d^4 x^4\right )}{8 d^7 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.03, size = 316, normalized size = 2. \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{8\,{d}^{7}} \left ( 2\,{\it csgn} \left ( d \right ){x}^{5}b{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+4\,{\it csgn} \left ( d \right ){x}^{3}a{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+5\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{2}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+12\,\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{c}^{2}{d}^{4}+15\,\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}^{4}{d}^{2}-12\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xa{c}^{2}-15\,{\it csgn} \left ( d \right ) d\sqrt{{d}^{2}{x}^{2}-{c}^{2}}xb{c}^{4}-12\,\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}^{4}{d}^{2}-15\,\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}^{6} \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 = 0.951779, size = 289, normalized size = 1.8 \begin{align*} \frac{b x^{5}}{4 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{5 \, b c^{2} x^{3}}{8 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} + \frac{a x^{3}}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{15 \, b c^{4} x}{8 \, \sqrt{d^{2} x^{2} - c^{2}} d^{6}} - \frac{3 \, a c^{2} x}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} + \frac{15 \, b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{8 \, \sqrt{d^{2}} d^{6}} + \frac{3 \, a 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51211, size = 390, normalized size = 2.42 \begin{align*} \frac{8 \, b c^{6} + 8 \, a c^{4} d^{2} - 8 \,{\left (b c^{4} d^{2} + a c^{2} d^{4}\right )} x^{2} +{\left (2 \, b d^{5} x^{5} +{\left (5 \, b c^{2} d^{3} + 4 \, a d^{5}\right )} x^{3} - 3 \,{\left (5 \, b c^{4} d + 4 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} + 3 \,{\left (5 \, b c^{6} + 4 \, a c^{4} d^{2} -{\left (5 \, b c^{4} d^{2} + 4 \, a c^{2} d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{8 \,{\left (d^{9} x^{2} - c^{2} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 99.7226, size = 233, normalized size = 1.45 \begin{align*} a \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 ) + b \left (\frac{c^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, - \frac{3}{2}, -1, 1 \\- \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, -1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{7}} + \frac{i c^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, 1 & \\- \frac{11}{4}, - \frac{9}{4} & - \frac{7}{2}, -3, -2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{7}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32636, size = 290, normalized size = 1.8 \begin{align*} -\frac{1}{688128} \,{\left (5 \, b c^{3} d^{35} + 4 \, a c d^{37}\right )} \log \left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right ) - \frac{{\left ({\left ({\left (2 \,{\left (5 \, b d^{35} - \frac{{\left (d x + c\right )} b d^{35}}{c}\right )}{\left (d x + c\right )} - \frac{25 \, b c^{2} d^{35} + 4 \, a d^{37}}{c}\right )}{\left (d x + c\right )} + \frac{35 \, b c^{3} d^{35} + 12 \, a c d^{37}}{c}\right )}{\left (d x + c\right )} - \frac{2 \,{\left (7 \, b c^{4} d^{35} + 2 \, a c^{2} d^{37}\right )}}{c}\right )} \sqrt{d x + c}}{2064384 \, \sqrt{d x - c}} - \frac{2 \,{\left (b c^{5} + a c^{3} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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