Optimal. Leaf size=164 \[ \frac{x^3 \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac{c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{8 d^7}+\frac{b x^5 \sqrt{d x-c} \sqrt{c+d x}}{6 d^2} \]
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Rubi [A] time = 0.119734, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {460, 100, 12, 90, 63, 217, 206} \[ \frac{x^3 \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac{c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{8 d^7}+\frac{b x^5 \sqrt{d x-c} \sqrt{c+d x}}{6 d^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 100
Rule 12
Rule 90
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2\right )}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx &=\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}-\frac{1}{6} \left (-6 a-\frac{5 b c^2}{d^2}\right ) \int \frac{x^4}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (5 b c^2+6 a d^2\right ) \int \frac{3 c^2 x^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{24 d^4}\\ &=\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (c^2 \left (5 b c^2+6 a d^2\right )\right ) \int \frac{x^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{8 d^4}\\ &=\frac{c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{16 d^6}+\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (c^2 \left (5 b c^2+6 a d^2\right )\right ) \int \frac{c^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{16 d^6}\\ &=\frac{c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{16 d^6}+\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (c^4 \left (5 b c^2+6 a d^2\right )\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx}{16 d^6}\\ &=\frac{c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{16 d^6}+\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (c^4 \left (5 b c^2+6 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )}{8 d^7}\\ &=\frac{c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{16 d^6}+\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{\left (c^4 \left (5 b c^2+6 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 )}{8 d^7}\\ &=\frac{c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt{-c+d x} \sqrt{c+d x}}{16 d^6}+\frac{\left (5 b c^2+6 a d^2\right ) x^3 \sqrt{-c+d x} \sqrt{c+d x}}{24 d^4}+\frac{b x^5 \sqrt{-c+d x} \sqrt{c+d x}}{6 d^2}+\frac{c^4 \left (5 b c^2+6 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{8 d^7}\\ \end{align*}
Mathematica [A] time = 0.115963, size = 148, normalized size = 0.9 \[ \frac{d x \left (d^2 x^2-c^2\right ) \left (6 a d^2 \left (3 c^2+2 d^2 x^2\right )+b \left (10 c^2 d^2 x^2+15 c^4+8 d^4 x^4\right )\right )+3 c^4 \sqrt{d^2 x^2-c^2} \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-c^2}}\right )}{48 d^7 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.028, size = 240, normalized size = 1.5 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{48\,{d}^{7}}\sqrt{dx-c}\sqrt{dx+c} \left ( 8\,{\it csgn} \left ( d \right ){x}^{5}b{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+12\,{\it csgn} \left ( d \right ){x}^{3}a{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+10\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{2}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+18\,{\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}+18\,\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}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960679, size = 289, normalized size = 1.76 \begin{align*} \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{5}}{6 \, d^{2}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2} x^{3}}{24 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a x^{3}}{4 \, d^{2}} + \frac{5 \, b c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{16 \, \sqrt{d^{2}} d^{6}} + \frac{3 \, a 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^{4}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{4} x}{16 \, d^{6}} + \frac{3 \, \sqrt{d^{2} x^{2} - c^{2}} a c^{2} x}{8 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58638, size = 251, normalized size = 1.53 \begin{align*} \frac{{\left (8 \, b d^{5} x^{5} + 2 \,{\left (5 \, b c^{2} d^{3} + 6 \, a d^{5}\right )} x^{3} + 3 \,{\left (5 \, b c^{4} d + 6 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} - 3 \,{\left (5 \, b c^{6} + 6 \, a c^{4} d^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{48 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 87.1952, size = 240, normalized size = 1.46 \begin{align*} \frac{a c^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{5}} - \frac{i a c^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{5}} + \frac{b c^{6}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{11}{4}, - \frac{9}{4} & - \frac{5}{2}, - \frac{5}{2}, -2, 1 \\-3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{7}} - \frac{i b c^{6}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, - \frac{13}{4}, -3, - \frac{11}{4}, - \frac{5}{2}, 1 & \\- \frac{13}{4}, - \frac{11}{4} & - \frac{7}{2}, -3, -3, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27013, size = 246, normalized size = 1.5 \begin{align*} -\frac{{\left (33 \, b c^{5} d^{36} + 30 \, a c^{3} d^{38} -{\left (85 \, b c^{4} d^{36} + 54 \, a c^{2} d^{38} - 2 \,{\left (55 \, b c^{3} d^{36} + 18 \, a c d^{38} -{\left (45 \, b c^{2} d^{36} + 6 \, a d^{38} + 4 \,{\left ({\left (d x + c\right )} b d^{36} - 5 \, b c d^{36}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c} + 6 \,{\left (5 \, b c^{6} d^{36} + 6 \, a c^{4} d^{38}\right )} \log \left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{34603008 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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