Optimal. Leaf size=175 \[ \frac{x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac{d x \sqrt{d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac{x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]
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Rubi [A] time = 0.121627, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1159, 388, 195, 217, 206} \[ \frac{x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac{d x \sqrt{d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac{x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]
Antiderivative was successfully verified.
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Rule 1159
Rule 388
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx &=\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac{\int \left (d+e x^2\right )^{3/2} \left (8 a e-(3 c d-8 b e) x^2\right ) \, dx}{8 e}\\ &=-\frac{(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}-\frac{1}{48} \left (-48 a-\frac{d (3 c d-8 b e)}{e^2}\right ) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=\frac{1}{192} \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac{(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac{1}{64} \left (d \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right )\right ) \int \sqrt{d+e x^2} \, dx\\ &=\frac{1}{128} d \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{192} \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac{(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac{1}{128} \left (d^2 \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{128} d \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{192} \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac{(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac{1}{128} \left (d^2 \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=\frac{1}{128} d \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{192} \left (48 a+\frac{d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac{(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac{d^2 \left (3 c d^2-8 b d e+48 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{128 e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.32968, size = 157, normalized size = 0.9 \[ \frac{\sqrt{d+e x^2} \left (\sqrt{e} x \left (8 e \left (6 a e \left (5 d+2 e x^2\right )+b \left (3 d^2+14 d e x^2+8 e^2 x^4\right )\right )+c \left (6 d^2 e x^2-9 d^3+72 d e^2 x^4+48 e^3 x^6\right )\right )+\frac{3 d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (8 e (6 a e-b d)+3 c d^2\right )}{\sqrt{\frac{e x^2}{d}+1}}\right )}{384 e^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 229, normalized size = 1.3 \begin{align*}{\frac{c{x}^{3}}{8\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{cdx}{16\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{c{d}^{2}x}{64\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,c{d}^{3}x}{128\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{4}}{128}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}}+{\frac{bx}{6\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{bdx}{24\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{b{d}^{2}x}{16\,e}\sqrt{e{x}^{2}+d}}-{\frac{{d}^{3}b}{16}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{ax}{4} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{e{x}^{2}+d}}+{\frac{3\,a{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}} \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 = 4.94813, size = 701, normalized size = 4.01 \begin{align*} \left [\frac{3 \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt{e} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) + 2 \,{\left (48 \, c e^{4} x^{7} + 8 \,{\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \,{\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \,{\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt{e x^{2} + d}}{768 \, e^{3}}, -\frac{3 \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (48 \, c e^{4} x^{7} + 8 \,{\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \,{\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \,{\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt{e x^{2} + d}}{384 \, e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 28.1383, size = 413, normalized size = 2.36 \begin{align*} \frac{a d^{\frac{3}{2}} x \sqrt{1 + \frac{e x^{2}}{d}}}{2} + \frac{a d^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 a \sqrt{d} e x^{3}}{8 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 \sqrt{e}} + \frac{a e^{2} x^{5}}{4 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{b d^{\frac{5}{2}} x}{16 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{17 b d^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{11 b \sqrt{d} e x^{5}}{24 \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{b d^{3} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 e^{\frac{3}{2}}} + \frac{b e^{2} x^{7}}{6 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 c d^{\frac{7}{2}} x}{128 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{5}{2}} x^{3}}{128 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{13 c d^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 c \sqrt{d} e x^{7}}{16 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{4} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 e^{\frac{5}{2}}} + \frac{c e^{2} x^{9}}{8 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1349, size = 196, normalized size = 1.12 \begin{align*} -\frac{1}{128} \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, c x^{2} e +{\left (9 \, c d e^{6} + 8 \, b e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} +{\left (3 \, c d^{2} e^{5} + 56 \, b d e^{6} + 48 \, a e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} - 3 \,{\left (3 \, c d^{3} e^{4} - 8 \, b d^{2} e^{5} - 80 \, a d e^{6}\right )} e^{\left (-6\right )}\right )} \sqrt{x^{2} e + d} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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