Optimal. Leaf size=97 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]
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Rubi [A] time = 0.0613475, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1159, 388, 217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]
Antiderivative was successfully verified.
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Rule 1159
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{\sqrt{d+e x^2}} \, dx &=\frac{c x^3 \sqrt{d+e x^2}}{4 e}+\frac{\int \frac{4 a e-(3 c d-4 b e) x^2}{\sqrt{d+e x^2}} \, dx}{4 e}\\ &=-\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e}-\frac{1}{8} \left (-8 a-\frac{d (3 c d-4 b e)}{e^2}\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=-\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e}-\frac{1}{8} \left (-8 a-\frac{d (3 c d-4 b e)}{e^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=-\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e}+\frac{\left (3 c d^2-4 b d e+8 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0629485, size = 82, normalized size = 0.85 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )+\sqrt{e} x \sqrt{d+e x^2} \left (4 b e-3 c d+2 c e x^2\right )}{8 e^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 122, normalized size = 1.3 \begin{align*}{\frac{c{x}^{3}}{4\,e}\sqrt{e{x}^{2}+d}}-{\frac{3\,cdx}{8\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}}+{\frac{bx}{2\,e}\sqrt{e{x}^{2}+d}}-{\frac{bd}{2}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{a\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 = 4.69738, size = 408, normalized size = 4.21 \begin{align*} \left [\frac{{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \sqrt{e} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) + 2 \,{\left (2 \, c e^{2} x^{3} -{\left (3 \, c d e - 4 \, b e^{2}\right )} x\right )} \sqrt{e x^{2} + d}}{16 \, e^{3}}, -\frac{{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (2 \, c e^{2} x^{3} -{\left (3 \, c d e - 4 \, b e^{2}\right )} x\right )} \sqrt{e x^{2} + d}}{8 \, e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.37356, size = 230, normalized size = 2.37 \begin{align*} a \left (\begin{cases} \frac{\sqrt{- \frac{d}{e}} \operatorname{asin}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e < 0 \\\frac{\sqrt{\frac{d}{e}} \operatorname{asinh}{\left (x \sqrt{\frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e > 0 \\\frac{\sqrt{- \frac{d}{e}} \operatorname{acosh}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{- d}} & \text{for}\: e > 0 \wedge d < 0 \end{cases}\right ) + \frac{b \sqrt{d} x \sqrt{1 + \frac{e x^{2}}{d}}}{2 e} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{3}{2}}} - \frac{3 c d^{\frac{3}{2}} x}{8 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c \sqrt{d} x^{3}}{8 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 e^{\frac{5}{2}}} + \frac{c x^{5}}{4 \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 = 1.17638, size = 107, normalized size = 1.1 \begin{align*} -\frac{1}{8} \,{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a 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}{8} \,{\left (2 \, c x^{2} e^{\left (-1\right )} -{\left (3 \, c d e - 4 \, b e^{2}\right )} e^{\left (-3\right )}\right )} \sqrt{x^{2} e + d} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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