Optimal. Leaf size=68 \[ -\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {d \sqrt {d+e x^2}}{3 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6221, 266, 43} \[ -\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {d \sqrt {d+e x^2}}{3 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6221
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{3} \sqrt {e} \int \frac {x^3}{\sqrt {d+e x^2}} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \operatorname {Subst}\left (\int \frac {x}{\sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \operatorname {Subst}\left (\int \left (-\frac {d}{e \sqrt {d+e x}}+\frac {\sqrt {d+e x}}{e}\right ) \, dx,x,x^2\right )\\ &=\frac {d \sqrt {d+e x^2}}{3 e^{3/2}}-\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 0.82 \[ \frac {1}{9} \left (\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2}}{e^{3/2}}+3 x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 65, normalized size = 0.96 \[ \frac {3 \, e^{2} x^{3} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) - 2 \, \sqrt {e x^{2} + d} {\left (e x^{2} - 2 \, d\right )} \sqrt {e}}{18 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 128, normalized size = 1.88 \[ \frac {x^{3} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{3}+\frac {e^{\frac {3}{2}} \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{3 d}-\frac {\sqrt {e}\, \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 99, normalized size = 1.46 \[ \frac {1}{3} \, x^{3} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d}{45 \, d e^{\frac {3}{2}}} + \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x^{2} + d} d^{2}}{45 \, d e^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 65, normalized size = 0.96 \[ \begin {cases} \frac {2 d \sqrt {d + e x^{2}}}{9 e^{\frac {3}{2}}} + \frac {x^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{3} - \frac {x^{2} \sqrt {d + e x^{2}}}{9 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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