Optimal. Leaf size=127 \[ \frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^3}-\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}} \]
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Rubi [A] time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6221, 321, 217, 206} \[ -\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^3}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 6221
Rubi steps
\begin {align*} \int x^5 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \int \frac {x^6}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {(5 d) \int \frac {x^4}{\sqrt {d+e x^2}} \, dx}{36 \sqrt {e}}\\ &=\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (5 d^2\right ) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx}{48 e^{3/2}}\\ &=-\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {\left (5 d^3\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{96 e^{5/2}}\\ &=-\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {\left (5 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{96 e^{5/2}}\\ &=-\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^3}+\frac {1}{6} x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 99, normalized size = 0.78 \[ \frac {15 d^3 \log \left (\sqrt {d+e x^2}+\sqrt {e} x\right )+\sqrt {e} x \sqrt {d+e x^2} \left (-15 d^2+10 d e x^2-8 e^2 x^4\right )+48 e^3 x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{288 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 86, normalized size = 0.68 \[ -\frac {2 \, {\left (8 \, e^{2} x^{5} - 10 \, d e x^{3} + 15 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {e} - 3 \, {\left (16 \, e^{3} x^{6} + 5 \, d^{3}\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{576 \, e^{3}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 172, normalized size = 1.35 \[ \frac {x^{6} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {\sqrt {e}\, x^{7} \sqrt {e \,x^{2}+d}}{48 d}-\frac {7 x^{5} \sqrt {e \,x^{2}+d}}{288 \sqrt {e}}+\frac {35 d \,x^{3} \sqrt {e \,x^{2}+d}}{1152 e^{\frac {3}{2}}}-\frac {5 d^{2} x \sqrt {e \,x^{2}+d}}{128 e^{\frac {5}{2}}}+\frac {5 d^{3} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{96 e^{3}}-\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{48 \sqrt {e}\, d}+\frac {5 x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{288 e^{\frac {3}{2}}}-\frac {5 d x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{384 e^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{12} \, x^{6} \log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{12} \, x^{6} \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{2} \, d \sqrt {e} \int -\frac {\sqrt {e x^{2} + d} x^{6}}{3 \, {\left (e^{2} x^{4} + d e x^{2} - {\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} \]
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 \[ \int x^5\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.03, size = 121, normalized size = 0.95 \[ \begin {cases} \frac {5 d^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{96 e^{3}} - \frac {5 d^{2} x \sqrt {d + e x^{2}}}{96 e^{\frac {5}{2}}} + \frac {5 d x^{3} \sqrt {d + e x^{2}}}{144 e^{\frac {3}{2}}} + \frac {x^{6} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{6} - \frac {x^{5} \sqrt {d + e x^{2}}}{36 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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