Optimal. Leaf size=99 \[ \frac {d^2 \sqrt {d+e x^2}}{5 (-e)^{5/2}}+\frac {\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}-\frac {2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5151, 266, 43} \[ \frac {d^2 \sqrt {d+e x^2}}{5 (-e)^{5/2}}+\frac {\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}-\frac {2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5151
Rubi steps
\begin {align*} \int x^4 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{5} \sqrt {-e} \int \frac {x^5}{\sqrt {d+e x^2}} \, dx\\ &=\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{10} \sqrt {-e} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{10} \sqrt {-e} \operatorname {Subst}\left (\int \left (\frac {d^2}{e^2 \sqrt {d+e x}}-\frac {2 d \sqrt {d+e x}}{e^2}+\frac {(d+e x)^{3/2}}{e^2}\right ) \, dx,x,x^2\right )\\ &=\frac {d^2 \sqrt {d+e x^2}}{5 (-e)^{5/2}}-\frac {2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac {\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}+\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 72, normalized size = 0.73 \[ \frac {\sqrt {d+e x^2} \left (8 d^2-4 d e x^2+3 e^2 x^4\right )}{75 (-e)^{5/2}}+\frac {1}{5} x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 68, normalized size = 0.69 \[ \frac {15 \, e^{3} x^{5} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (3 \, e^{2} x^{4} - 4 \, d e x^{2} + 8 \, d^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{75 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 102, normalized size = 1.03 \[ \frac {1}{5} \, x^{5} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) - \frac {1}{5} \, \sqrt {-x^{2} e^{2} - d e} d^{2} e^{\left (-3\right )} - \frac {1}{75} \, {\left (10 \, {\left (-x^{2} e^{2} - d e\right )}^{\frac {3}{2}} d e + 3 \, {\left (x^{2} e^{2} + d e\right )}^{2} \sqrt {-x^{2} e^{2} - d e}\right )} e^{\left (-5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 180, normalized size = 1.82 \[ \frac {x^{5} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{5}+\frac {\sqrt {-e}\, x^{6} \sqrt {e \,x^{2}+d}}{35 d}-\frac {6 \sqrt {-e}\, x^{4} \sqrt {e \,x^{2}+d}}{175 e}+\frac {8 \sqrt {-e}\, d \,x^{2} \sqrt {e \,x^{2}+d}}{175 e^{2}}-\frac {16 \sqrt {-e}\, d^{2} \sqrt {e \,x^{2}+d}}{175 e^{3}}-\frac {\sqrt {-e}\, x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{35 d e}+\frac {4 \sqrt {-e}\, x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{175 e^{2}}-\frac {8 \sqrt {-e}\, d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{525 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 139, normalized size = 1.40 \[ \frac {1}{5} \, x^{5} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (15 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} - 42 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2}\right )} \sqrt {-e}}{525 \, d e^{3}} + \frac {{\left (5 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x^{2} + d} d^{3}\right )} \sqrt {-e}}{175 \, d e^{3}} \]
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 \[ \int x^4\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.56, size = 97, normalized size = 0.98 \[ \begin {cases} - \frac {8 i d^{2} \sqrt {d + e x^{2}}}{75 e^{\frac {5}{2}}} + \frac {4 i d x^{2} \sqrt {d + e x^{2}}}{75 e^{\frac {3}{2}}} + \frac {i x^{5} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{5} - \frac {i x^{4} \sqrt {d + e x^{2}}}{25 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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