Optimal. Leaf size=84 \[ \frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}-\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 84, 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 = {805, 266, 43} \[ \frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}-\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 805
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {x^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (d^2-e^2 x\right )^{5/2}} \, dx,x,x^2\right )}{5 e}\\ &=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {d^2}{e^2 \left (d^2-e^2 x\right )^{5/2}}-\frac {1}{e^2 \left (d^2-e^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{5 e}\\ &=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.98 \[ \frac {8 d^4-8 d^3 e x-12 d^2 e^2 x^2+12 d e^3 x^3+3 e^4 x^4}{15 d e^5 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 171, normalized size = 2.04 \[ \frac {8 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 8 \, d^{5} - {\left (3 \, e^{4} x^{4} + 12 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} - 8 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{10} x^{5} - d^{2} e^{9} x^{4} - 2 \, d^{3} e^{8} x^{3} + 2 \, d^{4} e^{7} x^{2} + d^{5} e^{6} x - d^{6} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 64, normalized size = 0.76 \[ -\frac {{\left (8 \, d^{4} e^{\left (-5\right )} + {\left (3 \, x^{2} {\left (\frac {x}{d} + 5 \, e^{\left (-1\right )}\right )} - 20 \, d^{2} e^{\left (-3\right )}\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 77, normalized size = 0.92 \[ \frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (3 x^{4} e^{4}+12 x^{3} d \,e^{3}-12 d^{2} x^{2} e^{2}-8 d^{3} x e +8 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 159, normalized size = 1.89 \[ \frac {x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, d^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {3 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}} + \frac {d x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} + \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.70, size = 78, normalized size = 0.93 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (8\,d^4-8\,d^3\,e\,x-12\,d^2\,e^2\,x^2+12\,d\,e^3\,x^3+3\,e^4\,x^4\right )}{15\,d\,e^5\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 63.87, size = 418, normalized size = 4.98 \[ d \left (\begin {cases} - \frac {i x^{5}}{5 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {x^{5}}{5 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {8 d^{4}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {20 d^{2} e^{2} x^{2}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} + \frac {15 e^{4} x^{4}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 \left (d^{2}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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