Optimal. Leaf size=161 \[ \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {819, 780, 217, 203} \[ \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 819
Rubi steps
\begin {align*} \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3+7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5+35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7+105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 155, normalized size = 0.96 \[ \frac {96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-105 d^2 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-15 d e^5 x^5-15 e^6 x^6}{30 e^8 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 278, normalized size = 1.73 \[ \frac {96 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} + 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x - 96 \, d^{7} + 210 \, {\left (d^{2} e^{5} x^{5} - d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} + d^{6} e x - d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{6} x^{6} + 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} - 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} - d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} + 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x - d^{5} e^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 120, normalized size = 0.75 \[ -\frac {7}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-8\right )} \mathrm {sgn}\relax (d) - \frac {{\left (96 \, d^{7} e^{\left (-8\right )} + {\left (105 \, d^{6} e^{\left (-7\right )} - {\left (240 \, d^{5} e^{\left (-6\right )} + {\left (245 \, d^{4} e^{\left (-5\right )} - {\left (180 \, d^{3} e^{\left (-4\right )} + {\left (161 \, d^{2} e^{\left (-3\right )} - 15 \, {\left (x e^{\left (-1\right )} + 2 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 227, normalized size = 1.41 \[ -\frac {x^{7}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}-\frac {d \,x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {7 d^{2} x^{5}}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}+\frac {6 d^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {8 d^{5} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}-\frac {7 d^{2} x^{3}}{6 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}+\frac {16 d^{7}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8}}+\frac {7 d^{2} x}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}-\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 312, normalized size = 1.94 \[ -\frac {x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {7 \, d^{2} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )}}{30 \, e} - \frac {d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {7 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e^{3}} + \frac {6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {16 \, d^{7}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}} + \frac {14 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {49 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} - \frac {7 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{8}} \]
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 \[ \int \frac {x^7\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 66.42, size = 2004, normalized size = 12.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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