Optimal. Leaf size=147 \[ \frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 147, 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, 641, 217, 203} \[ \frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 641
Rule 819
Rubi steps
\begin {align*} \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^4 \left (5 d^3+6 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^2 \left (15 d^5+24 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^7+48 d^6 e x}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 142, normalized size = 0.97 \[ \frac {48 d^5-33 d^4 e x-87 d^3 e^2 x^2+52 d^2 e^3 x^3-15 d (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+38 d e^4 x^4-15 e^5 x^5}{15 e^7 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 263, normalized size = 1.79 \[ \frac {48 \, d e^{5} x^{5} - 48 \, d^{2} e^{4} x^{4} - 96 \, d^{3} e^{3} x^{3} + 96 \, d^{4} e^{2} x^{2} + 48 \, d^{5} e x - 48 \, d^{6} + 30 \, {\left (d e^{5} x^{5} - d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} + 2 \, d^{4} e^{2} x^{2} + d^{5} e x - d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{5} x^{5} - 38 \, d e^{4} x^{4} - 52 \, d^{2} e^{3} x^{3} + 87 \, d^{3} e^{2} x^{2} + 33 \, d^{4} e x - 48 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{12} x^{5} - d e^{11} x^{4} - 2 \, d^{2} e^{10} x^{3} + 2 \, d^{3} e^{9} x^{2} + d^{4} e^{8} x - d^{5} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 109, normalized size = 0.74 \[ -d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-7\right )} \mathrm {sgn}\relax (d) - \frac {{\left (48 \, d^{6} e^{\left (-7\right )} + {\left (15 \, d^{5} e^{\left (-6\right )} - {\left (120 \, d^{4} e^{\left (-5\right )} + {\left (35 \, d^{3} e^{\left (-4\right )} - {\left (90 \, d^{2} e^{\left (-3\right )} - {\left (15 \, x e^{\left (-1\right )} - 23 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 195, normalized size = 1.33 \[ -\frac {x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {d \,x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {6 d^{2} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {8 d^{4} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}-\frac {d \,x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}+\frac {16 d^{6}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7}}+\frac {d x}{\sqrt {-e^{2} x^{2}+d^{2}}\, e^{6}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 278, normalized size = 1.89 \[ \frac {1}{15} \, d x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{3 \, e^{2}} + \frac {6 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {8 \, d^{4} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}} + \frac {16 \, d^{6}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}} + \frac {4 \, d^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}} - \frac {7 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}} - \frac {d \arcsin \left (\frac {e x}{d}\right )}{e^{7}} \]
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 \[ \int \frac {x^6\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 62.08, size = 1821, normalized size = 12.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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