Optimal. Leaf size=148 \[ \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {796, 778, 192, 191} \begin {gather*} \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 778
Rule 796
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\int \frac {x \left (2 d^2 e-6 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d^2 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{21 d e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{105 d^3 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{315 d^5 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 126, normalized size = 0.85 \begin {gather*} \frac {-10 d^8+10 d^7 e x+35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+16 d e^7 x^7-16 e^8 x^8}{315 d^7 e^3 (d-e x)^4 (d+e x)^3 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 126, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-10 d^8+10 d^7 e x+35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+16 d e^7 x^7-16 e^8 x^8\right )}{315 d^7 e^3 (d-e x)^5 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 305, normalized size = 2.06 \begin {gather*} -\frac {10 \, e^{9} x^{9} - 10 \, d e^{8} x^{8} - 40 \, d^{2} e^{7} x^{7} + 40 \, d^{3} e^{6} x^{6} + 60 \, d^{4} e^{5} x^{5} - 60 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} + 40 \, d^{7} e^{2} x^{2} + 10 \, d^{8} e x - 10 \, d^{9} - {\left (16 \, e^{8} x^{8} - 16 \, d e^{7} x^{7} - 56 \, d^{2} e^{6} x^{6} + 56 \, d^{3} e^{5} x^{5} + 70 \, d^{4} e^{4} x^{4} - 70 \, d^{5} e^{3} x^{3} - 35 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 10 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{7} e^{12} x^{9} - d^{8} e^{11} x^{8} - 4 \, d^{9} e^{10} x^{7} + 4 \, d^{10} e^{9} x^{6} + 6 \, d^{11} e^{8} x^{5} - 6 \, d^{12} e^{7} x^{4} - 4 \, d^{13} e^{6} x^{3} + 4 \, d^{14} e^{5} x^{2} + d^{15} e^{4} x - d^{16} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 90, normalized size = 0.61 \begin {gather*} \frac {{\left ({\left ({\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, x^{2} e^{6}}{d^{7}} - \frac {9 \, e^{4}}{d^{5}}\right )} + \frac {63 \, e^{2}}{d^{3}}\right )} x^{2} - \frac {105}{d}\right )} x - 45 \, e^{\left (-1\right )}\right )} x^{2} + 10 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{315 \, {\left (x^{2} e^{2} - d^{2}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.82 \begin {gather*} -\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (16 e^{8} x^{8}-16 e^{7} x^{7} d -56 e^{6} x^{6} d^{2}+56 e^{5} x^{5} d^{3}+70 e^{4} x^{4} d^{4}-70 x^{3} d^{5} e^{3}-35 x^{2} d^{6} e^{2}-10 x \,d^{7} e +10 d^{8}\right )}{315 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}} d^{7} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 158, normalized size = 1.07 \begin {gather*} \frac {x^{2}}{7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e} + \frac {d x}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e^{2}} - \frac {2 \, d^{2}}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e^{3}} - \frac {x}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d e^{2}} - \frac {2 \, x}{105 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{2}} - \frac {8 \, x}{315 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e^{2}} - \frac {16 \, x}{315 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.74, size = 202, normalized size = 1.36 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}}{144\,d^3\,e^3\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{252\,e^3}-\frac {17\,x}{252\,d\,e^2}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {5}{144\,d^2\,e^3}+\frac {131\,x}{5040\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e^2\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^7\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 48.46, size = 1401, normalized size = 9.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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