Optimal. Leaf size=93 \[ \frac {d (d+e x)^3}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 (d+e x)^2}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 (d+e x)}{15 d e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1635, 789, 637} \[ \frac {d (d+e x)^3}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 (d+e x)^2}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 (d+e x)}{15 d e^3 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 637
Rule 789
Rule 1635
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {d (d+e x)^3}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\left (\frac {3 d^2}{e^2}+\frac {5 d x}{e}\right ) (d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d (d+e x)^3}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 (d+e x)^2}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 e^2}\\ &=\frac {d (d+e x)^3}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 (d+e x)^2}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 (d+e x)}{15 d e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 58, normalized size = 0.62 \[ \frac {(d+e x) \left (2 d^2-6 d e x+7 e^2 x^2\right )}{15 d e^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 106, normalized size = 1.14 \[ \frac {2 \, e^{3} x^{3} - 6 \, d e^{2} x^{2} + 6 \, d^{2} e x - 2 \, d^{3} - {\left (7 \, e^{2} x^{2} - 6 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{6} x^{3} - 3 \, d^{2} e^{5} x^{2} + 3 \, d^{3} e^{4} x - d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 72, normalized size = 0.77 \[ -\frac {{\left (2 \, d^{4} e^{\left (-3\right )} - {\left (5 \, d^{2} e^{\left (-1\right )} - {\left (x {\left (\frac {7 \, x e^{2}}{d} + 15 \, e\right )} + 5 \, d\right )} x\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.59 \[ \frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (7 e^{2} x^{2}-6 d e x +2 d^{2}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 154, normalized size = 1.66 \[ \frac {e x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {d^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {7 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {2 \, d^{4}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} + \frac {7 \, d x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} + \frac {7 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 49, normalized size = 0.53 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2-6\,d\,e\,x+7\,e^2\,x^2\right )}{15\,d\,e^3\,{\left (d-e\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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