Optimal. Leaf size=89 \[ \frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {789, 639, 191} \begin {gather*} \frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 639
Rule 789
Rubi steps
\begin {align*} \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.70 \begin {gather*} \frac {d^3-2 d^2 e x+8 d e^2 x^2-4 e^3 x^3}{15 d^3 e^2 (d-e x)^2 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 69, normalized size = 0.78 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^3-2 d^2 e x+8 d e^2 x^2-4 e^3 x^3\right )}{15 d^3 e^2 (d-e x)^3 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 117, normalized size = 1.31 \begin {gather*} \frac {e^{4} x^{4} - 2 \, d e^{3} x^{3} + 2 \, d^{3} e x - d^{4} + {\left (4 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 2 \, d^{2} e x - d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{6} x^{4} - 2 \, d^{4} e^{5} x^{3} + 2 \, d^{6} e^{3} x - d^{7} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 64, normalized size = 0.72 \begin {gather*} \frac {{\left ({\left (2 \, x {\left (\frac {2 \, x^{2} e^{3}}{d^{3}} - \frac {5 \, e}{d}\right )} - 5\right )} x^{2} - d^{2} e^{\left (-2\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 0.72 \begin {gather*} \frac {\left (-e x +d \right ) \left (e x +d \right )^{3} \left (-4 e^{3} x^{3}+8 d \,e^{2} x^{2}-2 d^{2} e x +d^{3}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 109, normalized size = 1.22 \begin {gather*} \frac {x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e} - \frac {4 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.86, size = 65, normalized size = 0.73 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^3-2\,d^2\,e\,x+8\,d\,e^2\,x^2-4\,e^3\,x^3\right )}{15\,d^3\,e^2\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \end {gather*}
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 \begin {gather*} \int \frac {x \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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