Optimal. Leaf size=97 \[ \frac {d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d+2 e x}{5 d e^4 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1649, 651}
\begin {gather*} \frac {d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d+2 e x}{5 d e^4 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 1649
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x) \left (\frac {2 d^3}{e^3}+\frac {5 d^2 x}{e^2}+\frac {5 d x^2}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {\frac {6 d^3}{e^3}+\frac {15 d^2 x}{e^2}}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d+2 e x}{5 d e^4 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 70, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^3-4 d^2 e x+d e^2 x^2+2 e^3 x^3\right )}{5 d e^4 (d-e x)^3 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(260\) vs.
\(2(85)=170\).
time = 0.07, size = 261, normalized size = 2.69
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (e x +d \right )^{3} \left (2 e^{3} x^{3}+d \,e^{2} x^{2}-4 d^{2} e x +2 d^{3}\right )}{5 e^{4} d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(65\) |
trager | \(\frac {\left (2 e^{3} x^{3}+d \,e^{2} x^{2}-4 d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 e^{4} d \left (-e x +d \right )^{3} \left (e x +d \right )}\) | \(67\) |
default | \(e^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+2 e d \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 142, normalized size = 1.46 \begin {gather*} \frac {d x^{3} e^{\left (-1\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {d^{2} x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {3 \, d^{3} x e^{\left (-3\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{4} e^{\left (-4\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {x^{4}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x e^{\left (-3\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x e^{\left (-3\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 109, normalized size = 1.12 \begin {gather*} \frac {2 \, x^{4} e^{4} - 4 \, d x^{3} e^{3} + 4 \, d^{3} x e - 2 \, d^{4} - {\left (2 \, x^{3} e^{3} + d x^{2} e^{2} - 4 \, d^{2} x e + 2 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d x^{4} e^{8} - 2 \, d^{2} x^{3} e^{7} + 2 \, d^{4} x e^{5} - d^{5} e^{4}\right )}} \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^{3} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.89, size = 66, normalized size = 0.68 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^3-4\,d^2\,e\,x+d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d\,e^4\,\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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