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 = 0.08, 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 = {1649, 803, 651}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 803
Rule 1649
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 = 0.32, size = 53, normalized size = 0.57 \begin {gather*} \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 (d-e x)^3} \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. \(364\) vs.
\(2(81)=162\).
time = 0.06, size = 365, normalized size = 3.92
method | result | size |
trager | \(\frac {\left (7 e^{2} x^{2}-6 d e x +2 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d \,e^{3} \left (-e x +d \right )^{3}}\) | \(50\) |
gosper | \(\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (7 e^{2} x^{2}-6 d e x +2 d^{2}\right )}{15 d \,e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(55\) |
default | \(e^{3} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+3 e^{2} d \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+3 e \,d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+d^{3} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\) | \(365\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 143, normalized size = 1.54 \begin {gather*} \frac {x^{4} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {d^{2} x^{2} e^{\left (-1\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {7 \, d^{3} x e^{\left (-2\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{4} e^{\left (-3\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d x^{3}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d x e^{\left (-2\right )}}{30 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {7 \, x e^{\left (-2\right )}}{15 \, \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 = 3.63, size = 100, normalized size = 1.08 \begin {gather*} \frac {2 \, x^{3} e^{3} - 6 \, d x^{2} e^{2} + 6 \, d^{2} x e - 2 \, d^{3} - {\left (7 \, x^{2} e^{2} - 6 \, d x e + 2 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d x^{3} e^{6} - 3 \, d^{2} x^{2} e^{5} + 3 \, d^{3} x e^{4} - d^{4} e^{3}\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^{2} \left (d + e x\right )^{3}}{\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 [A]
time = 0.69, size = 98, normalized size = 1.05 \begin {gather*} -\frac {4 \, {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} - 1\right )} e^{\left (-3\right )}}{15 \, d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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