Optimal. Leaf size=89 \[ \frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 833, 792,
223, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}+\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 792
Rule 833
Rule 864
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^3}\\ &=\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {\text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ &=\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 102, normalized size = 1.15 \begin {gather*} \frac {\left (2 d^2-d e x-4 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 e^4 (-d+e x) (d+e x)^2}+\frac {\log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^3 \sqrt {-e^2}} \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. \(203\) vs.
\(2(79)=158\).
time = 0.07, size = 204, normalized size = 2.29
method | result | size |
default | \(\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e}-\frac {d}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {x}{e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d^{3} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{4}}\) | \(204\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 91, normalized size = 1.02 \begin {gather*} -\arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} + \frac {4 \, x e^{\left (-3\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}}} - \frac {d e^{\left (-4\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} + \frac {d^{2}}{3 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x e^{5} + \sqrt {-x^{2} e^{2} + d^{2}} d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.63, size = 148, normalized size = 1.66 \begin {gather*} -\frac {2 \, x^{3} e^{3} + 2 \, d x^{2} e^{2} - 2 \, d^{2} x e - 2 \, d^{3} - 6 \, {\left (x^{3} e^{3} + d x^{2} e^{2} - d^{2} x e - d^{3}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (4 \, x^{2} e^{2} + d x e - 2 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (x^{3} e^{7} + d x^{2} e^{6} - d^{2} x e^{5} - d^{3} 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 (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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