Optimal. Leaf size=128 \[ \frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x^2 (4 d-5 e x)}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {(16 d-15 e x) \sqrt {d^2-e^2 x^2}}{6 e^6}-\frac {5 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]
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Rubi [A]
time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 833, 794,
223, 209} \begin {gather*} -\frac {5 d^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}+\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {(16 d-15 e x) \sqrt {d^2-e^2 x^2}}{6 e^6}-\frac {x^2 (4 d-5 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 794
Rule 833
Rule 864
Rubi steps
\begin {align*} \int \frac {x^5}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^5 (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {x^3 \left (4 d^3-5 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x^2 (4 d-5 e x)}{3 e^4 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {x \left (8 d^5-15 d^4 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{3 d^4 e^4}\\ &=\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x^2 (4 d-5 e x)}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {(16 d-15 e x) \sqrt {d^2-e^2 x^2}}{6 e^6}-\frac {\left (5 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^5}\\ &=\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x^2 (4 d-5 e x)}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {(16 d-15 e x) \sqrt {d^2-e^2 x^2}}{6 e^6}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5}\\ &=\frac {x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x^2 (4 d-5 e x)}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {(16 d-15 e x) \sqrt {d^2-e^2 x^2}}{6 e^6}-\frac {5 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 129, normalized size = 1.01 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (16 d^4+d^3 e x-23 d^2 e^2 x^2-3 d e^3 x^3+3 e^4 x^4\right )}{6 e^6 (-d+e x) (d+e x)^2}+\frac {5 d^2 \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 e^9} \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. \(349\) vs.
\(2(112)=224\).
time = 0.09, size = 350, normalized size = 2.73
method | result | size |
risch | \(-\frac {\left (-e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{6}}-\frac {5 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{5} \sqrt {e^{2}}}-\frac {25 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{12 e^{7} \left (x +\frac {d}{e}\right )}-\frac {d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{4 e^{7} \left (x -\frac {d}{e}\right )}+\frac {d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{6 e^{8} \left (x +\frac {d}{e}\right )^{2}}\) | \(202\) |
default | \(\frac {-\frac {x^{3}}{2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {3 d^{2} \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )}{2 e^{2}}}{e}-\frac {d \left (-\frac {x^{2}}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 d^{2}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2}}+\frac {d^{2} \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )}{e^{3}}-\frac {d^{3}}{e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {d^{2} x}{e^{5} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d^{5} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{6}}\) | \(350\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 139, normalized size = 1.09 \begin {gather*} -\frac {5}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} - \frac {x^{3} e^{\left (-3\right )}}{2 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {d x^{2} e^{\left (-4\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} + \frac {17 \, d^{2} x e^{\left (-5\right )}}{6 \, \sqrt {-x^{2} e^{2} + d^{2}}} - \frac {3 \, d^{3} e^{\left (-6\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} + \frac {d^{4}}{3 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x e^{7} + \sqrt {-x^{2} e^{2} + d^{2}} d e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.99, size = 179, normalized size = 1.40 \begin {gather*} -\frac {16 \, d^{2} x^{3} e^{3} + 16 \, d^{3} x^{2} e^{2} - 16 \, d^{4} x e - 16 \, d^{5} - 30 \, {\left (d^{2} x^{3} e^{3} + d^{3} x^{2} e^{2} - d^{4} x e - d^{5}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (3 \, x^{4} e^{4} - 3 \, d x^{3} e^{3} - 23 \, d^{2} x^{2} e^{2} + d^{3} x e + 16 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6 \, {\left (x^{3} e^{9} + d x^{2} e^{8} - d^{2} x e^{7} - d^{3} e^{6}\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^{5}}{\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^5}{{\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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