Optimal. Leaf size=204 \[ \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]
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Rubi [A]
time = 0.36, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1649,
1829, 655, 223, 209} \begin {gather*} \frac {18 d^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 655
Rule 866
Rule 1649
Rule 1829
Rubi steps
\begin {align*} \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^5}{e^5}+\frac {5 d^4 x}{e^4}-\frac {5 d^3 x^2}{e^3}+\frac {5 d^2 x^3}{e^2}-\frac {5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x)^2 \left (-\frac {60 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {30 d^3 x^2}{e^3}+\frac {15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x) \left (-\frac {240 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {15 d^3 x^2}{e^3}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\int \frac {\frac {720 d^6}{e^3}-\frac {885 d^5 x}{e^2}+\frac {180 d^4 x^2}{e}}{\sqrt {d^2-e^2 x^2}} \, dx}{45 d^3 e^2}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {\int \frac {-\frac {1620 d^6}{e}+1770 d^5 x}{\sqrt {d^2-e^2 x^2}} \, dx}{90 d^3 e^4}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 130, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{15 e^6 (d+e x)^3}+\frac {18 d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^7} \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. \(431\) vs.
\(2(182)=364\).
time = 0.08, size = 432, normalized size = 2.12
method | result | size |
risch | \(\frac {\left (e^{2} x^{2}-6 d e x +29 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{6}}+\frac {18 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{5} \sqrt {e^{2}}}+\frac {108 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{7} \left (x +\frac {d}{e}\right )}-\frac {17 d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{8} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d^{5} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{9} \left (x +\frac {d}{e}\right )^{3}}\) | \(209\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{6}}-\frac {4 d \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{e^{5}}+\frac {10 d^{2} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{6}}-\frac {5 d^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{9} \left (x +\frac {d}{e}\right )^{3}}-\frac {d^{5} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{9}}-\frac {10 d^{3} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{e^{7}}\) | \(432\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.10, size = 188, normalized size = 0.92 \begin {gather*} \frac {424 \, d^{3} x^{3} e^{3} + 1272 \, d^{4} x^{2} e^{2} + 1272 \, d^{5} x e + 424 \, d^{6} - 540 \, {\left (d^{3} x^{3} e^{3} + 3 \, d^{4} x^{2} e^{2} + 3 \, d^{5} x e + d^{6}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (5 \, x^{5} e^{5} - 15 \, d x^{4} e^{4} + 70 \, d^{2} x^{3} e^{3} + 674 \, d^{3} x^{2} e^{2} + 1002 \, d^{4} x e + 424 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{3} e^{9} + 3 \, d x^{2} e^{8} + 3 \, 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} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.64, size = 224, normalized size = 1.10 \begin {gather*} 18 \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (29 \, d^{2} e^{\left (-6\right )} + {\left (x e^{\left (-4\right )} - 6 \, d e^{\left (-5\right )}\right )} x\right )} - \frac {2 \, {\left (\frac {385 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-2\right )}}{x} + \frac {575 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{\left (-4\right )}}{x^{2}} + \frac {355 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-6\right )}}{x^{3}} + \frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-8\right )}}{x^{4}} + 93 \, d^{3}\right )} e^{\left (-6\right )}}{5 \, {\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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,\sqrt {d^2-e^2\,x^2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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