Optimal. Leaf size=162 \[ \frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
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Rubi [A]
time = 0.10, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {7 d^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \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 794
Rule 833
Rule 864
Rubi steps
\begin {align*} \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^7 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3-7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5-35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7-105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {\left (7 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^7}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 148, normalized size = 0.91 \begin {gather*} \frac {\frac {e \sqrt {d^2-e^2 x^2} \left (96 d^6-9 d^5 e x-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4+15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^2 (d+e x)^3}+105 d^2 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{30 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. \(650\) vs.
\(2(142)=284\).
time = 0.09, size = 651, normalized size = 4.02
method | result | size |
risch | \(\frac {\left (-e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{8}}+\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{7} \sqrt {e^{2}}}+\frac {773 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 e^{9} \left (x +\frac {d}{e}\right )}+\frac {31 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{48 e^{9} \left (x -\frac {d}{e}\right )}+\frac {d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{24 e^{10} \left (x -\frac {d}{e}\right )^{2}}-\frac {7 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{10} \left (x +\frac {d}{e}\right )^{2}}+\frac {d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 e^{11} \left (x +\frac {d}{e}\right )^{3}}\) | \(295\) |
default | \(\frac {-\frac {x^{5}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 d^{2} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\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^{2}}\right )}{2 e^{2}}}{e}-\frac {d \left (-\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 d^{2} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{2}}\right )}{e^{2}}+\frac {d^{2} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\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^{2}}\right )}{e^{3}}-\frac {d^{3} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{4}}+\frac {d^{4} \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )}{e^{5}}-\frac {d^{5}}{3 e^{8} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {d^{6} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{7}}-\frac {d^{7} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{8}}\) | \(651\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 265, normalized size = 1.64 \begin {gather*} \frac {d^{6}}{5 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{9} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{8}\right )}} - \frac {x^{5} e^{\left (-3\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {d x^{4} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {25 \, d^{2} x^{3} e^{\left (-5\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {65 \, d^{3} x^{2} e^{\left (-6\right )}}{6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {164 \, d^{4} x e^{\left (-7\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {53 \, d^{5} e^{\left (-8\right )}}{6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {7}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-8\right )} - \frac {7 \, d x^{2} e^{\left (-6\right )}}{6 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {229 \, d^{2} x e^{\left (-7\right )}}{30 \, \sqrt {-x^{2} e^{2} + d^{2}}} - \frac {14 \, d^{3} e^{\left (-8\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}}} - \frac {7}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} d e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.34, size = 255, normalized size = 1.57 \begin {gather*} \frac {96 \, d^{2} x^{5} e^{5} + 96 \, d^{3} x^{4} e^{4} - 192 \, d^{4} x^{3} e^{3} - 192 \, d^{5} x^{2} e^{2} + 96 \, d^{6} x e + 96 \, d^{7} - 210 \, {\left (d^{2} x^{5} e^{5} + d^{3} x^{4} e^{4} - 2 \, d^{4} x^{3} e^{3} - 2 \, d^{5} x^{2} e^{2} + d^{6} x e + d^{7}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (15 \, x^{6} e^{6} - 15 \, d x^{5} e^{5} - 176 \, d^{2} x^{4} e^{4} + 4 \, d^{3} x^{3} e^{3} + 249 \, d^{4} x^{2} e^{2} + 9 \, d^{5} x e - 96 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{5} e^{13} + d x^{4} e^{12} - 2 \, d^{2} x^{3} e^{11} - 2 \, d^{3} x^{2} e^{10} + d^{4} x e^{9} + d^{5} e^{8}\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^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{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^7}{{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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