Optimal. Leaf size=206 \[ \frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A]
time = 0.07, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {683, 685, 655,
223, 209} \begin {gather*} -\frac {231 d^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 \sqrt {d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac {77 d \sqrt {d^2-e^2 x^2} (d+e x)}{2 e}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 655
Rule 683
Rule 685
Rubi steps
\begin {align*} \int \frac {(d+e x)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {11}{5} \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {33}{5} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}-\frac {231}{5} \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-(77 d) \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^2\right ) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 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 )\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 131, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5446 d^5+12843 d^4 e x-8711 d^3 e^2 x^2+815 d^2 e^3 x^3+105 d e^4 x^4+10 e^5 x^5\right )}{30 e (-d+e x)^3}+\frac {231 d^3 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 \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. \(985\) vs.
\(2(178)=356\).
time = 0.52, size = 986, normalized size = 4.79
method | result | size |
risch | \(\frac {\left (2 e^{2} x^{2}+27 d x e +238 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e}-\frac {231 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {832 d^{4} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{3} \left (x -\frac {d}{e}\right )^{2}}-\frac {2768 d^{3} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{2} \left (x -\frac {d}{e}\right )}-\frac {64 d^{5} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{4} \left (x -\frac {d}{e}\right )^{3}}\) | \(216\) |
default | \(e^{9} \left (-\frac {x^{8}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {8 d^{2} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \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 )}{e^{2}}\right )}{3 e^{2}}\right )+9 d \,e^{8} \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\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}}}{e^{2}}\right )}{2 e^{2}}\right )+36 d^{2} e^{7} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \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 )}{e^{2}}\right )+84 d^{3} e^{6} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\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}}}{e^{2}}\right )+126 d^{4} e^{5} \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 )+126 d^{5} e^{4} \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 )+84 d^{6} e^{3} \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 )+36 d^{7} e^{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 )+\frac {9 d^{8}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{9} \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 )\) | \(986\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs.
\(2 (170) = 340\).
time = 0.52, size = 347, normalized size = 1.68 \begin {gather*} -\frac {x^{8} e^{7}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {9 \, d x^{7} e^{6}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {116 \, d^{2} x^{6} e^{5}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {358 \, d^{4} x^{4} e^{3}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {63 \, d^{5} x^{3} e^{2}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {1348 \, d^{6} x^{2} e}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2723 \, d^{8} e^{\left (-1\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {77}{10} \, {\left (\frac {15 \, x^{4} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {20 \, d^{2} x^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} e^{\left (-6\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}\right )} d^{3} x e^{6} - \frac {77}{2} \, {\left (\frac {3 \, x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}\right )} d^{3} x e^{4} - \frac {152 \, d^{7} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {231}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {619 \, d^{5} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {989 \, d^{3} x}{30 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 191, normalized size = 0.93 \begin {gather*} \frac {5446 \, d^{3} x^{3} e^{3} - 16338 \, d^{4} x^{2} e^{2} + 16338 \, d^{5} x e - 5446 \, d^{6} + 6930 \, {\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 (10 \, x^{5} e^{5} + 105 \, d x^{4} e^{4} + 815 \, d^{2} x^{3} e^{3} - 8711 \, d^{3} x^{2} e^{2} + 12843 \, d^{4} x e - 5446 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{3} e^{4} - 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} - d^{3} e\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 {\left (d + e x\right )^{9}}{\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 = 1.20, size = 223, normalized size = 1.08 \begin {gather*} -\frac {231}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (238 \, d^{2} e^{\left (-1\right )} + {\left (2 \, x e + 27 \, d\right )} x\right )} - \frac {32 \, {\left (\frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-2\right )}}{x} - \frac {850 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{\left (-4\right )}}{x^{2}} + \frac {480 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-6\right )}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-8\right )}}{x^{4}} - 133 \, d^{3}\right )} e^{\left (-1\right )}}{15 \, {\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 {{\left (d+e\,x\right )}^9}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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