Optimal. Leaf size=171 \[ -\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]
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Rubi [A]
time = 0.14, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1823, 847,
794, 201, 223, 209} \begin {gather*} -\frac {d^7 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 866
Rule 1823
Rubi steps
\begin {align*} \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^3 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx\\ &=-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^3 \left (-11 d^2 e^2+14 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{7 e^2}\\ &=\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^2 \left (-42 d^3 e^3+66 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{42 e^4}\\ &=-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x \left (-132 d^4 e^4+210 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{210 e^6}\\ &=-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^5 \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^3}\\ &=-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^3}\\ &=-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}\\ &=-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 133, normalized size = 0.78 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (-176 d^6+105 d^5 e x-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4-280 d e^5 x^5+120 e^6 x^6\right )-105 d^7 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{840 e^5} \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. \(564\) vs.
\(2(147)=294\).
time = 0.07, size = 565, normalized size = 3.30
method | result | size |
risch | \(-\frac {\left (-120 e^{6} x^{6}+280 d \,e^{5} x^{5}-144 d^{2} e^{4} x^{4}-70 d^{3} e^{3} x^{3}+88 d^{4} e^{2} x^{2}-105 e \,d^{5} x +176 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{840 e^{4}}-\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3} \sqrt {e^{2}}}\) | \(119\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{4}}-\frac {2 d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \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 )}{4}\right )}{6}\right )}{e^{3}}+\frac {3 d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 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}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \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 )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{4}}-\frac {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 {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 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}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \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 )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{e^{5}}\) | \(565\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 234, normalized size = 1.37 \begin {gather*} -\frac {1}{2} i \, d^{7} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-4\right )} - \frac {5}{8} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} + \frac {1}{2} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{5} x e^{\left (-3\right )} - \frac {5}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} x e^{\left (-3\right )} + \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{6} e^{\left (-4\right )} + \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{\left (-3\right )} - \frac {5}{12} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{\left (-4\right )} - \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{\left (-3\right )} + \frac {3}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{\left (-4\right )} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{4 \, {\left (x e^{5} + d e^{4}\right )}} - \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.86, size = 108, normalized size = 0.63 \begin {gather*} \frac {1}{840} \, {\left (210 \, d^{7} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (120 \, x^{6} e^{6} - 280 \, d x^{5} e^{5} + 144 \, d^{2} x^{4} e^{4} + 70 \, d^{3} x^{3} e^{3} - 88 \, d^{4} x^{2} e^{2} + 105 \, d^{5} x e - 176 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.32, size = 450, normalized size = 2.63 \begin {gather*} d^{2} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 270, normalized size = 1.58 \begin {gather*} \frac {{\left (13440 \, d^{8} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (105 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3780 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 189 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4992 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1981 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 700 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 105 \, d^{8} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{7}}{d^{7}}\right )} e^{\left (-12\right )}}{53760 \, d} \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 )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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