Optimal. Leaf size=140 \[ \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 140, 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 = {1653, 12, 799,
794, 201, 223, 209} \begin {gather*} \frac {d^7 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 201
Rule 209
Rule 223
Rule 794
Rule 799
Rule 1653
Rubi steps
\begin {align*} \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\int \frac {7 d e^3 x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx}{7 e^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {d \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx}{e}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\int x \left (d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2}\\ &=\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^5 \int \sqrt {d^2-e^2 x^2} \, dx}{8 e^2}\\ &=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^2}\\ &=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\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 )}{16 e^2}\\ &=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 133, normalized size = 0.95 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (96 d^6-105 d^5 e x+48 d^4 e^2 x^2+490 d^3 e^3 x^3-384 d^2 e^4 x^4-280 d e^5 x^5+240 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 )}{1680 e^4} \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. \(316\) vs.
\(2(120)=240\).
time = 0.06, size = 317, normalized size = 2.26
method | result | size |
risch | \(\frac {\left (240 e^{6} x^{6}-280 d \,e^{5} x^{5}-384 d^{2} e^{4} x^{4}+490 d^{3} e^{3} x^{3}+48 d^{4} e^{2} x^{2}-105 e \,d^{5} x +96 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1680 e^{3}}+\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{2} \sqrt {e^{2}}}\) | \(119\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{3}}-\frac {d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{3}}\) | \(317\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 186, normalized size = 1.33 \begin {gather*} -\frac {3}{8} i \, d^{7} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-3\right )} - \frac {5}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} + \frac {3}{8} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{5} x e^{\left (-2\right )} - \frac {5}{16} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} x e^{\left (-2\right )} + \frac {3}{4} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{6} e^{\left (-3\right )} + \frac {1}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{\left (-2\right )} - \frac {1}{6} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{\left (-2\right )} + \frac {1}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{\left (-3\right )} - \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.94, size = 109, normalized size = 0.78 \begin {gather*} -\frac {1}{1680} \, {\left (210 \, d^{7} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (240 \, x^{6} e^{6} - 280 \, d x^{5} e^{5} - 384 \, d^{2} x^{4} e^{4} + 490 \, d^{3} x^{3} e^{3} + 48 \, d^{4} x^{2} e^{2} - 105 \, d^{5} x e + 96 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.58, size = 653, normalized size = 4.66 \begin {gather*} d^{3} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - d^{2} e \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 ) - d e^{2} \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^{3} \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 = 1.37, size = 96, normalized size = 0.69 \begin {gather*} \frac {1}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{1680} \, {\left (96 \, d^{6} e^{\left (-3\right )} - {\left (105 \, d^{5} e^{\left (-2\right )} - 2 \, {\left (24 \, d^{4} e^{\left (-1\right )} + {\left (245 \, d^{3} - 4 \, {\left (48 \, d^{2} e - 5 \, {\left (6 \, x e^{3} - 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \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^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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