Optimal. Leaf size=173 \[ -\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {d^4 (256 d+165 e x) \sqrt {d^2-e^2 x^2}}{240 e^5}+\frac {11 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5} \]
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Rubi [A]
time = 0.14, antiderivative size = 173, 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 = {1823, 847, 794,
223, 209} \begin {gather*} \frac {11 d^6 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {d^4 (256 d+165 e x) \sqrt {d^2-e^2 x^2}}{240 e^5}-\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx &=-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x^4 \left (-11 d^2 e^2-12 d e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{6 e^2}\\ &=-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {x^3 \left (48 d^3 e^3+55 d^2 e^4 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{30 e^4}\\ &=-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x^2 \left (-165 d^4 e^4-192 d^3 e^5 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{120 e^6}\\ &=-\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {x \left (384 d^5 e^5+495 d^4 e^6 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{360 e^8}\\ &=-\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {d^4 (256 d+165 e x) \sqrt {d^2-e^2 x^2}}{240 e^5}+\frac {\left (11 d^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^4}\\ &=-\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {d^4 (256 d+165 e x) \sqrt {d^2-e^2 x^2}}{240 e^5}+\frac {\left (11 d^6\right ) \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^4}\\ &=-\frac {8 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {11 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {2 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {d^4 (256 d+165 e x) \sqrt {d^2-e^2 x^2}}{240 e^5}+\frac {11 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 125, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-256 d^5-165 d^4 e x-128 d^3 e^2 x^2-110 d^2 e^3 x^3-96 d e^4 x^4-40 e^5 x^5\right )}{240 e^5}+\frac {11 d^6 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{16 e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 295, normalized size = 1.71
method | result | size |
risch | \(-\frac {\left (40 e^{5} x^{5}+96 d \,e^{4} x^{4}+110 d^{2} e^{3} x^{3}+128 x^{2} d^{3} e^{2}+165 d^{4} x e +256 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{240 e^{5}}+\frac {11 d^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{4} \sqrt {e^{2}}}\) | \(108\) |
default | \(e^{2} \left (-\frac {x^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{6 e^{2}}+\frac {5 d^{2} \left (-\frac {x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4 e^{2}}+\frac {3 d^{2} \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )}{4 e^{2}}\right )}{6 e^{2}}\right )+2 d e \left (-\frac {x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )}{5 e^{2}}\right )+d^{2} \left (-\frac {x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4 e^{2}}+\frac {3 d^{2} \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )}{4 e^{2}}\right )\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 142, normalized size = 0.82 \begin {gather*} \frac {11}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} - \frac {2}{5} \, \sqrt {-x^{2} e^{2} + d^{2}} d x^{4} e^{\left (-1\right )} - \frac {11}{24} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} x^{3} e^{\left (-2\right )} - \frac {8}{15} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} x^{2} e^{\left (-3\right )} - \frac {11}{16} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4} x e^{\left (-4\right )} - \frac {16}{15} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} e^{\left (-5\right )} - \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.70, size = 98, normalized size = 0.57 \begin {gather*} -\frac {1}{240} \, {\left (330 \, d^{6} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (40 \, x^{5} e^{5} + 96 \, d x^{4} e^{4} + 110 \, d^{2} x^{3} e^{3} + 128 \, d^{3} x^{2} e^{2} + 165 \, d^{4} x e + 256 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-5\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 = 11.07, size = 558, normalized size = 3.23 \begin {gather*} d^{2} \left (\begin {cases} - \frac {3 i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{5}} + \frac {3 i d^{3} x}{8 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d x^{3}}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i 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 {3 d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{5}} - \frac {3 d^{3} x}{8 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d x^{3}}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {8 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{6}} - \frac {4 d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {5 i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{7}} + \frac {5 i d^{5} x}{16 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{3} x^{3}}{48 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d x^{5}}{24 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i 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 {5 d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{7}} - \frac {5 d^{5} x}{16 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{3} x^{3}}{48 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d x^{5}}{24 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 84, normalized size = 0.49 \begin {gather*} \frac {11}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{240} \, {\left (256 \, d^{5} e^{\left (-5\right )} + {\left (165 \, d^{4} e^{\left (-4\right )} + 2 \, {\left (64 \, d^{3} e^{\left (-3\right )} + {\left (55 \, d^{2} e^{\left (-2\right )} + 4 \, {\left (12 \, d e^{\left (-1\right )} + 5 \, 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^4\,{\left (d+e\,x\right )}^2}{\sqrt {d^2-e^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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