Optimal. Leaf size=147 \[ \frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {d^3 (64 d-45 e x) \sqrt {d^2-e^2 x^2}}{120 e^5}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 147, 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, 847, 794,
223, 209} \begin {gather*} \frac {3 d^5 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^5}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {d^3 (64 d-45 e x) \sqrt {d^2-e^2 x^2}}{120 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 847
Rule 864
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx &=\int \frac {x^4 (d-e x)}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {\int \frac {x^3 \left (4 d^2 e-5 d e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{5 e^2}\\ &=-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {\int \frac {x^2 \left (15 d^3 e^2-16 d^2 e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{20 e^4}\\ &=\frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {\int \frac {x \left (32 d^4 e^3-45 d^3 e^4 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{60 e^6}\\ &=\frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {d^3 (64 d-45 e x) \sqrt {d^2-e^2 x^2}}{120 e^5}+\frac {\left (3 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^4}\\ &=\frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {d^3 (64 d-45 e x) \sqrt {d^2-e^2 x^2}}{120 e^5}+\frac {\left (3 d^5\right ) \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^4}\\ &=\frac {4 d^2 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{4 e^2}+\frac {x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {d^3 (64 d-45 e x) \sqrt {d^2-e^2 x^2}}{120 e^5}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 111, normalized size = 0.76 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (64 d^4-45 d^3 e x+32 d^2 e^2 x^2-30 d e^3 x^3+24 e^4 x^4\right )+45 d^5 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{120 e^6} \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. \(295\) vs.
\(2(127)=254\).
time = 0.07, size = 296, normalized size = 2.01
method | result | size |
risch | \(\frac {\left (24 e^{4} x^{4}-30 d \,e^{3} x^{3}+32 d^{2} x^{2} e^{2}-45 d^{3} e x +64 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{120 e^{5}}+\frac {3 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{4} \sqrt {e^{2}}}\) | \(97\) |
default | \(\frac {-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{5 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{15 e^{4}}}{e}-\frac {d \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 e^{2}}+\frac {d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4 e^{2}}\right )}{e^{2}}-\frac {d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{5}}-\frac {d^{3} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{e^{4}}+\frac {d^{4} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{5}}\) | \(296\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 115, normalized size = 0.78 \begin {gather*} \frac {3}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} - \frac {5}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} x e^{\left (-4\right )} + \sqrt {-x^{2} e^{2} + d^{2}} d^{4} e^{\left (-5\right )} - \frac {1}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x^{2} e^{\left (-3\right )} + \frac {1}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x e^{\left (-4\right )} - \frac {7}{15} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.80, size = 89, normalized size = 0.61 \begin {gather*} -\frac {1}{120} \, {\left (90 \, d^{5} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (24 \, x^{4} e^{4} - 30 \, d x^{3} e^{3} + 32 \, d^{2} x^{2} e^{2} - 45 \, d^{3} x e + 64 \, d^{4}\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 77, normalized size = 0.52 \begin {gather*} \frac {3}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{120} \, {\left (64 \, d^{4} e^{\left (-5\right )} - {\left (45 \, d^{3} e^{\left (-4\right )} - 2 \, {\left (16 \, d^{2} e^{\left (-3\right )} + 3 \, {\left (4 \, x e^{\left (-1\right )} - 5 \, d e^{\left (-2\right )}\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\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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