Optimal. Leaf size=113 \[ \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 7, 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^5 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 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 )^{3/2}}{d+e x} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int \frac {5 d e^3 x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{5 e^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d \int \frac {x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int x \left (d^2 e-d e^2 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{e^2}\\ &=\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^3 \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \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^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 111, normalized size = 0.98 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (16 d^4-15 d^3 e x+8 d^2 e^2 x^2+30 d e^3 x^3-24 e^4 x^4\right )+15 d^5 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{120 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. \(237\) vs.
\(2(97)=194\).
time = 0.07, size = 238, normalized size = 2.11
method | result | size |
risch | \(\frac {\left (-24 e^{4} x^{4}+30 d \,e^{3} x^{3}+8 d^{2} x^{2} e^{2}-15 d^{3} e x +16 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{120 e^{3}}+\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{2} \sqrt {e^{2}}}\) | \(97\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e^{3}}-\frac {d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )\right )}{e^{3}}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 164, normalized size = 1.45 \begin {gather*} -\frac {1}{2} i \, d^{5} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-3\right )} - \frac {3}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} + \frac {1}{2} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{3} x e^{\left (-2\right )} - \frac {3}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} x e^{\left (-2\right )} + \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{4} e^{\left (-3\right )} - \frac {1}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x e^{\left (-2\right )} + \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{\left (-3\right )} - \frac {1}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.54, size = 88, normalized size = 0.78 \begin {gather*} -\frac {1}{120} \, {\left (30 \, 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} - 8 \, d^{2} x^{2} e^{2} + 15 \, d^{3} x e - 16 \, d^{4}\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 = 3.25, size = 279, normalized size = 2.47 \begin {gather*} d \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 ) - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 75, normalized size = 0.66 \begin {gather*} \frac {1}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{120} \, {\left (16 \, d^{4} e^{\left (-3\right )} - {\left (15 \, d^{3} e^{\left (-2\right )} - 2 \, {\left (4 \, d^{2} e^{\left (-1\right )} - 3 \, {\left (4 \, x e - 5 \, d\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 )}^{3/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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