Optimal. Leaf size=83 \[ -\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}-\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1823, 794, 223,
209} \begin {gather*} \frac {d^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}-\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 1823
Rubi steps
\begin {align*} \int \frac {x (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx &=-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x \left (-5 d^2 e^2-6 d e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{3 e^2}\\ &=-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}-\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {d^3 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e}\\ &=-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}-\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {d^3 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ &=-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}-\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 89, normalized size = 1.07 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^2-3 d e x-e^2 x^2\right )}{3 e^2}+\frac {d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 133, normalized size = 1.60
method | result | size |
risch | \(-\frac {\left (e^{2} x^{2}+3 d e x +5 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}+\frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e \sqrt {e^{2}}}\) | \(73\) |
default | \(e^{2} \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )+2 d e \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{e^{2}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 72, normalized size = 0.87 \begin {gather*} d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} - \sqrt {-x^{2} e^{2} + d^{2}} d x e^{\left (-1\right )} - \frac {5}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{\left (-2\right )} - \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.22, size = 67, normalized size = 0.81 \begin {gather*} -\frac {1}{3} \, {\left (6 \, d^{3} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (x^{2} e^{2} + 3 \, d x e + 5 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.42, size = 218, normalized size = 2.63 \begin {gather*} d^{2} \left (\begin {cases} \frac {x^{2}}{2 \sqrt {d^{2}}} & \text {for}\: e^{2} = 0 \\- \frac {\sqrt {d^{2} - e^{2} x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e^{3}} + \frac {i d x}{2 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e^{3}} - \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {2 d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \sqrt {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 = 1.14, size = 49, normalized size = 0.59 \begin {gather*} d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (5 \, d^{2} e^{\left (-2\right )} + {\left (3 \, d e^{\left (-1\right )} + x\right )} x\right )} \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\,{\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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