Optimal. Leaf size=83 \[ -\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.08, 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 = {1809, 780, 217, 203} \[ -\frac {d (5 d+3 e x) \sqrt {d^2-e^2 x^2}}{3 e^2}-\frac {1}{3} x^2 \sqrt {d^2-e^2 x^2}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 1809
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 \operatorname {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 = 0.05, size = 69, normalized size = 0.83 \[ \frac {3 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\sqrt {d^2-e^2 x^2} \left (5 d^2+3 d e x+e^2 x^2\right )}{3 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 71, normalized size = 0.86 \[ -\frac {6 \, d^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (e^{2} x^{2} + 3 \, d e x + 5 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 49, normalized size = 0.59 \[ d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\relax (d) - \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (5 \, d^{2} e^{\left (-2\right )} + {\left (3 \, d e^{\left (-1\right )} + x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 1.18 \[ \frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x^{2}}{3}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d x}{e}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}}{3 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 77, normalized size = 0.93 \[ -\frac {1}{3} \, \sqrt {-e^{2} x^{2} + d^{2}} x^{2} + \frac {d^{3} \arcsin \left (\frac {e x}{d}\right )}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d x}{e} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e^{2}} \]
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 \[ \int \frac {x\,{\left (d+e\,x\right )}^2}{\sqrt {d^2-e^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.62, size = 218, normalized size = 2.63 \[ d^{2} \left (\begin {cases} \frac {x^{2}}{2 \sqrt {d^{2}}} & \text {for}\: e^{2} = 0 \\- \frac {\sqrt {d^{2} - e^{2} x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e^{3}} - \frac {i d x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e^{3}} - \frac {d x}{2 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {x^{3}}{2 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {2 d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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