Optimal. Leaf size=132 \[ -\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}+\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2} \]
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Rubi [A] time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {797, 641, 195, 217, 203} \[ \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rule 797
Rubi steps
\begin {align*} \int x^2 (d+e x) \sqrt {d^2-e^2 x^2} \, dx &=-\frac {\int (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}+\frac {d^2 \int (d+e x) \sqrt {d^2-e^2 x^2} \, dx}{e^2}\\ &=-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}+\frac {d^3 \int \sqrt {d^2-e^2 x^2} \, dx}{e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\left (3 d^3\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2}+\frac {d^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\left (3 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}+\frac {d^5 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\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 )}{2 e^3}-\frac {\left (3 d^5\right ) \operatorname {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^2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\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 = 0.12, size = 112, normalized size = 0.85 \[ \frac {\sqrt {d^2-e^2 x^2} \left (15 d^4 \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-16 d^4-15 d^3 e x-8 d^2 e^2 x^2+30 d e^3 x^3+24 e^4 x^4\right )\right )}{120 e^3 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 95, normalized size = 0.72 \[ -\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (24 \, e^{4} x^{4} + 30 \, d e^{3} x^{3} - 8 \, d^{2} e^{2} x^{2} - 15 \, d^{3} e x - 16 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 74, normalized size = 0.56 \[ \frac {1}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\relax (d) - \frac {1}{120} \, {\left (16 \, d^{4} e^{\left (-3\right )} + {\left (15 \, d^{3} e^{\left (-2\right )} + 2 \, {\left (4 \, d^{2} e^{\left (-1\right )} - 3 \, {\left (4 \, x e + 5 \, d\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 125, normalized size = 0.95 \[ \frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, e^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} x}{8 e^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} x^{2}}{5 e}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d x}{4 e^{2}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2}}{15 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 104, normalized size = 0.79 \[ \frac {d^{5} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{8 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} x^{2}}{5 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e^{2}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{15 \, e^{3}} \]
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 \[ \int x^2\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.48, size = 279, normalized size = 2.11 \[ d \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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