Optimal. Leaf size=118 \[ -\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {3 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]
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Rubi [A] time = 0.10, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {850, 833, 780, 217, 203} \[ -\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {3 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 833
Rule 850
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx &=\int \frac {x^3 (d-e x)}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {\int \frac {x^2 \left (3 d^2 e-4 d e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}+\frac {\int \frac {x \left (8 d^3 e^2-9 d^2 e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{12 e^4}\\ &=-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {\left (3 d^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^3}\\ &=-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {\left (3 d^4\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^3}\\ &=-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {3 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 80, normalized size = 0.68 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-16 d^3+9 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )-9 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{24 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 83, normalized size = 0.70 \[ \frac {18 \, d^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 9 \, d^{2} e x - 16 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 66, normalized size = 0.56 \[ -\frac {3}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\relax (d) - \frac {1}{24} \, {\left (16 \, d^{3} e^{\left (-4\right )} - {\left (9 \, d^{2} e^{\left (-3\right )} + 2 \, {\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 185, normalized size = 1.57 \[ -\frac {d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e^{3}}+\frac {5 d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, e^{3}}+\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} x}{8 e^{3}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3}}{e^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} x}{4 e^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d}{3 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 101, normalized size = 0.86 \[ -\frac {3 \, d^{4} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{4}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} x}{8 \, e^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} x}{4 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, e^{4}} \]
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 \[ \int \frac {x^3\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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