Optimal. Leaf size=118 \[ \frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {d (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{6 e^5}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}-\frac {3 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5} \]
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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 = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {850, 819, 833, 780, 217, 203} \[ -\frac {d (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{6 e^5}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}+\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {3 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 819
Rule 833
Rule 850
Rubi steps
\begin {align*} \int \frac {x^4}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\int \frac {x^4 (d-e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x^2 \left (3 d^3-4 d^2 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}+\frac {\int \frac {x \left (8 d^4 e-9 d^3 e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{3 d^2 e^4}\\ &=\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}-\frac {d (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{6 e^5}-\frac {\left (3 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^4}\\ &=\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}-\frac {d (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{6 e^5}-\frac {\left (3 d^3\right ) \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^4}\\ &=\frac {x^3 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x^2 \sqrt {d^2-e^2 x^2}}{3 e^3}-\frac {d (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{6 e^5}-\frac {3 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 91, normalized size = 0.77 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-16 d^3-7 d^2 e x+d e^2 x^2-2 e^3 x^3\right )-9 d^3 (d+e x) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{6 e^5 (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 112, normalized size = 0.95 \[ -\frac {16 \, d^{3} e x + 16 \, d^{4} - 18 \, {\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (2 \, e^{3} x^{3} - d e^{2} x^{2} + 7 \, d^{2} e x + 16 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 147, normalized size = 1.25 \[ -\frac {3 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{4}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x^{2}}{3 e^{3}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d x}{2 e^{4}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3}}{\left (x +\frac {d}{e}\right ) e^{6}}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}}{3 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 113, normalized size = 0.96 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{e^{6} x + d e^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} x^{2}}{3 \, e^{3}} - \frac {3 \, d^{3} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{5}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d x}{2 \, e^{4}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e^{5}} \]
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 \[ \int \frac {x^4}{\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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