Optimal. Leaf size=144 \[ -\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4} \]
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Rubi [A] time = 0.19, antiderivative size = 144, 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 = {1809, 833, 780, 217, 203} \[ -\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx &=-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x^3 \left (-9 d^2 e^2-10 d e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{5 e^2}\\ &=-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {x^2 \left (30 d^3 e^3+36 d^2 e^4 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{20 e^4}\\ &=-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x \left (-72 d^4 e^4-90 d^3 e^5 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{60 e^6}\\ &=-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {\left (3 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^3}\\ &=-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\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 )}{4 e^3}\\ &=-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 92, normalized size = 0.64 \[ \frac {15 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\sqrt {d^2-e^2 x^2} \left (24 d^4+15 d^3 e x+12 d^2 e^2 x^2+10 d e^3 x^3+4 e^4 x^4\right )}{20 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 94, normalized size = 0.65 \[ -\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (4 \, e^{4} x^{4} + 10 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x + 24 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{20 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 73, normalized size = 0.51 \[ \frac {3}{4} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\relax (d) - \frac {1}{20} \, {\left (24 \, d^{4} e^{\left (-4\right )} + {\left (15 \, d^{3} e^{\left (-3\right )} + 2 \, {\left (6 \, d^{2} e^{\left (-2\right )} + {\left (5 \, d e^{\left (-1\right )} + 2 \, x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 149, normalized size = 1.03 \[ -\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x^{4}}{5}+\frac {3 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 \sqrt {e^{2}}\, e^{3}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d \,x^{3}}{2 e}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} x^{2}}{5 e^{2}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} x}{4 e^{3}}-\frac {6 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4}}{5 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 128, normalized size = 0.89 \[ -\frac {1}{5} \, \sqrt {-e^{2} x^{2} + d^{2}} x^{4} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d x^{3}}{2 \, e} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} x^{2}}{5 \, e^{2}} + \frac {3 \, d^{5} \arcsin \left (\frac {e x}{d}\right )}{4 \, e^{4}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{4 \, e^{3}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e^{4}} \]
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 \[ \int \frac {x^3\,{\left (d+e\,x\right )}^2}{\sqrt {d^2-e^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.87, size = 357, normalized size = 2.48 \[ d^{2} \left (\begin {cases} - \frac {2 d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {3 i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{5}} + \frac {3 i d^{3} x}{8 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d x^{3}}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{5}} - \frac {3 d^{3} x}{8 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d x^{3}}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {8 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{6}} - \frac {4 d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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