Optimal. Leaf size=159 \[ -\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}+\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \]
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Rubi [A] time = 0.11, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \]
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) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac {\int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac {d^2 \int (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac {d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=\frac {d^3 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 )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2}+\frac {\left (3 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac {3 d^5 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{8 e^2}+\frac {\left (3 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^2}+\frac {\left (3 d^7\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^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {3 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}-\frac {\left (5 d^7\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2}\\ &=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 135, normalized size = 0.85 \[ \frac {\sqrt {d^2-e^2 x^2} \left (105 d^6 \sin ^{-1}\left (\frac {e x}{d}\right )-\sqrt {1-\frac {e^2 x^2}{d^2}} \left (96 d^6+105 d^5 e x+48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+280 d e^5 x^5+240 e^6 x^6\right )\right )}{1680 e^3 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 116, normalized size = 0.73 \[ -\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (240 \, e^{6} x^{6} + 280 \, d e^{5} x^{5} - 384 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} + 48 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x + 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 96, normalized size = 0.60 \[ \frac {1}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\relax (d) - \frac {1}{1680} \, {\left (96 \, d^{6} e^{\left (-3\right )} + {\left (105 \, d^{5} e^{\left (-2\right )} + 2 \, {\left (24 \, d^{4} e^{\left (-1\right )} - {\left (245 \, d^{3} + 4 \, {\left (48 \, d^{2} e - 5 \, {\left (6 \, x e^{3} + 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 148, normalized size = 0.93 \[ \frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}\, e^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} x}{16 e^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} x}{24 e^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} x^{2}}{7 e}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d x}{6 e^{2}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2}}{35 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 127, normalized size = 0.80 \[ \frac {d^{7} \arcsin \left (\frac {e x}{d}\right )}{16 \, e^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{24 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{6 \, e^{2}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{35 \, e^{3}} \]
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 \[ \int x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.27, size = 653, normalized size = 4.11 \[ d^{3} \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 ) + d^{2} 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 ) - d e^{2} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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