Optimal. Leaf size=113 \[ \frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}+\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2} \]
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Rubi [A] time = 0.13, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1639, 12, 785, 780, 195, 217, 203} \begin {gather*} \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 195
Rule 203
Rule 217
Rule 780
Rule 785
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int \frac {5 d e^3 x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{5 e^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d \int \frac {x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int x \left (d^2 e-d e^2 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{e^2}\\ &=\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^3 \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\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 )}{8 e^2}\\ &=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 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}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 112, normalized size = 0.99 \begin {gather*} \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}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 114, normalized size = 1.01 \begin {gather*} \frac {d^5 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{8 e^4}+\frac {\sqrt {d^2-e^2 x^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 )}{120 e^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 94, normalized size = 0.83 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 222, normalized size = 1.96 \begin {gather*} \frac {d^{5} \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 )}{2 \sqrt {e^{2}}\, e^{2}}-\frac {3 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, e^{2}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} x}{8 e^{2}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3} x}{2 e^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d x}{4 e^{2}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{2}}{3 e^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.01, size = 174, normalized size = 1.54 \begin {gather*} -\frac {i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{3}} - \frac {3 \, d^{5} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{3}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{2 \, e^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{8 \, e^{2}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{3 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{3}} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.85, size = 279, normalized size = 2.47 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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