Optimal. Leaf size=136 \[ -\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^2}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e} \]
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Rubi [A] time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {793, 665, 195, 217, 203} \[ -\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 665
Rule 793
Rubi steps
\begin {align*} \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx}{3 e}\\ &=-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {(2 d) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{3 e}\\ &=-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {d^3 \int \sqrt {d^2-e^2 x^2} \, dx}{2 e}\\ &=-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {d^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e}\\ &=-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\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 )}{4 e}\\ &=-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 91, normalized size = 0.67 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-28 d^4+15 d^3 e x+16 d^2 e^2 x^2-30 d e^3 x^3+12 e^4 x^4\right )-15 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{60 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 94, normalized size = 0.69 \[ \frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (12 \, e^{4} x^{4} - 30 \, d e^{3} x^{3} + 16 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 28 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{60 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 198, normalized size = 1.46 \[ -\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{4 \sqrt {e^{2}}\, e}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3} x}{4 e}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d x}{6 e}-\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}}}{15 e^{2}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{3 \left (x +\frac {d}{e}\right )^{2} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.00, size = 167, normalized size = 1.23 \[ \frac {i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{2}} - \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{4 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{4 \, {\left (e^{3} x + d e^{2}\right )}} - \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{2 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{2}} \]
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 \[ \int \frac {x\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.57, size = 321, normalized size = 2.36 \[ d^{2} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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