Optimal. Leaf size=58 \[ \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {796, 12, 261} \[ \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 261
Rule 796
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {2 d^2 e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e}\\ &=\frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 52, normalized size = 0.90 \[ \frac {-2 d^2+2 d e x+e^2 x^2}{3 d e^3 (d-e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.83, size = 104, normalized size = 1.79 \[ -\frac {2 \, e^{3} x^{3} - 2 \, d e^{2} x^{2} - 2 \, d^{2} e x + 2 \, d^{3} - {\left (e^{2} x^{2} + 2 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{6} x^{3} - d^{2} e^{5} x^{2} - d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 51, normalized size = 0.88 \[ \frac {{\left (x^{2} {\left (\frac {x}{d} + 3 \, e^{\left (-1\right )}\right )} - 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 55, normalized size = 0.95 \[ -\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (-e^{2} x^{2}-2 d e x +2 d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 88, normalized size = 1.52 \[ \frac {x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {d x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.59, size = 55, normalized size = 0.95 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^2+2\,d\,e\,x+e^2\,x^2\right )}{3\,d\,e^3\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 9.88, size = 231, normalized size = 3.98 \[ d \left (\begin {cases} \frac {i x^{3}}{- 3 d^{5} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {x^{3}}{- 3 d^{5} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {2 d^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {3 e^{2} x^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________