Optimal. Leaf size=60 \[ \frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {855, 12, 261} \[ \frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 261
Rule 855
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {2 d x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d e}\\ &=-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e}\\ &=\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 60, normalized size = 1.00 \[ \frac {\sqrt {d^2-e^2 x^2} \left (2 d^2+2 d e x-e^2 x^2\right )}{3 d e^3 (d-e x) (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 103, normalized size = 1.72 \[ \frac {2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3} + {\left (e^{2} x^{2} - 2 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{6} x^{3} + d^{2} e^{5} x^{2} - d^{3} e^{4} x - d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 1, normalized size = 0.02 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 48, normalized size = 0.80 \[ \frac {\left (-e x +d \right ) \left (-e^{2} x^{2}+2 d e x +2 d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 86, normalized size = 1.43 \[ -\frac {d}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{3}\right )}} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} + \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.71, size = 56, normalized size = 0.93 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2+2\,d\,e\,x-e^2\,x^2\right )}{3\,d\,e^3\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________