Optimal. Leaf size=95 \[ -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 95, 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, 778, 191} \[ -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 778
Rule 855
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}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x (2 d+2 e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e^2}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 82, normalized size = 0.86 \[ \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4+2 d^3 e x-3 d^2 e^2 x^2+2 d e^3 x^3+2 e^4 x^4\right )}{15 d^3 e^3 (d-e x)^2 (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.81, size = 170, normalized size = 1.79 \[ \frac {2 \, e^{5} x^{5} + 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x + 2 \, d^{5} + {\left (2 \, e^{4} x^{4} + 2 \, d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 2 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{8} x^{5} + d^{4} e^{7} x^{4} - 2 \, d^{5} e^{6} x^{3} - 2 \, d^{6} e^{5} x^{2} + d^{7} e^{4} x + d^{8} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 70, normalized size = 0.74 \[ \frac {\left (-e x +d \right ) \left (2 x^{4} e^{4}+2 x^{3} d \,e^{3}-3 d^{2} x^{2} e^{2}+2 d^{3} x e +2 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 110, normalized size = 1.16 \[ -\frac {d}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{3}\right )}} - \frac {x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{2}} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {2 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.79, size = 78, normalized size = 0.82 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4+2\,d^3\,e\,x-3\,d^2\,e^2\,x^2+2\,d\,e^3\,x^3+2\,e^4\,x^4\right )}{15\,d^3\,e^3\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________