Optimal. Leaf size=89 \[ \frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}-\frac {d}{5 e^3 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {7}{15 e^3 (d+e x) \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {852, 1635, 778, 191} \[ -\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 778
Rule 852
Rule 1635
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^2 (d-e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\left (\frac {2 d^2}{e^2}-\frac {5 d x}{e}\right ) (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 e^2}\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 70, normalized size = 0.79 \[ \frac {\sqrt {d^2-e^2 x^2} \left (4 d^3+8 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x) (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 118, normalized size = 1.33 \[ \frac {4 \, e^{4} x^{4} + 8 \, d e^{3} x^{3} - 8 \, d^{3} e x - 4 \, d^{4} - {\left (e^{3} x^{3} + 2 \, d e^{2} x^{2} + 8 \, d^{2} e x + 4 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{7} x^{4} + 2 \, d^{3} e^{6} x^{3} - 2 \, d^{5} e^{4} x - d^{6} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 65, normalized size = 0.73 \[ \frac {\left (-e x +d \right ) \left (e^{3} x^{3}+2 d \,e^{2} x^{2}+8 d^{2} e x +4 d^{3}\right )}{15 \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 136, normalized size = 1.53 \[ -\frac {d}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{5} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{3}\right )}} + \frac {7}{15 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{3}\right )}} + \frac {x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.90, size = 66, normalized size = 0.74 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^3+8\,d^2\,e\,x+2\,d\,e^2\,x^2+e^3\,x^3\right )}{15\,d^2\,e^3\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________