Optimal. Leaf size=91 \[ -\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 191} \[ \frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {3 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d}\\ &=-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2}\\ &=\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 70, normalized size = 0.77 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3+d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x) (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 115, normalized size = 1.26 \[ -\frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{3} e x - 2 \, d^{4} + {\left (2 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} + d^{2} e x - 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} + 2 \, d^{5} e^{4} x^{3} - 2 \, d^{7} e^{2} x - d^{8} e\right )}} \]
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 = 66, normalized size = 0.73 \[ -\frac {\left (-e x +d \right ) \left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right )}{5 \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 136, normalized size = 1.49 \[ -\frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d e^{3} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} - \frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.85, size = 66, normalized size = 0.73 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^3+d^2\,e\,x+4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________