Optimal. Leaf size=89 \[ \frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.033264, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {789, 639, 191} \[ \frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 789
Rule 639
Rule 191
Rubi steps
\begin{align*} \int \frac{x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 \int \frac{d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0561763, size = 62, normalized size = 0.7 \[ \frac{-2 d^2 e x+d^3+8 d e^2 x^2-4 e^3 x^3}{15 d^3 e^2 (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 64, normalized size = 0.7 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3} \left ( -4\,{e}^{3}{x}^{3}+8\,d{e}^{2}{x}^{2}-2\,{d}^{2}ex+{d}^{3} \right ) }{15\,{d}^{3}{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01071, size = 147, normalized size = 1.65 \begin{align*} \frac{x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \, x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d e} - \frac{4 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87344, size = 228, normalized size = 2.56 \begin{align*} \frac{e^{4} x^{4} - 2 \, d e^{3} x^{3} + 2 \, d^{3} e x - d^{4} +{\left (4 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 2 \, d^{2} e x - d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{6} x^{4} - 2 \, d^{4} e^{5} x^{3} + 2 \, d^{6} e^{3} x - d^{7} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17491, size = 86, normalized size = 0.97 \begin{align*} \frac{{\left ({\left (2 \, x{\left (\frac{2 \, x^{2} e^{3}}{d^{3}} - \frac{5 \, e}{d}\right )} - 5\right )} x^{2} - d^{2} e^{\left (-2\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]