Optimal. Leaf size=115 \[ \frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0353934, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{5 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d}\\ &=-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d^2}\\ &=\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{21 d^4}\\ &=\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0612647, size = 93, normalized size = 0.81 \[ \frac{\sqrt{d^2-e^2 x^2} \left (24 d^3 e^2 x^2+4 d^2 e^3 x^3+9 d^4 e x-6 d^5-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 88, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{5}{x}^{5}+16\,{e}^{4}{x}^{4}d-4\,{e}^{3}{x}^{3}{d}^{2}-24\,{e}^{2}{x}^{2}{d}^{3}-9\,x{d}^{4}e+6\,{d}^{5} \right ) }{ \left ( 21\,ex+21\,d \right ){d}^{6}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.35041, size = 413, normalized size = 3.59 \begin{align*} -\frac{6 \, e^{6} x^{6} + 12 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} - 24 \, d^{3} e^{3} x^{3} - 6 \, d^{4} e^{2} x^{2} + 12 \, d^{5} e x + 6 \, d^{6} +{\left (8 \, e^{5} x^{5} + 16 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 24 \, d^{3} e^{2} x^{2} - 9 \, d^{4} e x + 6 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{21 \,{\left (d^{6} e^{7} x^{6} + 2 \, d^{7} e^{6} x^{5} - d^{8} e^{5} x^{4} - 4 \, d^{9} e^{4} x^{3} - d^{10} e^{3} x^{2} + 2 \, d^{11} e^{2} x + d^{12} e\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{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]