Optimal. Leaf size=166 \[ -\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0721729, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^5 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}+\frac{4 \int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx}{9 d}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}+\frac{4 \int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{21 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}+\frac{8 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{105 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}+\frac{8 \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{315 d^4}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0482122, size = 74, normalized size = 0.45 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (84 d^2 e^2 x^2+100 d^3 e x+83 d^4+40 d e^3 x^3+8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}+40\,{e}^{3}{x}^{3}d+84\,{e}^{2}{x}^{2}{d}^{2}+100\,x{d}^{3}e+83\,{d}^{4} \right ) }{315\,e{d}^{5} \left ( ex+d \right ) ^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{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 [A] time = 2.40017, size = 369, normalized size = 2.22 \begin{align*} -\frac{83 \, e^{5} x^{5} + 415 \, d e^{4} x^{4} + 830 \, d^{2} e^{3} x^{3} + 830 \, d^{3} e^{2} x^{2} + 415 \, d^{4} e x + 83 \, d^{5} +{\left (8 \, e^{4} x^{4} + 40 \, d e^{3} x^{3} + 84 \, d^{2} e^{2} x^{2} + 100 \, d^{3} e x + 83 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{315 \,{\left (d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} 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}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{5}}\, 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]