Optimal. Leaf size=118 \[ \frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.177683, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1805, 823, 12, 266, 63, 208} \[ \frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 852
Rule 1805
Rule 823
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac{(d-e x)^2}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^2+8 d e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-15 d^4 e^2+16 d^3 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\int -\frac{15 d^6 e^4}{x \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{10} e^4}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{d^4 e^2}\\ &=\frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end{align*}
Mathematica [A] time = 0.0979339, size = 95, normalized size = 0.81 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (22 d^2 e x+26 d^3-17 d e^2 x^2-16 e^3 x^3\right )}{(d-e x) (d+e x)^3}-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 \log (x)}{15 d^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.062, size = 187, normalized size = 1.6 \begin{align*}{\frac{1}{{d}^{4}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{4}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{8}{15\,{d}^{3}e} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{16\,ex}{15\,{d}^{5}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}+{\frac{1}{5\,{d}^{2}{e}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-2}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6897, size = 350, normalized size = 2.97 \begin{align*} \frac{26 \, e^{4} x^{4} + 52 \, d e^{3} x^{3} - 52 \, d^{3} e x - 26 \, d^{4} + 15 \,{\left (e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (16 \, e^{3} x^{3} + 17 \, d e^{2} x^{2} - 22 \, d^{2} e x - 26 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{5} e^{4} x^{4} + 2 \, d^{6} e^{3} x^{3} - 2 \, d^{8} e x - d^{9}\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}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{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]