Optimal. Leaf size=96 \[ d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.161523, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {852, 1809, 815, 844, 217, 203, 266, 63, 208} \[ d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 852
Rule 1809
Rule 815
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x (d+e x)^2} \, dx &=\int \frac{(d-e x)^2 \sqrt{d^2-e^2 x^2}}{x} \, dx\\ &=-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int \frac{\left (-3 d^2 e^2+6 d e^3 x\right ) \sqrt{d^2-e^2 x^2}}{x} \, dx}{3 e^2}\\ &=d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int \frac{6 d^4 e^4-6 d^3 e^5 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{6 e^4}\\ &=d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^4 \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-\left (d^3 e\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{2} d^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-\left (d^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}-d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{d^4 \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 )}{e^2}\\ &=d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}-d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.109484, size = 96, normalized size = 1. \[ \sqrt{d^2-e^2 x^2} \left (\frac{2 d^2}{3}-d e x+\frac{e^2 x^2}{3}\right )-d^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+d^3 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.067, size = 290, normalized size = 3. \begin{align*}{\frac{1}{5\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{1}{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{d}^{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}-{{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}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ex}{3\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-de\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }x-{{d}^{3}e\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{3\,{d}^{2}{e}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-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 = 1.70291, size = 196, normalized size = 2.04 \begin{align*} 2 \, d^{3} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{3} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + \frac{1}{3} \,{\left (e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 12.045, size = 272, normalized size = 2.83 \begin{align*} d^{2} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]