Optimal. Leaf size=113 \[ \frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0938843, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 641, 217, 203} \[ \frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 850
Rule 819
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac{x^4 (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{x^2 \left (3 d^3-4 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{3 d^5-8 d^4 e x}{\sqrt{d^2-e^2 x^2}} \, dx}{3 d^4 e^4}\\ &=\frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^4}\\ &=\frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4}\\ &=\frac{x^3 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}\\ \end{align*}
Mathematica [A] time = 0.150518, size = 93, normalized size = 0.82 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (5 d^2 e x+8 d^3-7 d e^2 x^2-3 e^3 x^3\right )}{(d-e x) (d+e x)^2}+3 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{3 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.062, size = 179, normalized size = 1.6 \begin{align*} -{\frac{{x}^{2}}{{e}^{3}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+3\,{\frac{{d}^{2}}{{e}^{5}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}-2\,{\frac{dx}{{e}^{4}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}+{\frac{d}{{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{3}}{3\,{e}^{6}} \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{2\,dx}{3\,{e}^{4}}{\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(-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 = 1.71407, size = 346, normalized size = 3.06 \begin{align*} \frac{8 \, d e^{3} x^{3} + 8 \, d^{2} e^{2} x^{2} - 8 \, d^{3} e x - 8 \, d^{4} - 6 \,{\left (d e^{3} x^{3} + d^{2} e^{2} x^{2} - d^{3} e x - d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (3 \, e^{3} x^{3} + 7 \, d e^{2} x^{2} - 5 \, d^{2} e x - 8 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (e^{8} x^{3} + d e^{7} x^{2} - d^{2} e^{6} x - d^{3} e^{5}\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^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]