Optimal. Leaf size=118 \[ -\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{3 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0989571, antiderivative size = 118, 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, 833, 780, 217, 203} \[ -\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{3 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 850
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3 \sqrt{d^2-e^2 x^2}}{d+e x} \, dx &=\int \frac{x^3 (d-e x)}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{\int \frac{x^2 \left (3 d^2 e-4 d e^2 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{\int \frac{x \left (8 d^3 e^2-9 d^2 e^3 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{12 e^4}\\ &=-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{\left (3 d^4\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^3}\\ &=-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{\left (3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}\\ &=-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{3 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4}\\ \end{align*}
Mathematica [A] time = 0.115075, size = 80, normalized size = 0.68 \[ \frac{\sqrt{d^2-e^2 x^2} \left (9 d^2 e x-16 d^3-8 d e^2 x^2+6 e^3 x^3\right )-9 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{24 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.064, size = 185, normalized size = 1.6 \begin{align*} -{\frac{x}{4\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{2}x}{8\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{5\,{d}^{4}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d}{3\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}}{{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{{d}^{4}}{{e}^{3}}\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}}}}} \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 = 1.54405, size = 177, normalized size = 1.5 \begin{align*} \frac{18 \, d^{4} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 9 \, d^{2} e x - 16 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, e^{4}} \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^{3} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15272, size = 89, normalized size = 0.75 \begin{align*} -\frac{3}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{24} \,{\left (16 \, d^{3} e^{\left (-4\right )} -{\left (9 \, d^{2} e^{\left (-3\right )} + 2 \,{\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]