Optimal. Leaf size=148 \[ \frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0466568, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {641, 195, 217, 203} \[ \frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+d \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{8} \left (7 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{48} \left (35 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{64} \left (35 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{128} \left (35 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{128} \left (35 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}\\ \end{align*}
Mathematica [A] time = 0.263427, size = 114, normalized size = 0.77 \[ \frac{1}{384} d \sqrt{d^2-e^2 x^2} \left (-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left (\frac{e x}{d}\right )}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}+279 d^6 x-48 e^6 x^7\right )-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 131, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}+{\frac{dx}{8} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{3}x}{48} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{5}x}{192} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{7}x}{128}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{35\,{d}^{9}}{128}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52541, size = 166, normalized size = 1.12 \begin{align*} \frac{35 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}}} + \frac{35}{128} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x + \frac{35}{192} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x + \frac{7}{48} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x + \frac{1}{8} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}}}{9 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18489, size = 311, normalized size = 2.1 \begin{align*} -\frac{630 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (128 \, e^{8} x^{8} + 144 \, d e^{7} x^{7} - 512 \, d^{2} e^{6} x^{6} - 600 \, d^{3} e^{5} x^{5} + 768 \, d^{4} e^{4} x^{4} + 978 \, d^{5} e^{3} x^{3} - 512 \, d^{6} e^{2} x^{2} - 837 \, d^{7} e x + 128 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1152 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 24.1894, size = 1290, normalized size = 8.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22357, size = 161, normalized size = 1.09 \begin{align*} \frac{35}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{1152} \,{\left (128 \, d^{8} e^{\left (-1\right )} -{\left (837 \, d^{7} + 2 \,{\left (256 \, d^{6} e -{\left (489 \, d^{5} e^{2} + 4 \,{\left (96 \, d^{4} e^{3} -{\left (75 \, d^{3} e^{4} + 2 \,{\left (32 \, d^{2} e^{5} -{\left (8 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\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]