Optimal. Leaf size=256 \[ -\frac{211144 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (1685 x+1484) x^{7/2}}{27 \sqrt{3 x^2+5 x+2}}+\frac{45820}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{167336 \sqrt{3 x^2+5 x+2} x^{3/2}}{2835}+\frac{211144 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{1521056 (3 x+2) \sqrt{x}}{76545 \sqrt{3 x^2+5 x+2}}+\frac{1521056 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{76545 \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18789, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {818, 832, 839, 1189, 1100, 1136} \[ \frac{2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (1685 x+1484) x^{7/2}}{27 \sqrt{3 x^2+5 x+2}}+\frac{45820}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{167336 \sqrt{3 x^2+5 x+2} x^{3/2}}{2835}+\frac{211144 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{1521056 (3 x+2) \sqrt{x}}{76545 \sqrt{3 x^2+5 x+2}}-\frac{211144 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}+\frac{1521056 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{76545 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 818
Rule 832
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(-407-340 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{4}{27} \int \frac{x^{5/2} \left (5194+\frac{11455 x}{2}\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{8}{567} \int \frac{\left (-\frac{57275}{2}-\frac{62751 x}{2}\right ) x^{3/2}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{167336 x^{3/2} \sqrt{2+5 x+3 x^2}}{2835}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{16 \int \frac{\sqrt{x} \left (\frac{188253}{2}+\frac{395895 x}{4}\right )}{\sqrt{2+5 x+3 x^2}} \, dx}{8505}\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{211144 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{167336 x^{3/2} \sqrt{2+5 x+3 x^2}}{2835}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{32 \int \frac{-\frac{395895}{4}-\frac{142599 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{76545}\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{211144 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{167336 x^{3/2} \sqrt{2+5 x+3 x^2}}{2835}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{64 \operatorname{Subst}\left (\int \frac{-\frac{395895}{4}-\frac{142599 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{76545}\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{211144 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{167336 x^{3/2} \sqrt{2+5 x+3 x^2}}{2835}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{1521056 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{25515}-\frac{422288 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{5103}\\ &=\frac{2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1521056 \sqrt{x} (2+3 x)}{76545 \sqrt{2+5 x+3 x^2}}-\frac{4 x^{7/2} (1484+1685 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{211144 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{167336 x^{3/2} \sqrt{2+5 x+3 x^2}}{2835}+\frac{45820}{567} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{1521056 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{76545 \sqrt{2+5 x+3 x^2}}-\frac{211144 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.279296, size = 187, normalized size = 0.73 \[ \frac{-1646104 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (18225 x^7-70956 x^6+262710 x^5-2106756 x^4-2967300 x^3+5504080 x^2+8876240 x+3042112\right )-1521056 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{76545 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 312, normalized size = 1.2 \begin{align*} -{\frac{2}{ \left ( 229635+229635\,x \right ) \left ( 2+3\,x \right ) } \left ( 1328364\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+1140792\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+54675\,{x}^{7}+2213940\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+1901320\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-212868\,{x}^{6}+885576\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +760528\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +788130\,{x}^{5}-26854524\,{x}^{4}-77349420\,{x}^{3}-67906368\,{x}^{2}-19002960\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \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{{\left (5 \, x - 2\right )} x^{\frac{13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (5 \, x^{7} - 2 \, x^{6}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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*} \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]