Optimal. Leaf size=196 \[ \frac{9 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{66 \sqrt{x} (3 x+2)}{5 \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}+\frac{3 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}-\frac{66 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5 \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135523, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {834, 839, 1189, 1100, 1136} \[ \frac{66 \sqrt{x} (3 x+2)}{5 \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}+\frac{3 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}+\frac{9 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 834
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{2-5 x}{x^{7/2} \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1}{5} \int \frac{45+9 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}+\frac{3 \sqrt{2+5 x+3 x^2}}{x^{3/2}}+\frac{1}{15} \int \frac{198+\frac{135 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}+\frac{3 \sqrt{2+5 x+3 x^2}}{x^{3/2}}-\frac{66 \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{1}{15} \int \frac{-\frac{135}{2}-297 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}+\frac{3 \sqrt{2+5 x+3 x^2}}{x^{3/2}}-\frac{66 \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{2}{15} \operatorname{Subst}\left (\int \frac{-\frac{135}{2}-297 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}+\frac{3 \sqrt{2+5 x+3 x^2}}{x^{3/2}}-\frac{66 \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}+9 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{198}{5} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{66 \sqrt{x} (2+3 x)}{5 \sqrt{2+5 x+3 x^2}}-\frac{2 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}+\frac{3 \sqrt{2+5 x+3 x^2}}{x^{3/2}}-\frac{66 \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{66 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5 \sqrt{2+5 x+3 x^2}}+\frac{9 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.174913, size = 150, normalized size = 0.77 \[ \frac{-87 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+90 x^3+138 x^2+132 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+40 x-8}{10 x^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 124, normalized size = 0.6 \begin{align*} -{\frac{1}{10} \left ( 51\,\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}-22\,\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}+396\,{x}^{4}+570\,{x}^{3}+126\,{x}^{2}-40\,x+8 \right ){x}^{-{\frac{5}{2}}}{\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{5 \, x - 2}{\sqrt{3 \, x^{2} + 5 \, x + 2} x^{\frac{7}{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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} \sqrt{x}}{3 \, x^{6} + 5 \, x^{5} + 2 \, x^{4}}, x\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{2}{x^{\frac{7}{2}} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{5}{x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}}\, 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*} \int -\frac{5 \, x - 2}{\sqrt{3 \, x^{2} + 5 \, x + 2} x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]