Optimal. Leaf size=224 \[ \frac{168145 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{112266 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{33} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{5/2}+\frac{730}{891} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{3/2}+\frac{12130 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{2 x+3}}{6237}+\frac{(280359 x+250447) \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}}{56133}-\frac{32567 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{16038 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.156656, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{33} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{5/2}+\frac{730}{891} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{3/2}+\frac{12130 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{2 x+3}}{6237}+\frac{(280359 x+250447) \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}}{56133}+\frac{168145 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{112266 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{32567 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{16038 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2} \, dx &=-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{2}{33} \int (3+2 x)^{3/2} \left (280+\frac{365 x}{2}\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{4}{891} \int \sqrt{3+2 x} \left (\frac{24555}{4}+\frac{18195 x}{4}\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{12130 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{8 \int \frac{\left (\frac{327705}{4}+\frac{467265 x}{8}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{18711}\\ &=\frac{\sqrt{3+2 x} (250447+280359 x) \sqrt{2+5 x+3 x^2}}{56133}+\frac{12130 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{4 \int \frac{\frac{3868215}{8}+\frac{3419535 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{841995}\\ &=\frac{\sqrt{3+2 x} (250447+280359 x) \sqrt{2+5 x+3 x^2}}{56133}+\frac{12130 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{168145 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{224532}-\frac{32567 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{32076}\\ &=\frac{\sqrt{3+2 x} (250447+280359 x) \sqrt{2+5 x+3 x^2}}{56133}+\frac{12130 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{\left (168145 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{112266 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (32567 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{16038 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{\sqrt{3+2 x} (250447+280359 x) \sqrt{2+5 x+3 x^2}}{56133}+\frac{12130 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{730}{891} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{32567 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{16038 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{168145 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{112266 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.39761, size = 208, normalized size = 0.93 \[ -\frac{-127082 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (734832 x^7+789264 x^6-18348768 x^5-80563032 x^4-147414969 x^3-137602437 x^2-64194200 x-11846900\right ) \sqrt{2 x+3}+227969 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{336798 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 156, normalized size = 0.7 \begin{align*} -{\frac{1}{20207880\,{x}^{3}+63991620\,{x}^{2}+63991620\,x+20207880}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 14696640\,{x}^{7}+15785280\,{x}^{6}-366975360\,{x}^{5}+59824\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -227969\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -1611260640\,{x}^{4}-2948299380\,{x}^{3}-2765726880\,{x}^{2}-1306680900\,x-246056760 \right ) } \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 \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}\,{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 (-{\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, 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 - 45 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 51 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 8 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 4 x^{3} \sqrt{2 x + 3} \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 -\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]