Optimal. Leaf size=170 \[ \frac{1039 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{378 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{21} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{945} \sqrt{2 x+3} (2169 x+2327) \sqrt{3 x^2+5 x+2}-\frac{697 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{270 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.123973, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2}{21} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{945} \sqrt{2 x+3} (2169 x+2327) \sqrt{3 x^2+5 x+2}+\frac{1039 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{378 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{697 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{270 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2} \, dx &=-\frac{2}{21} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}+\frac{2}{21} \int \frac{\left (182+\frac{241 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{1}{945} \sqrt{3+2 x} (2327+2169 x) \sqrt{2+5 x+3 x^2}-\frac{2}{21} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{945} \int \frac{\frac{4721}{2}+\frac{4879 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{945} \sqrt{3+2 x} (2327+2169 x) \sqrt{2+5 x+3 x^2}-\frac{2}{21} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{697}{540} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{1039}{756} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{945} \sqrt{3+2 x} (2327+2169 x) \sqrt{2+5 x+3 x^2}-\frac{2}{21} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{\left (697 \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 )}{270 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (1039 \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 )}{378 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{1}{945} \sqrt{3+2 x} (2327+2169 x) \sqrt{2+5 x+3 x^2}-\frac{2}{21} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{697 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{270 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{1039 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{378 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.311076, size = 198, normalized size = 1.16 \[ -\frac{-1762 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (4860 x^5-15552 x^4-121239 x^3-200865 x^2-128926 x-28888\right ) \sqrt{2 x+3}+4879 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{5670 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 146, normalized size = 0.9 \begin{align*}{\frac{1}{340200\,{x}^{3}+1077300\,{x}^{2}+1077300\,x+340200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -97200\,{x}^{5}+316\,\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 ) +4879\,\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 ) +311040\,{x}^{4}+2424780\,{x}^{3}+4310040\,{x}^{2}+3066420\,x+772920 \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} \sqrt{2 \, x + 3}{\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 (-\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}, 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 - 5 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int x \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} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]