Optimal. Leaf size=207 \[ -\frac{12857 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{56 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{(5 x+53) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{5/2}}+\frac{(879 x+2291) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{3/2}}-\frac{(3117 x+10763) \sqrt{3 x^2+5 x+2}}{140 \sqrt{2 x+3}}+\frac{2333 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{40 \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.132171, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {812, 843, 718, 424, 419} \[ -\frac{(5 x+53) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{5/2}}+\frac{(879 x+2291) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{3/2}}-\frac{(3117 x+10763) \sqrt{3 x^2+5 x+2}}{140 \sqrt{2 x+3}}-\frac{12857 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{56 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{2333 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{40 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 812
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx &=-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac{1}{14} \int \frac{(-245-293 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{5/2}} \, dx\\ &=\frac{(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac{1}{140} \int \frac{(-7939-9351 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx\\ &=-\frac{(10763+3117 x) \sqrt{2+5 x+3 x^2}}{140 \sqrt{3+2 x}}+\frac{(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac{1}{840} \int \frac{-124041-146979 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{(10763+3117 x) \sqrt{2+5 x+3 x^2}}{140 \sqrt{3+2 x}}+\frac{(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac{6999}{80} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{12857}{112} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{(10763+3117 x) \sqrt{2+5 x+3 x^2}}{140 \sqrt{3+2 x}}+\frac{(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac{\left (12857 \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 )}{56 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (2333 \sqrt{3} \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 )}{40 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(10763+3117 x) \sqrt{2+5 x+3 x^2}}{140 \sqrt{3+2 x}}+\frac{(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac{(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac{2333 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{40 \sqrt{2+5 x+3 x^2}}-\frac{12857 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{56 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.449724, size = 202, normalized size = 0.98 \[ \frac{-10422 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )-3240 x^7+12744 x^6+41220 x^5+335988 x^4+1717690 x^3+3262382 x^2+2556580 x+48993 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+701136}{840 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 311, normalized size = 1.5 \begin{align*} -{\frac{1}{8400} \left ( 61168\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+195972\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+32400\,{x}^{7}+183504\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+587916\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-127440\,{x}^{6}+137628\,\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 ) +440937\,\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 ) -412200\,{x}^{5}+8398440\,{x}^{4}+37695260\,{x}^{3}+60462880\,{x}^{2}+42044540\,x+10626120 \right ) \left ( 3+2\,x \right ) ^{-{\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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\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{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81}, 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 (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]