Optimal. Leaf size=175 \[ \frac{41860 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2 (139 x+121) (2 x+3)^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{28 (1177 x+1018) \sqrt{2 x+3}}{27 \sqrt{3 x^2+5 x+2}}-\frac{31892 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.108222, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {818, 843, 718, 424, 419} \[ -\frac{2 (139 x+121) (2 x+3)^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{28 (1177 x+1018) \sqrt{2 x+3}}{27 \sqrt{3 x^2+5 x+2}}+\frac{41860 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{31892 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 818
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (3+2 x)^{5/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(3+2 x)^{3/2} (-238+133 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3+2 x)^{5/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{28 \sqrt{3+2 x} (1018+1177 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{4}{27} \int \frac{-6727-7973 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^{5/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{28 \sqrt{3+2 x} (1018+1177 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{15946}{27} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{20930}{27} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^{5/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{28 \sqrt{3+2 x} (1018+1177 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{\left (31892 \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 )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (41860 \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 )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 (3+2 x)^{5/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{28 \sqrt{3+2 x} (1018+1177 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{31892 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{41860 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.374799, size = 196, normalized size = 1.12 \[ -\frac{-\frac{6776 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}+\frac{63784 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}-\frac{6 \sqrt{2 x+3} \left (47766 x^3+118690 x^2+96107 x+25237\right )}{3 x^2+5 x+2}+\frac{31892 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{81 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.026, size = 308, normalized size = 1.8 \begin{align*}{\frac{2}{405\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 23919\,\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}+7476\,\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}+39865\,\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}+12460\,\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}+15946\,\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 ) +4984\,\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 ) +1432980\,{x}^{4}+5710170\,{x}^{3}+8224260\,{x}^{2}+5081925\,x+1135665 \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{3+2\,x}}}} \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 (2 \, x + 3\right )}^{\frac{7}{2}}{\left (x - 5\right )}}{{\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 (8 \, x^{4} - 4 \, x^{3} - 126 \, x^{2} - 243 \, x - 135\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{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 (2 \, x + 3\right )}^{\frac{7}{2}}{\left (x - 5\right )}}{{\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]