Optimal. Leaf size=218 \[ -\frac{11346991 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{378125 \sqrt{33}}+\frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{896 (3 x+2)^{7/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4439 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 \sqrt{5 x+3}}-\frac{932783 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{332750}-\frac{21713939 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1663750}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1512500 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0813794, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{896 (3 x+2)^{7/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4439 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 \sqrt{5 x+3}}-\frac{932783 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{332750}-\frac{21713939 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1663750}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{378125 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1512500 \sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^{7/2} \left (\frac{485}{2}+411 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-23785-\frac{80763 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{4439 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 \sqrt{3+5 x}}-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2 \int \frac{\left (-\frac{1710201}{4}-\frac{2798349 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=\frac{4439 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 \sqrt{3+5 x}}-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{932783 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{332750}+\frac{2 \int \frac{\sqrt{2+3 x} \left (\frac{240978825}{8}+\frac{195425451 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{499125}\\ &=\frac{4439 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 \sqrt{3+5 x}}-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{21713939 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1663750}-\frac{932783 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{332750}-\frac{2 \int \frac{-\frac{8597342907}{8}-\frac{13580003439 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7486875}\\ &=\frac{4439 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 \sqrt{3+5 x}}-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{21713939 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1663750}-\frac{932783 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{332750}+\frac{11346991 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{756250}+\frac{1508889271 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{16637500}\\ &=\frac{4439 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 \sqrt{3+5 x}}-\frac{896 (2+3 x)^{7/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{21713939 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1663750}-\frac{932783 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{332750}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1512500 \sqrt{33}}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{378125 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.282548, size = 107, normalized size = 0.49 \[ \frac{-759987865 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{5 \sqrt{6 x+4} \left (48514950 x^4+286777260 x^3-1463754851 x^2-376752444 x+356556921\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}+1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{24956250 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.024, size = 238, normalized size = 1.1 \begin{align*}{\frac{1}{49912500\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1519975730\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3017778542\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-759987865\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +1508889271\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -1455448500\,{x}^{5}-9573616800\,{x}^{4}+38177100330\,{x}^{3}+40577670340\,{x}^{2}-3161658750\,x-7131138420 \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 \frac{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\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 (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, 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\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]