Optimal. Leaf size=218 \[ \frac{18177329 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3781250 \sqrt{33}}+\frac{7 (3 x+2)^{9/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{7/2}}{1815 (5 x+3)^{3/2}}-\frac{4553 \sqrt{1-2 x} (3 x+2)^{5/2}}{99825 \sqrt{5 x+3}}+\frac{380188 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{831875}+\frac{17427983 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8318750}+\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}} \]
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Rubi [A] time = 0.0810254, 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} \[ \frac{7 (3 x+2)^{9/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{7/2}}{1815 (5 x+3)^{3/2}}-\frac{4553 \sqrt{1-2 x} (3 x+2)^{5/2}}{99825 \sqrt{5 x+3}}+\frac{380188 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{831875}+\frac{17427983 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8318750}+\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}}+\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}} \]
Antiderivative was successfully verified.
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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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{11} \int \frac{(2+3 x)^{7/2} \left (\frac{353}{2}+312 x\right )}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2 \int \frac{(2+3 x)^{5/2} \left (\frac{38081}{4}+\frac{32493 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4553 \sqrt{1-2 x} (2+3 x)^{5/2}}{99825 \sqrt{3+5 x}}-\frac{4 \int \frac{(2+3 x)^{3/2} \left (\frac{1361397}{8}+285141 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{99825}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4553 \sqrt{1-2 x} (2+3 x)^{5/2}}{99825 \sqrt{3+5 x}}+\frac{380188 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{831875}+\frac{4 \int \frac{\left (-\frac{48291975}{4}-\frac{156851847 x}{8}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2495625}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4553 \sqrt{1-2 x} (2+3 x)^{5/2}}{99825 \sqrt{3+5 x}}+\frac{17427983 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8318750}+\frac{380188 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{831875}-\frac{4 \int \frac{\frac{6892999929}{16}+\frac{5444240679 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{37434375}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4553 \sqrt{1-2 x} (2+3 x)^{5/2}}{99825 \sqrt{3+5 x}}+\frac{17427983 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8318750}+\frac{380188 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{831875}-\frac{18177329 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7562500}-\frac{604915631 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{41593750}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4553 \sqrt{1-2 x} (2+3 x)^{5/2}}{99825 \sqrt{3+5 x}}+\frac{17427983 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8318750}+\frac{380188 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{831875}+\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}}+\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.248555, size = 141, normalized size = 0.65 \[ \frac{609979405 \sqrt{2-4 x} (5 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \left (-242574750 x^4-1255998150 x^3+1267558775 x^2+2667846028 x+904528061\right ) \sqrt{5 x+3}-1209831262 \sqrt{2-4 x} (5 x+3)^2 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{249562500 \sqrt{1-2 x} (5 x+3)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 229, normalized size = 1.1 \begin{align*}{\frac{1}{1497375000\,{x}^{2}+249562500\,x-499125000}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 6049156310\,\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}-3049897025\,\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}+3629493786\,\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 ) -1829938215\,\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 ) +7277242500\,{x}^{5}+42531439500\,{x}^{4}-12906800250\,{x}^{3}-105386556340\,{x}^{2}-80492762390\,x-18090561220 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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