Optimal. Leaf size=249 \[ -\frac{776112041 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{230343750 \sqrt{33}}+\frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac{601 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac{18034 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{625625}-\frac{11725073 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{56306250}-\frac{776112041 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{506756250}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{460687500 \sqrt{33}} \]
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Rubi [A] time = 0.0970553, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac{601 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac{18034 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{625625}-\frac{11725073 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{56306250}-\frac{776112041 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{506756250}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{230343750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{460687500 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx &=\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{2}{65} \int \left (-\frac{71}{2}-\frac{89 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{4 \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2} \left (-1082+\frac{1803 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{10725}\\ &=\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{4 \int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{661845}{8}+\frac{243459 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{482625}\\ &=-\frac{18034 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{625625}+\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (-8651787-\frac{105525657 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{16891875}\\ &=-\frac{11725073 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{56306250}-\frac{18034 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{625625}+\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{9078479379}{16}+\frac{6985008369 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{253378125}\\ &=-\frac{776112041 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{506756250}-\frac{11725073 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{56306250}-\frac{18034 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{625625}+\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{4 \int \frac{-\frac{36751750227}{2}-\frac{464411639007 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2280403125}\\ &=-\frac{776112041 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{506756250}-\frac{11725073 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{56306250}-\frac{18034 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{625625}+\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{776112041 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{460687500}+\frac{51601293223 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5067562500}\\ &=-\frac{776112041 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{506756250}-\frac{11725073 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{56306250}-\frac{18034 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{625625}+\frac{601 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{460687500 \sqrt{33}}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{230343750 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.294363, size = 115, normalized size = 0.46 \[ \frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (5197919174 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (7016625000 x^5+12374775000 x^4+3047388750 x^3-5775295500 x^2-3548873565 x+325972172\right )\right )}{7601343750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{456080625000\,{x}^{3}+349661812500\,{x}^{2}-106418812500\,x-91216125000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -6314962500000\,{x}^{8}-15978768750000\,{x}^{7}-9807753375000\,{x}^{6}+25989595870\,\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 ) -51601293223\,\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 ) +6956762962500\,{x}^{5}+10046351241000\,{x}^{4}+1491065725050\,{x}^{3}-2009737437330\,{x}^{2}-570343085580\,x+58674990960 \right ) } \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{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\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 (-{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\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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