Optimal. Leaf size=160 \[ -\frac {4721 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5250}-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac {102}{175} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {4721 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1050}-\frac {78472 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2625} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac {102}{175} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {4721 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1050}-\frac {4721 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250}-\frac {78472 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {1}{7} \int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {129}{2}+102 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {102}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{175} \int \frac {\left (-4602-\frac {14163 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {4721 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1050}-\frac {102}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {\int \frac {\frac {596157}{4}+235416 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1575}\\ &=-\frac {4721 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1050}-\frac {102}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {51931 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10500}+\frac {78472 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2625}\\ &=-\frac {4721 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1050}-\frac {102}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {78472 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2625}-\frac {4721 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 100, normalized size = 0.62 \[ \frac {313888 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (31619 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (2250 x^2+5910 x+7457\right )\right )}{15750 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 150, normalized size = 0.94 \[ \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-2025000 x^{5}-6871500 x^{4}-10316700 x^{3}-3499230 x^{2}+2629770 x -313888 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+158095 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1342260\right )}{945000 x^{3}+724500 x^{2}-220500 x -189000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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