Optimal. Leaf size=156 \[ -\frac {536 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{77 \sqrt {33}}+\frac {89020 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 \sqrt {5 x+3}}-\frac {1340 \sqrt {1-2 x} \sqrt {3 x+2}}{231 (5 x+3)^{3/2}}+\frac {6 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {17804 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}} \]
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Rubi [A] time = 0.05, antiderivative size = 156, 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 = {104, 152, 158, 113, 119} \[ \frac {89020 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 \sqrt {5 x+3}}-\frac {1340 \sqrt {1-2 x} \sqrt {3 x+2}}{231 (5 x+3)^{3/2}}+\frac {6 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}-\frac {17804 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {2}{7} \int \frac {40-45 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}-\frac {4}{231} \int \frac {\frac {3245}{2}-1005 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}+\frac {8 \int \frac {21135+\frac {66765 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2541}\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}+\frac {268}{77} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {17804}{847} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}-\frac {17804 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}-\frac {536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 99, normalized size = 0.63 \[ \frac {2 \left (2 \sqrt {2} \left (4451 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-2240 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \left (667650 x^2+823580 x+253409\right )}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )}{2541} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2250 \, x^{6} + 5925 \, x^{5} + 5305 \, x^{4} + 1111 \, x^{3} - 1035 \, x^{2} - 648 \, x - 108}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.40 \[ \frac {2 \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (1335300 x^{3}+979510 x^{2}-44510 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+22400 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-316762 x -26706 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+13440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-253409\right )}{2541 \left (5 x +3\right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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