Optimal. Leaf size=156 \[ -\frac {824 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {41570 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 \sqrt {5 x+3}}+\frac {824 \sqrt {3 x+2}}{17787 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2}}{231 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {8314 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}} \]
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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} \[ -\frac {41570 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 \sqrt {5 x+3}}+\frac {824 \sqrt {3 x+2}}{17787 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2}}{231 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {824 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}+\frac {8314 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}} \]
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}{(1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {161}{2}-45 x}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {824 \sqrt {2+3 x}}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {7865}{4}+1545 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {824 \sqrt {2+3 x}}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {41570 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {30615}{4}+\frac {62355 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{195657}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {824 \sqrt {2+3 x}}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {41570 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}+\frac {412 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5929}-\frac {8314 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{65219}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {824 \sqrt {2+3 x}}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {41570 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}+\frac {8314 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {824 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 98, normalized size = 0.63 \[ \frac {2 \left (\sqrt {2} \left (10955 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-4157 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {\sqrt {3 x+2} \left (-83140 x^2+74076 x-14559\right )}{(1-2 x)^{3/2} \sqrt {5 x+3}}\right )}{195657} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{600 \, x^{6} + 220 \, x^{5} - 534 \, x^{4} - 135 \, x^{3} + 166 \, x^{2} + 21 \, x - 18}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 228, normalized size = 1.46 \[ -\frac {2 \left (249420 x^{3}-55948 x^{2}-8314 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+21910 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-104475 x +4157 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10955 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+29118\right ) \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{195657 \left (15 x^{2}+19 x +6\right ) \left (2 x -1\right )^{2}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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