Optimal. Leaf size=222 \[ -\frac {1282376 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{972405}-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {42623864 \sqrt {5 x+3} \sqrt {1-2 x}}{972405 \sqrt {3 x+2}}+\frac {613276 \sqrt {5 x+3} \sqrt {1-2 x}}{138915 (3 x+2)^{3/2}}+\frac {13136 \sqrt {5 x+3} \sqrt {1-2 x}}{19845 (3 x+2)^{5/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{7/2}}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405} \]
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Rubi [A] time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, 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 = {97, 150, 152, 158, 113, 119} \[ -\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {42623864 \sqrt {5 x+3} \sqrt {1-2 x}}{972405 \sqrt {3 x+2}}+\frac {613276 \sqrt {5 x+3} \sqrt {1-2 x}}{138915 (3 x+2)^{3/2}}+\frac {13136 \sqrt {5 x+3} \sqrt {1-2 x}}{19845 (3 x+2)^{5/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{7/2}}-\frac {1282376 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {-299+\frac {745 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}-\frac {8 \int \frac {-\frac {87639}{4}+24630 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{19845}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}-\frac {16 \int \frac {-\frac {3794793}{4}+\frac {2299785 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{416745}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {32 \int \frac {-\frac {101193495}{8}-\frac {79919745 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2917215}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}+\frac {7053068 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{972405}+\frac {42623864 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{972405}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {1282376 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 111, normalized size = 0.50 \[ \frac {4 \left (\sqrt {2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5366165 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (1726266492 x^4+4661331894 x^3+4722182964 x^2+2127363207 x+359554583\right )}{2 (3 x+2)^{9/2}}\right )}{2917215} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 504, normalized size = 2.27 \[ \frac {2 \left (51787994760 x^{6}+145018756296 x^{5}-1726266492 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+869318730 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+140113086174 x^{4}-4603377312 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2318183280 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+36035458056 x^{3}-4603377312 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2318183280 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-25330919565 x^{2}-2045945472 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1030303680 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-18067605114 x -340990912 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+171717280 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3235991247\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{2917215 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {9}{2}}} \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]
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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