Optimal. Leaf size=222 \[ -\frac {234856 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{83349}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac {10 \sqrt {5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac {7810384 \sqrt {5 x+3} \sqrt {1-2 x}}{83349 \sqrt {3 x+2}}+\frac {112436 \sqrt {5 x+3} \sqrt {1-2 x}}{11907 (3 x+2)^{3/2}}+\frac {832 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}-\frac {7810384 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83349} \]
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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} \[ -\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac {10 \sqrt {5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac {7810384 \sqrt {5 x+3} \sqrt {1-2 x}}{83349 \sqrt {3 x+2}}+\frac {112436 \sqrt {5 x+3} \sqrt {1-2 x}}{11907 (3 x+2)^{3/2}}+\frac {832 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}-\frac {234856 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83349}-\frac {7810384 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83349} \]
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)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {\sqrt {1-2 x} \left (-435+\frac {255 x}{2}\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {8 \int \frac {\frac {79845}{4}-\frac {45525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{8505}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {112436 \sqrt {1-2 x} \sqrt {3+5 x}}{11907 (2+3 x)^{3/2}}+\frac {16 \int \frac {869010-\frac {2108175 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{178605}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {112436 \sqrt {1-2 x} \sqrt {3+5 x}}{11907 (2+3 x)^{3/2}}+\frac {7810384 \sqrt {1-2 x} \sqrt {3+5 x}}{83349 \sqrt {2+3 x}}+\frac {32 \int \frac {\frac {92710725}{8}+\frac {36611175 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1250235}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {112436 \sqrt {1-2 x} \sqrt {3+5 x}}{11907 (2+3 x)^{3/2}}+\frac {7810384 \sqrt {1-2 x} \sqrt {3+5 x}}{83349 \sqrt {2+3 x}}+\frac {1291708 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{83349}+\frac {7810384 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{83349}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {112436 \sqrt {1-2 x} \sqrt {3+5 x}}{11907 (2+3 x)^{3/2}}+\frac {7810384 \sqrt {1-2 x} \sqrt {3+5 x}}{83349 \sqrt {2+3 x}}-\frac {7810384 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83349}-\frac {234856 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83349}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 111, normalized size = 0.50 \[ \frac {4 \left (\sqrt {2} \left (1952596 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-983815 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (316320552 x^4+854146674 x^3+865270206 x^2+389804925 x+65886031\right )}{2 (3 x+2)^{9/2}}\right )}{250047} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 504, normalized size = 2.27 \[ \frac {2 \left (9489616560 x^{6}+26573361876 x^{5}-316320552 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+159378030 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+25673661234 x^{4}-843521472 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+425008080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6602638302 x^{3}-843521472 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+425008080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4641436149 x^{2}-374898432 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+188892480 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3310586232 x -62483072 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+31482080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-592974279\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{250047 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {9}{2}}} \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]
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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