Optimal. Leaf size=222 \[ -\frac {95264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{250047}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{567 (3 x+2)^{7/2}}+\frac {1532 (5 x+3)^{3/2} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}+\frac {3545996 \sqrt {5 x+3} \sqrt {1-2 x}}{250047 \sqrt {3 x+2}}-\frac {104036 \sqrt {5 x+3} \sqrt {1-2 x}}{35721 (3 x+2)^{3/2}}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047} \]
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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} \[ -\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{567 (3 x+2)^{7/2}}+\frac {1532 (5 x+3)^{3/2} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}+\frac {3545996 \sqrt {5 x+3} \sqrt {1-2 x}}{250047 \sqrt {3 x+2}}-\frac {104036 \sqrt {5 x+3} \sqrt {1-2 x}}{35721 (3 x+2)^{3/2}}-\frac {95264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047} \]
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} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {1905}{2}+\frac {15 x}{2}\right )}{(2+3 x)^{7/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {8 \int \frac {\left (\frac {91845}{4}-14325 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{8505}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {16 \int \frac {\frac {3022395}{8}-\frac {1057575 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{535815}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {32 \int \frac {\frac {41857275}{8}+\frac {66487425 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3750705}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {523952 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{250047}+\frac {3545996 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{250047}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {95264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 111, normalized size = 0.50 \[ \frac {4 \left (\sqrt {2} \left (886499 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-493535 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (143612838 x^4+386630766 x^3+391601529 x^2+176436240 x+29785139\right )}{2 (3 x+2)^{9/2}}\right )}{750141} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 504, normalized size = 2.27 \[ \frac {2 \left (4308385140 x^{6}+12029761494 x^{5}-143612838 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+79952670 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+11615422626 x^{4}-382967568 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+213207120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2988214893 x^{3}-382967568 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+213207120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2101550871 x^{2}-170207808 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+94758720 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1498570743 x -28367968 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+15793120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-268066251\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750141 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {9}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]
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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