Optimal. Leaf size=222 \[ \frac {135334 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{25515}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt {3 x+2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {135334 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{5103}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515} \]
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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, 154, 158, 113, 119} \[ \frac {5260}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt {3 x+2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {135334 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{5103}+\frac {135334 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {\left (\frac {235}{2}-\frac {6575 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (83425-\frac {1173675 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}\\ &=-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\left (\frac {1656225}{8}-\frac {5075025 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42525}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {2298225}{2}-\frac {33929925 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{382725}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {452399 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{25515}-\frac {744337 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{25515}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}+\frac {135334 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 112, normalized size = 0.50 \[ \frac {-2685410 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {5 x+3} \left (24300 x^4-25110 x^3+5949 x^2+108285 x+56963\right )}{(3 x+2)^{3/2}}+452399 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{76545} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 234, normalized size = 1.05 \[ \frac {\left (7290000 x^{6}-6804000 x^{5}-1155600 x^{4}+34923870 x^{3}+19802040 x^{2}-1357197 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8056230 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-8036760 x -904798 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5370820 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5126670\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{76545 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \]
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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