Optimal. Leaf size=253 \[ -\frac {595387 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{4921875}-\frac {524}{225} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{7/2}-\frac {442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {59662 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{7875}+\frac {373022 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{196875}+\frac {500501 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{984375}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875} \]
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Rubi [A] time = 0.10, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {524}{225} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{7/2}-\frac {442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {59662 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{7875}+\frac {373022 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{196875}+\frac {500501 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{984375}-\frac {595387 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875} \]
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} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {1}{2}-36 x\right ) (1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {627}{2}-\frac {5895 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (111060-\frac {1342395 x}{4}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {(2+3 x)^{3/2} \left (-\frac {4470615}{8}+\frac {8392995 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{354375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {13705875}{8}-\frac {67567635 x}{8}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8859375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {-\frac {349379055}{16}-\frac {71895465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{132890625}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {1065118 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4921875}+\frac {6549257 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9843750}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {595387 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 117, normalized size = 0.46 \[ \frac {17517535 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {3 x+2} \left (4725000 x^5+1327500 x^4-5654250 x^3+470675 x^2+4026600 x+1215489\right )}{(5 x+3)^{3/2}}+2130236 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{29531250} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 239, normalized size = 0.94 \[ -\frac {\left (-850500000 x^{7}-380700000 x^{6}+1261440000 x^{5}+164556000 x^{4}-1078163250 x^{3}-311345520 x^{2}+10651180 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+87587675 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+205131330 x +6390708 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+52552605 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+72929340\right ) \sqrt {3 x +2}\, \sqrt {-2 x +1}}{29531250 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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