Optimal. Leaf size=187 \[ -\frac {160297 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{30250 \sqrt {33}}+\frac {7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {665 (3 x+2)^{5/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3284 \sqrt {1-2 x} (3 x+2)^{3/2}}{19965 \sqrt {5 x+3}}-\frac {153319 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{66550}-\frac {5327983 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac {7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {665 (3 x+2)^{5/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3284 \sqrt {1-2 x} (3 x+2)^{3/2}}{19965 \sqrt {5 x+3}}-\frac {153319 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{66550}-\frac {160297 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}-\frac {5327983 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^{5/2} \left (\frac {359}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-\frac {23357}{2}-\frac {40023 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {\left (-\frac {425475}{2}-\frac {1379871 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}+\frac {2 \int \frac {\frac {60716097}{8}+\frac {47951847 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{299475}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}+\frac {160297 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{60500}+\frac {5327983 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{332750}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}-\frac {5327983 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}-\frac {160297 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 102, normalized size = 0.55 \[ \frac {-5366165 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {5 \sqrt {6 x+4} \left (1078110 x^3-11321446 x^2-3117099 x+2438391\right )}{(1-2 x)^{3/2} \sqrt {5 x+3}}+10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{998250 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 233, normalized size = 1.25 \[ \frac {\left (-32343300 x^{4}+318081180 x^{3}+319941890 x^{2}-21311932 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10732330 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10809750 x +10655966 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5366165 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-48767820\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}}{1996500 \left (15 x^{2}+19 x +6\right ) \left (2 x -1\right )^{2}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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