Optimal. Leaf size=160 \[ -\frac {178 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1029}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 \sqrt {3 x+2}}-\frac {97 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {458 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 \sqrt {3 x+2}}-\frac {97 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{3/2}}-\frac {178 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}+\frac {458 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {1}{7} \int \frac {-\frac {181}{2}-160 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {2}{147} \int \frac {-\frac {247}{2}-\frac {485 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}-\frac {4 \int \frac {\frac {395}{4}+\frac {1145 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}-\frac {458 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1029}+\frac {979 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}+\frac {458 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}-\frac {178 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 97, normalized size = 0.61 \[ \frac {\sqrt {2} \left (3395 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {3 \sqrt {10 x+6} \left (1374 x^2+908 x+11\right )}{\sqrt {1-2 x} (3 x+2)^{3/2}}-458 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )}{3087} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 219, normalized size = 1.37 \[ -\frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (41220 x^{3}+51972 x^{2}-1374 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10185 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+16674 x -916 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6790 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+198\right )}{3087 \left (3 x +2\right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \]
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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