Optimal. Leaf size=155 \[ \frac {4817 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{250 \sqrt {33}}+\frac {\sqrt {5 x+3} (3 x+2)^{5/2}}{\sqrt {1-2 x}}+\frac {9}{5} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {419}{50} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 155, 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 = {97, 154, 158, 113, 119} \[ \frac {\sqrt {5 x+3} (3 x+2)^{5/2}}{\sqrt {1-2 x}}+\frac {9}{5} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {419}{50} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}+\frac {4817 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250 \sqrt {33}}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{3/2} \left (\frac {55}{2}+45 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{25} \int \frac {\left (-\frac {3875}{2}-\frac {6285 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {1}{375} \int \frac {\frac {276495}{4}+109185 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {4817}{500} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {7279}{125} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4817 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 110, normalized size = 0.71 \[ \frac {14665 \sqrt {2-4 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-30 \sqrt {3 x+2} \sqrt {5 x+3} \left (90 x^2+328 x-799\right )-29116 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1500 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 145, normalized size = 0.94 \[ -\frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-40500 x^{4}-198900 x^{3}+156390 x^{2}+396390 x -29116 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+14665 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+143820\right )}{1500 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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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