Optimal. Leaf size=156 \[ -\frac {496 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}-\frac {326 \sqrt {1-2 x} (3 x+2)^{3/2}}{825 \sqrt {5 x+3}}+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{1375}-\frac {169 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}} \]
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Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}-\frac {326 \sqrt {1-2 x} (3 x+2)^{3/2}}{825 \sqrt {5 x+3}}+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{1375}-\frac {496 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {169 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}} \]
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 {\sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {11}{2}-18 x\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (\frac {675}{4}-\frac {2061 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}-\frac {4 \int \frac {-\frac {5823}{4}-\frac {1521 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}+\frac {169 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6875}+\frac {248}{625} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}-\frac {169 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {496 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 102, normalized size = 0.65 \[ \frac {8015 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {10 \sqrt {1-2 x} \sqrt {3 x+2} \left (2475 x^2+1825 x+193\right )}{(5 x+3)^{3/2}}+169 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{20625} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 224, normalized size = 1.44 \[ -\frac {\left (-148500 x^{4}-134250 x^{3}+19670 x^{2}+845 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+40075 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+34570 x +507 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+24045 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3860\right ) \sqrt {-2 x +1}\, \sqrt {3 x +2}}{20625 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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