Optimal. Leaf size=158 \[ -\frac {314 \sqrt {33} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{21875}+\frac {2}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {23}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {859 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{4375}-\frac {61151 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750} \]
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Rubi [A] time = 0.05, antiderivative size = 158, 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 = {101, 154, 158, 113, 119} \[ \frac {2}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {23}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {859 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{4375}-\frac {314 \sqrt {33} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{21875}-\frac {61151 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {2}{35} \int \frac {\left (-\frac {27}{2}-\frac {23 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {23}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {2}{875} \int \frac {\sqrt {2+3 x} \left (\frac {3275}{4}+\frac {2577 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {859 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4375}-\frac {23}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {2 \int \frac {-\frac {116289}{4}-\frac {183453 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{13125}\\ &=-\frac {859 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4375}-\frac {23}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {5181 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{21875}+\frac {61151 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{43750}\\ &=-\frac {859 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4375}-\frac {23}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {61151 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750}-\frac {314 \sqrt {33} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{21875}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 97, normalized size = 0.61 \[ \frac {-30065 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (2250 x^2+2655 x-89\right )+61151 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{65625 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 150, normalized size = 0.95 \[ \frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2025000 x^{5}+3942000 x^{4}+1279350 x^{3}-1023960 x^{2}-459210 x -61151 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+30065 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+16020\right )}{3937500 x^{3}+3018750 x^{2}-918750 x -787500} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}}{\sqrt {5\,x+3}} \,d x \]
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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