Optimal. Leaf size=160 \[ -\frac {68}{343} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {5256 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 \sqrt {3 x+2}}+\frac {54 \sqrt {1-2 x} \sqrt {5 x+3}}{539 (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 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 = {104, 152, 158, 113, 119} \[ \frac {5256 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 \sqrt {3 x+2}}+\frac {54 \sqrt {1-2 x} \sqrt {5 x+3}}{539 (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}-\frac {68}{343} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {87}{2}-45 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}-\frac {4 \int \frac {-522+\frac {405 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {8 \int \frac {-\frac {25335}{4}-9855 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11319}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {102}{343} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {5256 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3773}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {68}{343} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 99, normalized size = 0.62 \[ \frac {2 \left (3 \sqrt {2} \left (292 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-105 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {5 x+3} \left (-15768 x^2-3006 x+5543\right )}{\sqrt {1-2 x} (3 x+2)^{3/2}}\right )}{3773} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.37 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (78840 x^{3}+62334 x^{2}-2628 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+945 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-18697 x -1752 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+630 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-16629\right )}{3773 \left (3 x +2\right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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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