Optimal. Leaf size=187 \[ -\frac {992 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2401 \sqrt {33}}+\frac {338 \sqrt {1-2 x} \sqrt {5 x+3}}{26411 \sqrt {3 x+2}}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 (3 x+2)^{3/2}}+\frac {326 \sqrt {5 x+3}}{1617 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac {338 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 152, 158, 113, 119} \[ \frac {338 \sqrt {1-2 x} \sqrt {5 x+3}}{26411 \sqrt {3 x+2}}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 (3 x+2)^{3/2}}+\frac {326 \sqrt {5 x+3}}{1617 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac {992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}-\frac {338 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {2}{21} \int \frac {-22-\frac {75 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {4203}{4}+\frac {7335 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {14247}{8}+\frac {10305 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{33957}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {29115}{8}+\frac {7605 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{237699}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}+\frac {338 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{26411}+\frac {496 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2401}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}-\frac {338 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}-\frac {992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 103, normalized size = 0.55 \[ \frac {2 \left (\sqrt {2} \left (8015 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {\sqrt {5 x+3} \left (6084 x^3-21264 x^2+727 x+7965\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}\right )}{79233} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{216 \, x^{6} + 108 \, x^{5} - 198 \, x^{4} - 71 \, x^{3} + 66 \, x^{2} + 12 \, x - 8}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.66 \[ -\frac {2 \left (-30420 x^{4}+88068 x^{3}+1014 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+48090 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+60157 x^{2}+169 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8015 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-42006 x -338 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-16030 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-23895\right ) \sqrt {-2 x +1}}{79233 \left (3 x +2\right )^{\frac {3}{2}} \left (2 x -1\right )^{2} \sqrt {5 x +3}} \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \]
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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