Optimal. Leaf size=142 \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1561 (3 x+2)^3}{726 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {7723 \sqrt {1-2 x} (3 x+2)^2}{39930 \sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (16227780 x+39109961)}{2129600}+\frac {243189 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \begin {gather*} \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1561 (3 x+2)^3}{726 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {7723 \sqrt {1-2 x} (3 x+2)^2}{39930 \sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (16227780 x+39109961)}{2129600}+\frac {243189 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (211+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-17212-\frac {117321 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {\left (-\frac {1244193}{4}-\frac {4056945 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}
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Mathematica [C] time = 3.99, size = 246, normalized size = 1.73 \begin {gather*} \frac {1}{20} \left (\frac {1673 \left (50000 (1-2 x)^{3/2} \left (6 x^2+x-2\right )^3 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {5}{11} (2 x-1)\right )+33 \sqrt {55} \left (\sqrt {10-20 x} \sqrt {5 x+3} \left (21600 x^5-43740 x^4+79209 x^3+272474 x^2+678368 x+129582\right )-3993 \left (513 x^3+2538 x^2+936 x+334\right ) \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )\right )}{52707600 \sqrt {22} (1-2 x)^3}+\frac {153 \left (326700 x^2-824990 x+120879 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-612430\right )}{96800 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {3 (3 x+2)^4}{(1-2 x)^{3/2} \sqrt {5 x+3}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.20, size = 141, normalized size = 0.99 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {1920 (1-2 x)^4}{(5 x+3)^4}-\frac {4855205025 (1-2 x)^3}{(5 x+3)^3}-\frac {3236506646 (1-2 x)^2}{(5 x+3)^2}-\frac {412972000 (1-2 x)}{5 x+3}+26891200\right )}{6388800 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {243189 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1600 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.66, size = 111, normalized size = 0.78 \begin {gather*} -\frac {971053677 \, \sqrt {10} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (77623920 \, x^{4} + 536898780 \, x^{3} - 1790987404 \, x^{2} - 525679641 \, x + 435258129\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{127776000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 144, normalized size = 1.01 \begin {gather*} \frac {243189}{16000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1663750 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (323433 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 271 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 3237172310 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 53407238379 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3993000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{831875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 168, normalized size = 1.18 \begin {gather*} \frac {\sqrt {-2 x +1}\, \left (-1552478400 \sqrt {-10 x^{2}-x +3}\, x^{4}+19421073540 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-10737975600 \sqrt {-10 x^{2}-x +3}\, x^{3}-7768429416 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+35819748080 \sqrt {-10 x^{2}-x +3}\, x^{2}-6797375739 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+10513592820 \sqrt {-10 x^{2}-x +3}\, x +2913161031 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-8705162580 \sqrt {-10 x^{2}-x +3}\right )}{127776000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 112, normalized size = 0.79 \begin {gather*} \frac {243 \, x^{3}}{40 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {243189}{32000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {7209 \, x^{2}}{160 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {751566017 \, x}{6388800 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {638622829}{6388800 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {16807}{528 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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