Optimal. Leaf size=142 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac {2051 \sqrt {5 x+3} (3 x+2)^3}{726 \sqrt {1-2 x}}-\frac {23909 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{4840}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (50124540 x+120791143)}{774400}+\frac {8261577 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6400 \sqrt {10}} \]
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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 = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 150, 153, 147, 54, 216} \begin {gather*} \frac {7 \sqrt {5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac {2051 \sqrt {5 x+3} (3 x+2)^3}{726 \sqrt {1-2 x}}-\frac {23909 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{4840}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (50124540 x+120791143)}{774400}+\frac {8261577 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6400 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 150
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (281+\frac {927 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2051 (2+3 x)^3 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-32787-\frac {215181 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {23909 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{4840}-\frac {2051 (2+3 x)^3 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {\int \frac {(2+3 x) \left (\frac {11526957}{4}+\frac {37593405 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10890}\\ &=-\frac {23909 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{4840}-\frac {2051 (2+3 x)^3 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (120791143+50124540 x)}{774400}+\frac {8261577 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{12800}\\ &=-\frac {23909 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{4840}-\frac {2051 (2+3 x)^3 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (120791143+50124540 x)}{774400}+\frac {8261577 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6400 \sqrt {5}}\\ &=-\frac {23909 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{4840}-\frac {2051 (2+3 x)^3 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (120791143+50124540 x)}{774400}+\frac {8261577 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 95, normalized size = 0.67 \begin {gather*} \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (18817920 x^4+101146320 x^3+359461476 x^2-1261070176 x+452899509\right )+2998952451 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{23232000 \sqrt {1-2 x} (2 x-1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 141, normalized size = 0.99 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {74973803595 (1-2 x)^4}{(5 x+3)^4}-\frac {79971970640 (1-2 x)^3}{(5 x+3)^3}-\frac {26390006196 (1-2 x)^2}{(5 x+3)^2}-\frac {2189712000 (1-2 x)}{5 x+3}+107564800\right )}{2323200 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}-\frac {8261577 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{6400 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 101, normalized size = 0.71 \begin {gather*} -\frac {2998952451 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\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 (18817920 \, x^{4} + 101146320 \, x^{3} + 359461476 \, x^{2} - 1261070176 \, x + 452899509\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{46464000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 97, normalized size = 0.68 \begin {gather*} \frac {8261577}{64000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 119 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 27809 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 9996528778 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 164942367909 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1452000000 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 154, normalized size = 1.08 \begin {gather*} \frac {\left (-376358400 \sqrt {-10 x^{2}-x +3}\, x^{4}-2022926400 \sqrt {-10 x^{2}-x +3}\, x^{3}+11995809804 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-7189229520 \sqrt {-10 x^{2}-x +3}\, x^{2}-11995809804 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+25221403520 \sqrt {-10 x^{2}-x +3}\, x +2998952451 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9057990180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{46464000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 108, normalized size = 0.76 \begin {gather*} -\frac {81}{40} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {8261577}{128000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {4131}{320} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {326943}{6400} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {16807 \, \sqrt {-10 \, x^{2} - x + 3}}{528 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1020425 \, \sqrt {-10 \, x^{2} - x + 3}}{5808 \, {\left (2 \, x - 1\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}\,\sqrt {5\,x+3}} \,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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