Optimal. Leaf size=180 \[ \frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {245529161 \sqrt {5 x+3} \sqrt {1-2 x}}{169344 (3 x+2)}+\frac {2347559 \sqrt {5 x+3} \sqrt {1-2 x}}{12096 (3 x+2)^2}+\frac {67187 \sqrt {5 x+3} \sqrt {1-2 x}}{2160 (3 x+2)^3}+\frac {2023 \sqrt {5 x+3} \sqrt {1-2 x}}{360 (3 x+2)^4}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \begin {gather*} \frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {245529161 \sqrt {5 x+3} \sqrt {1-2 x}}{169344 (3 x+2)}+\frac {2347559 \sqrt {5 x+3} \sqrt {1-2 x}}{12096 (3 x+2)^2}+\frac {67187 \sqrt {5 x+3} \sqrt {1-2 x}}{2160 (3 x+2)^3}+\frac {2023 \sqrt {5 x+3} \sqrt {1-2 x}}{360 (3 x+2)^4}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (\frac {421}{2}-190 x\right ) \sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}-\frac {1}{180} \int \frac {-\frac {79903}{4}+28825 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}-\frac {\int \frac {-\frac {14846615}{8}+2351545 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{3780}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}-\frac {\int \frac {-\frac {1768979345}{16}+\frac {410822825 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{52920}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {\int -\frac {98318061765}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{370440}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}+\frac {104040277 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}+\frac {104040277 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 135, normalized size = 0.75 \begin {gather*} \frac {1}{35} \left (\frac {78167 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (15707 x^2+21638 x+7488\right )}{(3 x+2)^3}-19965 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{18816}+\frac {963 \sqrt {5 x+3} (1-2 x)^{7/2}}{56 (3 x+2)^4}+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{(3 x+2)^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 138, normalized size = 0.77 \begin {gather*} \frac {1331 \sqrt {1-2 x} \left (\frac {3531495 (1-2 x)^4}{(5 x+3)^4}+\frac {61736570 (1-2 x)^3}{(5 x+3)^3}+\frac {490263424 (1-2 x)^2}{(5 x+3)^2}+\frac {1876789670 (1-2 x)}{5 x+3}+2815184505\right )}{94080 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 131, normalized size = 0.73 \begin {gather*} -\frac {1560604155 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.08, size = 426, normalized size = 2.37 \begin {gather*} \frac {104040277}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1331 \, \sqrt {10} {\left (706299 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 493892560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 156884295680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 24022907776000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1441374466560000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5765497866240000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9408 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (379226809665 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1264089365550 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+154683371430 \sqrt {-10 x^{2}-x +3}\, x^{4}+1685452487400 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+419390813940 \sqrt {-10 x^{2}-x +3}\, x^{3}+1123634991600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+426661359656 \sqrt {-10 x^{2}-x +3}\, x^{2}+374544997200 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+193043476304 \sqrt {-10 x^{2}-x +3}\, x +49939332960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+32779018944 \sqrt {-10 x^{2}-x +3}\right )}{1317120 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 184, normalized size = 1.02 \begin {gather*} \frac {104040277}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{45 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {637 \, \sqrt {-10 \, x^{2} - x + 3}}{120 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {67187 \, \sqrt {-10 \, x^{2} - x + 3}}{2160 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {2347559 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {245529161 \, \sqrt {-10 \, x^{2} - x + 3}}{169344 \, {\left (3 \, x + 2\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 (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,\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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