Optimal. Leaf size=173 \[ -\frac {16985 \sqrt {5 x+3}}{316932 \sqrt {1-2 x}}+\frac {605 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {\sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {3 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^3}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {25365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 173, 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 = {99, 151, 152, 12, 93, 204} \begin {gather*} -\frac {16985 \sqrt {5 x+3}}{316932 \sqrt {1-2 x}}+\frac {605 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {\sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {3 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^3}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {25365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {2}{21} \int \frac {-\frac {71}{2}-60 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {2}{441} \int \frac {-\frac {1059}{4}-405 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {\int \frac {-\frac {14385}{8}-315 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{3087}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {24465}{16}+\frac {190575 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{21609}\\ &=-\frac {16985 \sqrt {3+5 x}}{316932 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {2 \int \frac {17577945}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1663893}\\ &=-\frac {16985 \sqrt {3+5 x}}{316932 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {25365 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38416}\\ &=-\frac {16985 \sqrt {3+5 x}}{316932 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {25365 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{19208}\\ &=-\frac {16985 \sqrt {3+5 x}}{316932 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {25365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 100, normalized size = 0.58 \begin {gather*} -\frac {-7 \sqrt {5 x+3} \left (1834380 x^4+235980 x^3-1465461 x^2-39530 x+302352\right )-837045 \sqrt {7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4437048 (1-2 x)^{3/2} (3 x+2)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 138, normalized size = 0.80 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {837045 (1-2 x)^4}{(5 x+3)^4}+\frac {8385160 (1-2 x)^3}{(5 x+3)^3}+\frac {33658149 (1-2 x)^2}{(5 x+3)^2}+\frac {2521344 (1-2 x)}{5 x+3}+87808\right )}{633864 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {25365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.48, size = 131, normalized size = 0.76 \begin {gather*} -\frac {837045 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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 (1834380 \, x^{4} + 235980 \, x^{3} - 1465461 \, x^{2} - 39530 \, x + 302352\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8874096 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.15, size = 349, normalized size = 2.02 \begin {gather*} \frac {5073}{537824} \, \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 {32 \, {\left (361 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2178 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{13865775 \, {\left (2 \, x - 1\right )}^{2}} - \frac {297 \, \sqrt {10} {\left (603 \, {\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} - 235200 \, {\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 {37240000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {148960000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 305, normalized size = 1.76 \begin {gather*} \frac {\left (90400860 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+90400860 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25681320 \sqrt {-10 x^{2}-x +3}\, x^{4}-37667025 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3303720 \sqrt {-10 x^{2}-x +3}\, x^{3}-48548610 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-20516454 \sqrt {-10 x^{2}-x +3}\, x^{2}+3348180 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-553420 \sqrt {-10 x^{2}-x +3}\, x +6696360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4232928 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{8874096 \left (3 x +2\right )^{3} \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 = 1.24, size = 240, normalized size = 1.39 \begin {gather*} \frac {25365}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {84925 \, x}{316932 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {131015}{633864 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {375 \, x}{1372 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{189 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {11}{196 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {377}{3528 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {3215}{74088 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \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 {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,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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