Optimal. Leaf size=107 \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac {73 (3 x+2)^2}{3630 \sqrt {1-2 x} (5 x+3)^2}-\frac {2133933 x+1287116}{2196150 \sqrt {1-2 x} (5 x+3)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{366025 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 149, 144, 63, 206} \begin {gather*} \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac {73 (3 x+2)^2}{3630 \sqrt {1-2 x} (5 x+3)^2}-\frac {2133933 x+1287116}{2196150 \sqrt {1-2 x} (5 x+3)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{366025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 144
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1}{33} \int \frac {(2+3 x)^2 (43+96 x)}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=-\frac {73 (2+3 x)^2}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {\int \frac {(2+3 x) (3056+6117 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=-\frac {73 (2+3 x)^2}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1287116+2133933 x}{2196150 \sqrt {1-2 x} (3+5 x)}+\frac {14423 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{732050}\\ &=-\frac {73 (2+3 x)^2}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1287116+2133933 x}{2196150 \sqrt {1-2 x} (3+5 x)}-\frac {14423 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{732050}\\ &=-\frac {73 (2+3 x)^2}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1287116+2133933 x}{2196150 \sqrt {1-2 x} (3+5 x)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{366025 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 95, normalized size = 0.89 \begin {gather*} -\frac {-4634 (5 x+3)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {5}{11} (1-2 x)\right )+1548 (2 x-1) (5 x+3)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )-33 \left (490050 x^3+567140 x^2+151043 x-7696\right )}{998250 (1-2 x)^{3/2} (5 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 79, normalized size = 0.74 \begin {gather*} \frac {-17356125 (1-2 x)^3+92891843 (1-2 x)^2-156673825 (1-2 x)+79893275}{2196150 (5 (1-2 x)-11)^2 (1-2 x)^{3/2}}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{366025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 99, normalized size = 0.93 \begin {gather*} \frac {43269 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (34712250 \, x^{3} + 40823468 \, x^{2} + 11479257 \, x - 311208\right )} \sqrt {-2 \, x + 1}}{120788250 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 89, normalized size = 0.83 \begin {gather*} \frac {14423}{40262750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {343 \, {\left (81 \, x - 2\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {125 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 277 \, \sqrt {-2 \, x + 1}}{133100 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.62 \begin {gather*} -\frac {14423 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{20131375}+\frac {2401}{7986 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {9261}{29282 \sqrt {-2 x +1}}+\frac {\frac {5 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}-\frac {277 \sqrt {-2 x +1}}{33275}}{\left (-10 x -6\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 92, normalized size = 0.86 \begin {gather*} \frac {14423}{40262750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17356125 \, {\left (2 \, x - 1\right )}^{3} + 92891843 \, {\left (2 \, x - 1\right )}^{2} + 313347650 \, x - 76780550}{2196150 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 71, normalized size = 0.66 \begin {gather*} \frac {\frac {51793\,x}{9075}+\frac {8444713\,{\left (2\,x-1\right )}^2}{4991250}+\frac {46283\,{\left (2\,x-1\right )}^3}{146410}-\frac {12691}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}}-\frac {14423\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{20131375} \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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