Optimal. Leaf size=100 \[ -\frac {\sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac {84 \sqrt {1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac {63 \sqrt {1-2 x} (75 x+352)}{30250}-\frac {2667 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 100, 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, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac {84 \sqrt {1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac {63 \sqrt {1-2 x} (75 x+352)}{30250}-\frac {2667 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac {1}{110} \int \frac {(-147-189 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac {\int \frac {(-5502-4725 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac {63 \sqrt {1-2 x} (352+75 x)}{30250}+\frac {2667 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{30250}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac {63 \sqrt {1-2 x} (352+75 x)}{30250}-\frac {2667 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{30250}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{110 (3+5 x)^2}-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{3025 (3+5 x)}-\frac {63 \sqrt {1-2 x} (352+75 x)}{30250}-\frac {2667 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 63, normalized size = 0.63 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} \left (163350 x^3+784080 x^2+764745 x+211864\right )}{(5 x+3)^2}-5334 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1663750} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 79, normalized size = 0.79 \begin {gather*} \frac {\left (81675 (1-2 x)^3-1029105 (1-2 x)^2+3342675 (1-2 x)-3242701\right ) \sqrt {1-2 x}}{30250 (5 (1-2 x)-11)^2}-\frac {2667 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 79, normalized size = 0.79 \begin {gather*} \frac {2667 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (163350 \, x^{3} + 784080 \, x^{2} + 764745 \, x + 211864\right )} \sqrt {-2 \, x + 1}}{1663750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 86, normalized size = 0.86 \begin {gather*} \frac {27}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2667}{1663750} \, \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 {1107}{1250} \, \sqrt {-2 \, x + 1} + \frac {1335 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2959 \, \sqrt {-2 \, x + 1}}{302500 \, {\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 = 0.01, size = 66, normalized size = 0.66 \begin {gather*} -\frac {2667 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{831875}+\frac {27 \left (-2 x +1\right )^{\frac {3}{2}}}{250}-\frac {1107 \sqrt {-2 x +1}}{1250}+\frac {\frac {267 \left (-2 x +1\right )^{\frac {3}{2}}}{15125}-\frac {269 \sqrt {-2 x +1}}{6875}}{\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 = 1.34, size = 92, normalized size = 0.92 \begin {gather*} \frac {27}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2667}{1663750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1107}{1250} \, \sqrt {-2 \, x + 1} + \frac {1335 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2959 \, \sqrt {-2 \, x + 1}}{75625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 74, normalized size = 0.74 \begin {gather*} \frac {27\,{\left (1-2\,x\right )}^{3/2}}{250}-\frac {1107\,\sqrt {1-2\,x}}{1250}-\frac {\frac {269\,\sqrt {1-2\,x}}{171875}-\frac {267\,{\left (1-2\,x\right )}^{3/2}}{378125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2667{}\mathrm {i}}{831875} \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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