Optimal. Leaf size=120 \[ -\frac {\sqrt {1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac {201 \sqrt {1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac {1512 \sqrt {1-2 x} (3 x+2)^2}{75625}-\frac {189 \sqrt {1-2 x} (2875 x+8976)}{756250}-\frac {22113 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}} \]
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Rubi [A] time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 153, 147, 63, 206} \[ -\frac {\sqrt {1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac {201 \sqrt {1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac {1512 \sqrt {1-2 x} (3 x+2)^2}{75625}-\frac {189 \sqrt {1-2 x} (2875 x+8976)}{756250}-\frac {22113 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {1}{110} \int \frac {(-150-183 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {201 \sqrt {1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac {\int \frac {(-6237-3024 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {1512 \sqrt {1-2 x} (2+3 x)^2}{75625}-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {201 \sqrt {1-2 x} (2+3 x)^3}{6050 (3+5 x)}+\frac {\int \frac {(2+3 x) (348138+543375 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{151250}\\ &=-\frac {1512 \sqrt {1-2 x} (2+3 x)^2}{75625}-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {201 \sqrt {1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac {189 \sqrt {1-2 x} (8976+2875 x)}{756250}+\frac {22113 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{756250}\\ &=-\frac {1512 \sqrt {1-2 x} (2+3 x)^2}{75625}-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {201 \sqrt {1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac {189 \sqrt {1-2 x} (8976+2875 x)}{756250}-\frac {22113 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{756250}\\ &=-\frac {1512 \sqrt {1-2 x} (2+3 x)^2}{75625}-\frac {\sqrt {1-2 x} (2+3 x)^4}{110 (3+5 x)^2}-\frac {201 \sqrt {1-2 x} (2+3 x)^3}{6050 (3+5 x)}-\frac {189 \sqrt {1-2 x} (8976+2875 x)}{756250}-\frac {22113 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 68, normalized size = 0.57 \[ \frac {-\frac {55 \sqrt {1-2 x} \left (7350750 x^4+32506650 x^3+76970520 x^2+63610155 x+16525496\right )}{(5 x+3)^2}-44226 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{41593750} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 84, normalized size = 0.70 \[ \frac {22113 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (7350750 \, x^{4} + 32506650 \, x^{3} + 76970520 \, x^{2} + 63610155 \, x + 16525496\right )} \sqrt {-2 \, x + 1}}{41593750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 102, normalized size = 0.85 \[ -\frac {243}{2500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {513}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {22113}{41593750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {39393}{12500} \, \sqrt {-2 \, x + 1} + \frac {333 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 737 \, \sqrt {-2 \, x + 1}}{302500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.62 \[ -\frac {22113 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{20796875}-\frac {243 \left (-2 x +1\right )^{\frac {5}{2}}}{2500}+\frac {513 \left (-2 x +1\right )^{\frac {3}{2}}}{625}-\frac {39393 \sqrt {-2 x +1}}{12500}+\frac {\frac {333 \left (-2 x +1\right )^{\frac {3}{2}}}{75625}-\frac {67 \sqrt {-2 x +1}}{6875}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 101, normalized size = 0.84 \[ -\frac {243}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {513}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {22113}{41593750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {39393}{12500} \, \sqrt {-2 \, x + 1} + \frac {333 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 737 \, \sqrt {-2 \, x + 1}}{75625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 83, normalized size = 0.69 \[ \frac {513\,{\left (1-2\,x\right )}^{3/2}}{625}-\frac {39393\,\sqrt {1-2\,x}}{12500}-\frac {243\,{\left (1-2\,x\right )}^{5/2}}{2500}-\frac {\frac {67\,\sqrt {1-2\,x}}{171875}-\frac {333\,{\left (1-2\,x\right )}^{3/2}}{1890625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22113{}\mathrm {i}}{20796875} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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