Optimal. Leaf size=107 \[ -\frac {71 \sqrt {1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}+\frac {9 \sqrt {1-2 x} (3044+5093 x)}{13310 (3+5 x)}-\frac {111 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \]
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Rubi [A]
time = 0.02, 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 = {100, 154, 151,
65, 212} \begin {gather*} \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {71 \sqrt {1-2 x} (3 x+2)^2}{1210 (5 x+3)^2}+\frac {9 \sqrt {1-2 x} (5093 x+3044)}{13310 (5 x+3)}-\frac {111 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 151
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(2+3 x)^2 (47+102 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\int \frac {(2+3 x) (3636+6945 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}+\frac {9 \sqrt {1-2 x} (3044+5093 x)}{13310 (3+5 x)}+\frac {111 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}+\frac {9 \sqrt {1-2 x} (3044+5093 x)}{13310 (3+5 x)}-\frac {111 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2662}\\ &=-\frac {71 \sqrt {1-2 x} (2+3 x)^2}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^2}+\frac {9 \sqrt {1-2 x} (3044+5093 x)}{13310 (3+5 x)}-\frac {111 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 63, normalized size = 0.59 \begin {gather*} \frac {\frac {11 \left (146824+411911 x+149298 x^2-215622 x^3\right )}{\sqrt {1-2 x} (3+5 x)^2}-222 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{146410} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 66, normalized size = 0.62
method | result | size |
risch | \(-\frac {215622 x^{3}-149298 x^{2}-411911 x -146824}{13310 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {111 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}\) | \(51\) |
derivativedivides | \(\frac {81 \sqrt {1-2 x}}{250}+\frac {\frac {271 \left (1-2 x \right )^{\frac {3}{2}}}{33275}-\frac {273 \sqrt {1-2 x}}{15125}}{\left (-6-10 x \right )^{2}}-\frac {111 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {2401}{2662 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {81 \sqrt {1-2 x}}{250}+\frac {\frac {271 \left (1-2 x \right )^{\frac {3}{2}}}{33275}-\frac {273 \sqrt {1-2 x}}{15125}}{\left (-6-10 x \right )^{2}}-\frac {111 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {2401}{2662 \sqrt {1-2 x}}\) | \(66\) |
trager | \(\frac {\left (215622 x^{3}-149298 x^{2}-411911 x -146824\right ) \sqrt {1-2 x}}{13310 \left (3+5 x \right )^{2} \left (-1+2 x \right )}+\frac {111 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{146410}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 92, normalized size = 0.86 \begin {gather*} \frac {111}{146410} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {81}{250} \, \sqrt {-2 \, x + 1} + \frac {7505835 \, {\left (2 \, x - 1\right )}^{2} + 66039512 \, x + 3295369}{332750 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 89, normalized size = 0.83 \begin {gather*} \frac {111 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11 \, {\left (215622 \, x^{3} - 149298 \, x^{2} - 411911 \, x - 146824\right )} \sqrt {-2 \, x + 1}}{146410 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.27, size = 86, normalized size = 0.80 \begin {gather*} \frac {111}{146410} \, \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 {81}{250} \, \sqrt {-2 \, x + 1} + \frac {2401}{2662 \, \sqrt {-2 \, x + 1}} + \frac {1355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3003 \, \sqrt {-2 \, x + 1}}{665500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 71, normalized size = 0.66 \begin {gather*} \frac {81\,\sqrt {1-2\,x}}{250}-\frac {111\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{73205}+\frac {\frac {3001796\,x}{378125}+\frac {1501167\,{\left (2\,x-1\right )}^2}{1663750}+\frac {299579}{756250}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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