Optimal. Leaf size=80 \[ \frac {81}{280} (1-2 x)^{7/2}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {136419 \sqrt {1-2 x}}{5000}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {88, 63, 206} \[ \frac {81}{280} (1-2 x)^{7/2}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {136419 \sqrt {1-2 x}}{5000}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx &=\int \left (\frac {136419}{5000 \sqrt {1-2 x}}-\frac {34371 \sqrt {1-2 x}}{1000}+\frac {2889}{200} (1-2 x)^{3/2}-\frac {81}{40} (1-2 x)^{5/2}+\frac {1}{625 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}+\frac {1}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}-\frac {1}{625} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 0.70 \[ -\frac {3 \sqrt {1-2 x} \left (3375 x^3+11790 x^2+19095 x+26872\right )}{4375}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 55, normalized size = 0.69 \[ -\frac {3}{4375} \, {\left (3375 \, x^{3} + 11790 \, x^{2} + 19095 \, x + 26872\right )} \sqrt {-2 \, x + 1} + \frac {1}{34375} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 90, normalized size = 1.12 \[ -\frac {81}{280} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2889}{1000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {11457}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{34375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {136419}{5000} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{34375}+\frac {11457 \left (-2 x +1\right )^{\frac {3}{2}}}{1000}-\frac {2889 \left (-2 x +1\right )^{\frac {5}{2}}}{1000}+\frac {81 \left (-2 x +1\right )^{\frac {7}{2}}}{280}-\frac {136419 \sqrt {-2 x +1}}{5000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 73, normalized size = 0.91 \[ \frac {81}{280} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2889}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {11457}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{34375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {136419}{5000} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 57, normalized size = 0.71 \[ \frac {11457\,{\left (1-2\,x\right )}^{3/2}}{1000}-\frac {136419\,\sqrt {1-2\,x}}{5000}-\frac {2889\,{\left (1-2\,x\right )}^{5/2}}{1000}+\frac {81\,{\left (1-2\,x\right )}^{7/2}}{280}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{34375} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 55.93, size = 114, normalized size = 1.42 \[ \frac {81 \left (1 - 2 x\right )^{\frac {7}{2}}}{280} - \frac {2889 \left (1 - 2 x\right )^{\frac {5}{2}}}{1000} + \frac {11457 \left (1 - 2 x\right )^{\frac {3}{2}}}{1000} - \frac {136419 \sqrt {1 - 2 x}}{5000} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} > \frac {5}{11} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} < \frac {5}{11} \end {cases}\right )}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
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