Optimal. Leaf size=93 \[ -\frac {\sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac {84 \sqrt {1-2 x} (3 x+2)^2}{1375}-\frac {21 \sqrt {1-2 x} (375 x+1144)}{6875}-\frac {266 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 93, 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, 153, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac {84 \sqrt {1-2 x} (3 x+2)^2}{1375}-\frac {21 \sqrt {1-2 x} (375 x+1144)}{6875}-\frac {266 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {1}{55} \int \frac {(-77-84 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}+\frac {\int \frac {(2+3 x) (4858+7875 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1375}\\ &=-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}+\frac {133 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6875}\\ &=-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}-\frac {133 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6875}\\ &=-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}-\frac {266 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 63, normalized size = 0.68 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} \left (22275 x^3+82665 x^2+171765 x+78112\right )}{5 x+3}-266 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 79, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {1-2 x} \left (22275 (1-2 x)^3-232155 (1-2 x)^2+1084545 (1-2 x)-1499561\right )}{27500 (5 (1-2 x)-11)}-\frac {266 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 69, normalized size = 0.74 \begin {gather*} \frac {133 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (22275 \, x^{3} + 82665 \, x^{2} + 171765 \, x + 78112\right )} \sqrt {-2 \, x + 1}}{378125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 90, normalized size = 0.97 \begin {gather*} -\frac {81}{500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {333}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {133}{378125} \, \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 {12393}{2500} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{6875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.68 \begin {gather*} -\frac {266 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{378125}-\frac {81 \left (-2 x +1\right )^{\frac {5}{2}}}{500}+\frac {333 \left (-2 x +1\right )^{\frac {3}{2}}}{250}-\frac {12393 \sqrt {-2 x +1}}{2500}+\frac {2 \sqrt {-2 x +1}}{34375 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 80, normalized size = 0.86 \begin {gather*} -\frac {81}{500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {333}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {133}{378125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {12393}{2500} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{6875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 64, normalized size = 0.69 \begin {gather*} \frac {333\,{\left (1-2\,x\right )}^{3/2}}{250}-\frac {12393\,\sqrt {1-2\,x}}{2500}-\frac {2\,\sqrt {1-2\,x}}{34375\,\left (2\,x+\frac {6}{5}\right )}-\frac {81\,{\left (1-2\,x\right )}^{5/2}}{500}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,266{}\mathrm {i}}{378125} \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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