Optimal. Leaf size=117 \[ \frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{\sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{\sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)} \]
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Rubi [A] time = 0.113681, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {711, 68} \[ \frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{\sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{\sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)} \]
Antiderivative was successfully verified.
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Rule 711
Rule 68
Rubi steps
\begin{align*} \int \frac{(1+4 x)^m}{1-5 x+3 x^2} \, dx &=\int \left (-\frac{6 (1+4 x)^m}{\sqrt{13} \left (5+\sqrt{13}-6 x\right )}-\frac{6 (1+4 x)^m}{\sqrt{13} \left (-5+\sqrt{13}+6 x\right )}\right ) \, dx\\ &=-\frac{6 \int \frac{(1+4 x)^m}{5+\sqrt{13}-6 x} \, dx}{\sqrt{13}}-\frac{6 \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx}{\sqrt{13}}\\ &=\frac{3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13-2 \sqrt{13}}\right )}{\sqrt{13} \left (13-2 \sqrt{13}\right ) (1+m)}-\frac{3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13+2 \sqrt{13}}\right )}{\sqrt{13} \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.107085, size = 94, normalized size = 0.8 \[ \frac{(4 x+1)^{m+1} \left (\left (13+2 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )+\left (2 \sqrt{13}-13\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )\right )}{39 \sqrt{13} (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.245, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m}}{3\,{x}^{2}-5\,x+1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (4 x + 1\right )^{m}}{3 x^{2} - 5 x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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