Optimal. Leaf size=165 \[ -\frac{3 \left (416-135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (416+135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{123 (4 x+1)^{m+1}}{16 (m+1)}+\frac{9 (4 x+1)^{m+2}}{16 (m+2)} \]
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Rubi [A] time = 0.152348, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1628, 68} \[ -\frac{3 \left (416-135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (416+135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{123 (4 x+1)^{m+1}}{16 (m+1)}+\frac{9 (4 x+1)^{m+2}}{16 (m+2)} \]
Antiderivative was successfully verified.
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Rule 1628
Rule 68
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3 (1+4 x)^m}{1-5 x+3 x^2} \, dx &=\int \left (\frac{123}{4} (1+4 x)^m+\frac{9}{4} (1+4 x)^{1+m}+\frac{\left (192+\frac{810}{\sqrt{13}}\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (192-\frac{810}{\sqrt{13}}\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{123 (1+4 x)^{1+m}}{16 (1+m)}+\frac{9 (1+4 x)^{2+m}}{16 (2+m)}+\frac{1}{13} \left (6 \left (416-135 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx+\frac{1}{13} \left (6 \left (416+135 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx\\ &=\frac{123 (1+4 x)^{1+m}}{16 (1+m)}+\frac{9 (1+4 x)^{2+m}}{16 (2+m)}-\frac{3 \left (416-135 \sqrt{13}\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13-2 \sqrt{13}}\right )}{13 \left (13-2 \sqrt{13}\right ) (1+m)}-\frac{3 \left (416+135 \sqrt{13}\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13+2 \sqrt{13}}\right )}{13 \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.115976, size = 117, normalized size = 0.71 \[ \frac{(4 x+1)^{m+1} \left (16 \left (71 \sqrt{13}-146\right ) (m+2) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )-16 \left (146+71 \sqrt{13}\right ) (m+2) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )+117 (4 m (3 x+11)+12 x+85)\right )}{624 \left (m^2+3 m+2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.305, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m} \left ( 2+3\,x \right ) ^{3}}{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}{\left (3 \, x + 2\right )}^{3}}{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 (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}{\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 (3 x + 2\right )^{3} \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}{\left (3 \, x + 2\right )}^{3}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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