Optimal. Leaf size=164 \[ \frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{85 (m+1)}+\frac{3 \left (13+9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{442 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (13-9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{442 \left (13+2 \sqrt{13}\right ) (m+1)} \]
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Rubi [A] time = 0.200582, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {960, 68, 830} \[ \frac{3 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{85 (m+1)}+\frac{3 \left (13+9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{442 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (13-9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{442 \left (13+2 \sqrt{13}\right ) (m+1)} \]
Antiderivative was successfully verified.
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Rule 960
Rule 68
Rule 830
Rubi steps
\begin{align*} \int \frac{(1+4 x)^m}{(2+3 x) \left (1-5 x+3 x^2\right )} \, dx &=\int \left (\frac{3 (1+4 x)^m}{17 (2+3 x)}+\frac{(7-3 x) (1+4 x)^m}{17 \left (1-5 x+3 x^2\right )}\right ) \, dx\\ &=\frac{1}{17} \int \frac{(7-3 x) (1+4 x)^m}{1-5 x+3 x^2} \, dx+\frac{3}{17} \int \frac{(1+4 x)^m}{2+3 x} \, dx\\ &=\frac{3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{85 (1+m)}+\frac{1}{17} \int \left (\frac{\left (-3+\frac{27}{\sqrt{13}}\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (-3-\frac{27}{\sqrt{13}}\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{85 (1+m)}-\frac{1}{221} \left (3 \left (13-9 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx-\frac{1}{221} \left (3 \left (13+9 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx\\ &=\frac{3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{85 (1+m)}+\frac{3 \left (13+9 \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 )}{442 \left (13-2 \sqrt{13}\right ) (1+m)}+\frac{3 \left (13-9 \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 )}{442 \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.166683, size = 110, normalized size = 0.67 \[ \frac{(4 x+1)^{m+1} \left (234 \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )+5 \left (31+11 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )+5 \left (31-11 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )\right )}{6630 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.312, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m}}{ \left ( 2+3\,x \right ) \left ( 3\,{x}^{2}-5\,x+1 \right ) }}\, 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} - 5 \, x + 1\right )}{\left (3 \, x + 2\right )}}\,{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}}{9 \, x^{3} - 9 \, x^{2} - 7 \, x + 2}, 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}}{\left (3 x + 2\right ) \left (3 x^{2} - 5 x + 1\right )}\, 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} - 5 \, x + 1\right )}{\left (3 \, x + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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