Optimal. Leaf size=199 \[ \frac{27 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}+\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{12 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{425 (m+1)} \]
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Rubi [A] time = 0.221183, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {960, 68, 830} \[ \frac{27 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}+\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{12 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{425 (m+1)} \]
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)^2 \left (1-5 x+3 x^2\right )} \, dx &=\int \left (\frac{3 (1+4 x)^m}{17 (2+3 x)^2}+\frac{27 (1+4 x)^m}{289 (2+3 x)}+\frac{(46-27 x) (1+4 x)^m}{289 \left (1-5 x+3 x^2\right )}\right ) \, dx\\ &=\frac{1}{289} \int \frac{(46-27 x) (1+4 x)^m}{1-5 x+3 x^2} \, dx+\frac{27}{289} \int \frac{(1+4 x)^m}{2+3 x} \, dx+\frac{3}{17} \int \frac{(1+4 x)^m}{(2+3 x)^2} \, dx\\ &=\frac{27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac{12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{425 (1+m)}+\frac{1}{289} \int \left (\frac{\left (-27+\frac{141}{\sqrt{13}}\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (-27-\frac{141}{\sqrt{13}}\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac{12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{425 (1+m)}-\frac{\left (3 \left (117-47 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx}{3757}-\frac{\left (3 \left (117+47 \sqrt{13}\right )\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx}{3757}\\ &=\frac{27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac{3 \left (117+47 \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 )}{7514 \left (13-2 \sqrt{13}\right ) (1+m)}+\frac{3 \left (117-47 \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 )}{7514 \left (13+2 \sqrt{13}\right ) (1+m)}+\frac{12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac{3}{5} (1+4 x)\right )}{425 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.134042, size = 152, normalized size = 0.76 \[ \frac{(4 x+1)^{m+1} \left (10530 \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )+25 \left (211+65 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )-1625 \sqrt{13} \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )+5275 \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )+15912 \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )\right )}{563550 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.34, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m}}{ \left ( 2+3\,x \right ) ^{2} \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 )}^{2}}\,{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}}{27 \, x^{4} - 9 \, x^{3} - 39 \, x^{2} - 8 \, x + 4}, 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 )^{2} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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