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 = 0.20, antiderivative size = 164, normalized size of antiderivative = 1.00, 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 68
Rule 830
Rule 960
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*}
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Mathematica [A] time = 0.19, 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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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x + 1\right )}^{m}}{9 \, x^{3} - 9 \, x^{2} - 7 \, x + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )} {\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x +1\right )^{m}}{\left (3 x +2\right ) \left (3 x^{2}-5 x +1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )} {\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (4\,x+1\right )}^m}{\left (3\,x+2\right )\,\left (3\,x^2-5\,x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x + 1\right )^{m}}{\left (3 x + 2\right ) \left (3 x^{2} - 5 x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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