Optimal. Leaf size=117 \[ \frac {3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13-2 \sqrt {13}}\right )}{\sqrt {13} \left (13-2 \sqrt {13}\right ) (1+m)}-\frac {3 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13+2 \sqrt {13}}\right )}{\sqrt {13} \left (13+2 \sqrt {13}\right ) (1+m)} \]
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Rubi [A]
time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {725, 70}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 725
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*}
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Mathematica [A]
time = 0.11, size = 94, normalized size = 0.80 \begin {gather*} \frac {(1+4 x)^{1+m} \left (\left (13+2 \sqrt {13}\right ) \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13-2 \sqrt {13}}\right )+\left (-13+2 \sqrt {13}\right ) \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13+2 \sqrt {13}}\right )\right )}{39 \sqrt {13} (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (1+4 x \right )^{m}}{3 x^{2}-5 x +1}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (4 x + 1\right )^{m}}{3 x^{2} - 5 x + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (4\,x+1\right )}^m}{3\,x^2-5\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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