Optimal. Leaf size=202 \[ -\frac{\left (13689-\sqrt{13} \left (-1570 \sqrt{13} m+4474 m+297\right )\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{169 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{\left (\sqrt{13} \left (1570 \sqrt{13} m+4474 m+297\right )+13689\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{169 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(844-2355 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )}+\frac{9 (4 x+1)^{m+1}}{4 (m+1)} \]
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Rubi [A] time = 0.294886, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1648, 1628, 68} \[ -\frac{\left (13689-\sqrt{13} \left (-1570 \sqrt{13} m+4474 m+297\right )\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{169 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{\left (\sqrt{13} \left (1570 \sqrt{13} m+4474 m+297\right )+13689\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{169 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(844-2355 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )}+\frac{9 (4 x+1)^{m+1}}{4 (m+1)} \]
Antiderivative was successfully verified.
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Rule 1648
Rule 1628
Rule 68
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4 (1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx &=\frac{(844-2355 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \frac{(1+4 x)^m \left (13 (4617+3376 m)-39 (1521+3140 m) x-13689 x^2\right )}{1-5 x+3 x^2} \, dx\\ &=\frac{(844-2355 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \left (-4563 (1+4 x)^m+\frac{\left (-82134-122460 m-6 \sqrt{13} (297+4474 m)\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (-82134-122460 m+6 \sqrt{13} (297+4474 m)\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{9 (1+4 x)^{1+m}}{4 (1+m)}+\frac{(844-2355 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \left (-82134-122460 m+6 \sqrt{13} (297+4474 m)\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx+\frac{1}{507} \left (82134+122460 m+6 \sqrt{13} (297+4474 m)\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx\\ &=\frac{9 (1+4 x)^{1+m}}{4 (1+m)}+\frac{(844-2355 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{\left (13689-\sqrt{13} \left (297+4474 m-1570 \sqrt{13} m\right )\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13-2 \sqrt{13}}\right )}{169 \left (13-2 \sqrt{13}\right ) (1+m)}-\frac{\left (13689+\sqrt{13} \left (297+4474 m+1570 \sqrt{13} m\right )\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13+2 \sqrt{13}}\right )}{169 \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.308564, size = 251, normalized size = 1.24 \[ \frac{(4 x+1)^{m+1} \left (-\frac{1053 \left (128 \sqrt{13}-117\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )}{\left (2 \sqrt{13}-13\right ) (m+1)}-\frac{1053 \left (117+128 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )}{\left (13+2 \sqrt{13}\right ) (m+1)}-\frac{\left (2 \left (5731+667 \sqrt{13}\right ) m-14679 \left (\sqrt{13}-2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )-\left (2 \left (667 \sqrt{13}-5731\right ) m-14679 \left (2+\sqrt{13}\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )}{m+1}+\frac{13689}{4 m+4}+\frac{39 (844-2355 x)}{3 x^2-5 x+1}\right )}{1521} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m} \left ( 2+3\,x \right ) ^{4}}{ \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, 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 )}^{4}}{{\left (3 \, x^{2} - 5 \, x + 1\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 (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}{\left (4 \, x + 1\right )}^{m}}{9 \, x^{4} - 30 \, x^{3} + 31 \, x^{2} - 10 \, x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{4}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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