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 = 0.29, antiderivative size = 202, normalized size of antiderivative = 1.00, 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 68
Rule 1628
Rule 1648
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*}
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Mathematica [A] time = 0.44, 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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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\left (4 \, x + 1\right )}^{m}}{9 \, x^{4} - 30 \, x^{3} + 31 \, x^{2} - 10 \, x + 1}, x\right ) \]
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 \[ \int \frac {{\left (4 \, x + 1\right )}^{m} {\left (3 \, x + 2\right )}^{4}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x +2\right )^{4} \left (4 x +1\right )^{m}}{\left (3 x^{2}-5 x +1\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (4 \, x + 1\right )}^{m} {\left (3 \, x + 2\right )}^{4}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (3\,x+2\right )}^4\,{\left (4\,x+1\right )}^m}{{\left (3\,x^2-5\,x+1\right )}^2} \,d x \]
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 \[ \int \frac {\left (3 x + 2\right )^{4} \left (4 x + 1\right )^{m}}{\left (3 x^{2} - 5 x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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