Optimal. Leaf size=202 \[ \frac {9 (1+4 x)^{1+m}}{4 (1+m)}+\frac {(844-2355 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac {\left (13689-\sqrt {13} \left (297+4474 m-1570 \sqrt {13} m\right )\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13-2 \sqrt {13}}\right )}{169 \left (13-2 \sqrt {13}\right ) (1+m)}-\frac {\left (13689+\sqrt {13} \left (297+4474 m+1570 \sqrt {13} m\right )\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13+2 \sqrt {13}}\right )}{169 \left (13+2 \sqrt {13}\right ) (1+m)} \]
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Rubi [A]
time = 0.19, 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 = {1662, 1642, 70}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 1642
Rule 1662
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 = 0.35, size = 251, normalized size = 1.24 \begin {gather*} \frac {(1+4 x)^{1+m} \left (\frac {13689}{4+4 m}+\frac {39 (844-2355 x)}{1-5 x+3 x^2}-\frac {1053 \left (-117+128 \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 ) (1+m)}-\frac {1053 \left (117+128 \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 ) (1+m)}-\frac {-\left (\left (-14679 \left (2+\sqrt {13}\right )+2 \left (-5731+667 \sqrt {13}\right ) m\right ) \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13-2 \sqrt {13}}\right )\right )+\left (-14679 \left (-2+\sqrt {13}\right )+2 \left (5731+667 \sqrt {13}\right ) m\right ) \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13+2 \sqrt {13}}\right )}{1+m}\right )}{1521} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (2+3 x \right )^{4} \left (1+4 x \right )^{m}}{\left (3 x^{2}-5 x +1\right )^{2}}\, dx\]
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 \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 (3 x + 2\right )^{4} \left (4 x + 1\right )^{m}}{\left (3 x^{2} - 5 x + 1\right )^{2}}\, 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 = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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