Optimal. Leaf size=199 \[ \frac {27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {3 \left (117+47 \sqrt {13}\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13-2 \sqrt {13}}\right )}{7514 \left (13-2 \sqrt {13}\right ) (1+m)}+\frac {3 \left (117-47 \sqrt {13}\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13+2 \sqrt {13}}\right )}{7514 \left (13+2 \sqrt {13}\right ) (1+m)}+\frac {12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{425 (1+m)} \]
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Rubi [A]
time = 0.14, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {974, 70, 844}
\begin {gather*} \frac {27 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {3}{5} (4 x+1)\right )}{1445 (m+1)}+\frac {3 \left (117+47 \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 )}{7514 \left (13-2 \sqrt {13}\right ) (m+1)}+\frac {3 \left (117-47 \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 )}{7514 \left (13+2 \sqrt {13}\right ) (m+1)}+\frac {12 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {3}{5} (4 x+1)\right )}{425 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 844
Rule 974
Rubi steps
\begin {align*} \int \frac {(1+4 x)^m}{(2+3 x)^2 \left (1-5 x+3 x^2\right )} \, dx &=\int \left (\frac {3 (1+4 x)^m}{17 (2+3 x)^2}+\frac {27 (1+4 x)^m}{289 (2+3 x)}+\frac {(46-27 x) (1+4 x)^m}{289 \left (1-5 x+3 x^2\right )}\right ) \, dx\\ &=\frac {1}{289} \int \frac {(46-27 x) (1+4 x)^m}{1-5 x+3 x^2} \, dx+\frac {27}{289} \int \frac {(1+4 x)^m}{2+3 x} \, dx+\frac {3}{17} \int \frac {(1+4 x)^m}{(2+3 x)^2} \, dx\\ &=\frac {27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{425 (1+m)}+\frac {1}{289} \int \left (\frac {\left (-27+\frac {141}{\sqrt {13}}\right ) (1+4 x)^m}{-5-\sqrt {13}+6 x}+\frac {\left (-27-\frac {141}{\sqrt {13}}\right ) (1+4 x)^m}{-5+\sqrt {13}+6 x}\right ) \, dx\\ &=\frac {27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{425 (1+m)}-\frac {\left (3 \left (117-47 \sqrt {13}\right )\right ) \int \frac {(1+4 x)^m}{-5-\sqrt {13}+6 x} \, dx}{3757}-\frac {\left (3 \left (117+47 \sqrt {13}\right )\right ) \int \frac {(1+4 x)^m}{-5+\sqrt {13}+6 x} \, dx}{3757}\\ &=\frac {27 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {3 \left (117+47 \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 )}{7514 \left (13-2 \sqrt {13}\right ) (1+m)}+\frac {3 \left (117-47 \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 )}{7514 \left (13+2 \sqrt {13}\right ) (1+m)}+\frac {12 (1+4 x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{425 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 152, normalized size = 0.76 \begin {gather*} \frac {(1+4 x)^{1+m} \left (10530 \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )+25 \left (211+65 \sqrt {13}\right ) \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13-2 \sqrt {13}}\right )+5275 \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13+2 \sqrt {13}}\right )-1625 \sqrt {13} \, _2F_1\left (1,1+m;2+m;\frac {3+12 x}{13+2 \sqrt {13}}\right )+15912 \, _2F_1\left (2,1+m;2+m;-\frac {3}{5} (1+4 x)\right )\right )}{563550 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (1+4 x \right )^{m}}{\left (2+3 x \right )^{2} \left (3 x^{2}-5 x +1\right )}\, dx\]
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 \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}}{\left (3 x + 2\right )^{2} \cdot \left (3 x^{2} - 5 x + 1\right )}\, 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}{{\left (3\,x+2\right )}^2\,\left (3\,x^2-5\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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