Optimal. Leaf size=340 \[ \frac {9 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {3}{5} (4 x+1)\right )}{1445 (m+1)}-\frac {\left (\left (62+22 \sqrt {13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13-2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13-2 \sqrt {13}\right ) (m+1)}+\frac {9 \left (13+9 \sqrt {13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13-2 \sqrt {13}}\right )}{7514 \left (13-2 \sqrt {13}\right ) (m+1)}+\frac {\left (\left (62-22 \sqrt {13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13+2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13+2 \sqrt {13}\right ) (m+1)}+\frac {9 \left (13-9 \sqrt {13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13+2 \sqrt {13}}\right )}{7514 \left (13+2 \sqrt {13}\right ) (m+1)}+\frac {(43-33 x) (4 x+1)^{m+1}}{663 \left (3 x^2-5 x+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {960, 68, 822, 830} \[ \frac {9 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {3}{5} (4 x+1)\right )}{1445 (m+1)}-\frac {\left (\left (62+22 \sqrt {13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13-2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13-2 \sqrt {13}\right ) (m+1)}+\frac {9 \left (13+9 \sqrt {13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13-2 \sqrt {13}}\right )}{7514 \left (13-2 \sqrt {13}\right ) (m+1)}+\frac {\left (\left (62-22 \sqrt {13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13+2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13+2 \sqrt {13}\right ) (m+1)}+\frac {9 \left (13-9 \sqrt {13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {3 (4 x+1)}{13+2 \sqrt {13}}\right )}{7514 \left (13+2 \sqrt {13}\right ) (m+1)}+\frac {(43-33 x) (4 x+1)^{m+1}}{663 \left (3 x^2-5 x+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 822
Rule 830
Rule 960
Rubi steps
\begin {align*} \int \frac {(1+4 x)^m}{(2+3 x) \left (1-5 x+3 x^2\right )^2} \, dx &=\int \left (\frac {9 (1+4 x)^m}{289 (2+3 x)}+\frac {(7-3 x) (1+4 x)^m}{17 \left (1-5 x+3 x^2\right )^2}-\frac {3 (-7+3 x) (1+4 x)^m}{289 \left (1-5 x+3 x^2\right )}\right ) \, dx\\ &=-\left (\frac {3}{289} \int \frac {(-7+3 x) (1+4 x)^m}{1-5 x+3 x^2} \, dx\right )+\frac {9}{289} \int \frac {(1+4 x)^m}{2+3 x} \, dx+\frac {1}{17} \int \frac {(7-3 x) (1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx\\ &=\frac {(43-33 x) (1+4 x)^{1+m}}{663 \left (1-5 x+3 x^2\right )}+\frac {9 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}-\frac {\int \frac {(1+4 x)^m (13 (81+172 m)-1716 m x)}{1-5 x+3 x^2} \, dx}{8619}-\frac {3}{289} \int \left (\frac {\left (3-\frac {27}{\sqrt {13}}\right ) (1+4 x)^m}{-5-\sqrt {13}+6 x}+\frac {\left (3+\frac {27}{\sqrt {13}}\right ) (1+4 x)^m}{-5+\sqrt {13}+6 x}\right ) \, dx\\ &=\frac {(43-33 x) (1+4 x)^{1+m}}{663 \left (1-5 x+3 x^2\right )}+\frac {9 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}-\frac {\int \left (\frac {\left (-1716 m+6 \sqrt {13} (81+62 m)\right ) (1+4 x)^m}{-5-\sqrt {13}+6 x}+\frac {\left (-1716 m-6 \sqrt {13} (81+62 m)\right ) (1+4 x)^m}{-5+\sqrt {13}+6 x}\right ) \, dx}{8619}-\frac {\left (9 \left (13-9 \sqrt {13}\right )\right ) \int \frac {(1+4 x)^m}{-5-\sqrt {13}+6 x} \, dx}{3757}-\frac {\left (9 \left (13+9 \sqrt {13}\right )\right ) \int \frac {(1+4 x)^m}{-5+\sqrt {13}+6 x} \, dx}{3757}\\ &=\frac {(43-33 x) (1+4 x)^{1+m}}{663 \left (1-5 x+3 x^2\right )}+\frac {9 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {9 \left (13+9 \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 {9 \left (13-9 \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 {\left (2 \left (81+\left (62-22 \sqrt {13}\right ) m\right )\right ) \int \frac {(1+4 x)^m}{-5-\sqrt {13}+6 x} \, dx}{221 \sqrt {13}}+\frac {\left (2 \left (81+\left (62+22 \sqrt {13}\right ) m\right )\right ) \int \frac {(1+4 x)^m}{-5+\sqrt {13}+6 x} \, dx}{221 \sqrt {13}}\\ &=\frac {(43-33 x) (1+4 x)^{1+m}}{663 \left (1-5 x+3 x^2\right )}+\frac {9 (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {3}{5} (1+4 x)\right )}{1445 (1+m)}+\frac {9 \left (13+9 \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 {\left (81+\left (62+22 \sqrt {13}\right ) m\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13-2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13-2 \sqrt {13}\right ) (1+m)}+\frac {9 \left (13-9 \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 {\left (81+\left (62-22 \sqrt {13}\right ) m\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {3 (1+4 x)}{13+2 \sqrt {13}}\right )}{221 \sqrt {13} \left (13+2 \sqrt {13}\right ) (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.77, size = 274, normalized size = 0.81 \[ \frac {(4 x+1)^{m+1} \left (\frac {9126 \, _2F_1\left (1,m+1;m+2;-\frac {3}{5} (4 x+1)\right )}{m+1}+\frac {1755 \left (13+9 \sqrt {13}\right ) \, _2F_1\left (1,m+1;m+2;\frac {12 x+3}{13-2 \sqrt {13}}\right )}{\left (13-2 \sqrt {13}\right ) (m+1)}+\frac {1755 \left (13-9 \sqrt {13}\right ) \, _2F_1\left (1,m+1;m+2;\frac {12 x+3}{13+2 \sqrt {13}}\right )}{\left (13+2 \sqrt {13}\right ) (m+1)}+\frac {510 \sqrt {13} \left (\frac {\left (\left (62+22 \sqrt {13}\right ) m+81\right ) \, _2F_1\left (1,m+1;m+2;\frac {12 x+3}{13-2 \sqrt {13}}\right )}{2 \sqrt {13}-13}+\frac {\left (\left (62-22 \sqrt {13}\right ) m+81\right ) \, _2F_1\left (1,m+1;m+2;\frac {12 x+3}{13+2 \sqrt {13}}\right )}{13+2 \sqrt {13}}\right )}{m+1}+\frac {2210 (43-33 x)}{3 x^2-5 x+1}\right )}{1465230} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x + 1\right )}^{m}}{27 \, x^{5} - 72 \, x^{4} + 33 \, x^{3} + 32 \, x^{2} - 17 \, x + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2} {\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x +1\right )^{m}}{\left (3 x +2\right ) \left (3 x^{2}-5 x +1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2} {\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (4\,x+1\right )}^m}{\left (3\,x+2\right )\,{\left (3\,x^2-5\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x + 1\right )^{m}}{\left (3 x + 2\right ) \left (3 x^{2} - 5 x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________