Integrand size = 34, antiderivative size = 84 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=-\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}+\frac {97 \text {arctanh}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )}{3226944} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1016, 754, 820, 738, 212} \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\frac {97 \text {arctanh}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right )}{3226944}-\frac {388 x+275}{98 (10-3 x) \sqrt {12 x^2+17 x+6}}+\frac {3137 \sqrt {12 x^2+17 x+6}}{38416 (10-3 x)} \]
[In]
[Out]
Rule 212
Rule 738
Rule 754
Rule 820
Rule 1016
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}} \, dx \\ & = -\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}-\frac {1}{882} \int \frac {-\frac {14859}{2}-10476 x}{(10-3 x)^2 \sqrt {6+17 x+12 x^2}} \, dx \\ & = -\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}+\frac {97 \int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx}{76832} \\ & = -\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}-\frac {97 \text {Subst}\left (\int \frac {1}{7056-x^2} \, dx,x,\frac {-206-291 x}{\sqrt {6+17 x+12 x^2}}\right )}{38416} \\ & = -\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}+\frac {97 \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )}{3226944} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\frac {\left (88978+98767 x-37644 x^2\right ) \sqrt {6+17 x+12 x^2}}{38416 (-10+3 x) (2+3 x) (3+4 x)}+\frac {97 \text {arctanh}\left (\frac {6 \sqrt {6+17 x+12 x^2}}{7 (2+3 x)}\right )}{1613472} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {37644 x^{2}-98767 x -88978}{38416 \left (3 x -10\right ) \sqrt {12 x^{2}+17 x +6}}+\frac {97 \,\operatorname {arctanh}\left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{3226944}\) | \(57\) |
trager | \(-\frac {\left (37644 x^{2}-98767 x -88978\right ) \sqrt {12 x^{2}+17 x +6}}{38416 \left (36 x^{3}-69 x^{2}-152 x -60\right )}-\frac {97 \ln \left (-\frac {84 \sqrt {12 x^{2}+17 x +6}-206-291 x}{3 x -10}\right )}{3226944}\) | \(74\) |
default | \(-\frac {\left (12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}\right )^{\frac {3}{2}}}{72 \left (x +\frac {2}{3}\right )^{2}}+\frac {\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}}{288}+\frac {\ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}\right ) \sqrt {12}}{6912}-\frac {\left (12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}\right )^{\frac {3}{2}}}{67765824 \left (x -\frac {10}{3}\right )}-\frac {97 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{45177216}-\frac {7057 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}\right ) \sqrt {12}}{813189888}+\frac {97 \,\operatorname {arctanh}\left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{3226944}+\frac {\left (17+24 x \right ) \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{135531648}+\frac {32 \left (12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}\right )^{\frac {3}{2}}}{2401 \left (x +\frac {3}{4}\right )^{2}}+\frac {384 \sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}}{117649}-\frac {16 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}\right ) \sqrt {12}}{117649}\) | \(245\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\frac {97 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac {291 \, x + 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 97 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac {291 \, x - 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 168 \, {\left (37644 \, x^{2} - 98767 \, x - 88978\right )} \sqrt {12 \, x^{2} + 17 \, x + 6}}{6453888 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )}} \]
[In]
[Out]
\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\int \frac {\sqrt {\left (3 x + 2\right ) \left (4 x + 3\right )}}{\left (3 x - 10\right )^{2} \left (3 x + 2\right )^{2} \left (4 x + 3\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\int { \frac {\sqrt {12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )}^{2} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (70) = 140\).
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\frac {1}{9680832} \, \sqrt {3} {\left (\sqrt {3} {\left (175672 \, \sqrt {3} + 97 \, \log \left (\frac {7 \, \sqrt {3} - 12}{7 \, \sqrt {3} + 12}\right )\right )} \mathrm {sgn}\left (\frac {1}{3 \, x + 2}\right ) - {\left (97 \, \sqrt {3} \log \left (\frac {{\left | -28 \, \sqrt {3} + 24 \, \sqrt {\frac {1}{3 \, x + 2} + 4} \right |}}{4 \, {\left (7 \, \sqrt {3} + 6 \, \sqrt {\frac {1}{3 \, x + 2} + 4}\right )}}\right ) + 134456 \, \sqrt {\frac {1}{3 \, x + 2} + 4} + \frac {28 \, {\left (\frac {221183}{3 \, x + 2} - 18436\right )}}{12 \, {\left (\frac {1}{3 \, x + 2} + 4\right )}^{\frac {3}{2}} - 49 \, \sqrt {\frac {1}{3 \, x + 2} + 4}}\right )} \mathrm {sgn}\left (\frac {1}{3 \, x + 2}\right )\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx=\int \frac {\sqrt {12\,x^2+17\,x+6}}{{\left (3\,x+2\right )}^2\,{\left (-12\,x^2+31\,x+30\right )}^2} \,d x \]
[In]
[Out]