Integrand size = 24, antiderivative size = 444 \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 E\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \]
-1/51759*(176-23*(1-6/x)^2)*x^2/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+1/317282 67*(45401-3722*(1-6/x)^2)*(1-6/x)*x^2/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+37 22/31728267*(613-182*(1-6/x)^2+(-1+6/x)^4)*(1-6/x)*x^2/((6-x)^2/x^2+613^(1 /2))/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+3722/31728267*x^2*(cos(2*arctan(1/6 13*(6-x)*613^(3/4)/x))^2)^(1/2)/cos(2*arctan(1/613*(6-x)*613^(3/4)/x))*Ell ipticE(sin(2*arctan(1/613*(6-x)*613^(3/4)/x)),1/1226*(751538+111566*613^(1 /2))^(1/2))*((6-x)^2/x^2+613^(1/2))*((613-182*(1-6/x)^2+(-1+6/x)^4)/((6-x) ^2/x^2+613^(1/2))^2)^(1/2)*613^(1/4)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)-1/1 26913068*x^2*(cos(2*arctan(1/613*(6-x)*613^(3/4)/x))^2)^(1/2)/cos(2*arctan (1/613*(6-x)*613^(3/4)/x))*EllipticF(sin(2*arctan(1/613*(6-x)*613^(3/4)/x) ),1/1226*(751538+111566*613^(1/2))^(1/2))*(7444-145*613^(1/2))*((6-x)^2/x^ 2+613^(1/2))*((613-182*(1-6/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+613^(1/2))^2)^(1 /2)*613^(1/4)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 15.99 (sec) , antiderivative size = 4974, normalized size of antiderivative = 11.20 \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\text {Result too large to show} \]
(2*(106926 + 592639*x - 232005*x^2 - 44664*x^3 + 81441*EllipticF[ArcSin[Sq rt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6 *#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15* #1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], ((Root[9 - 6*#1 - 44*# 1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*# 1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Roo t[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((Root[9 - 6*#1 - 44*# 1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*# 1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Roo t[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]*Sqrt[(x - Root[9 - 6* #1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1 ^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3 , 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/((x - Root[9 - 6*# 1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1 ^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*...
Time = 0.78 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2504, 27, 7270, 2202, 1576, 27, 1158, 2206, 27, 1497, 27, 1409, 1496}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (3 x^4+15 x^3-44 x^2-6 x+9\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2504 |
\(\displaystyle -1296 \int \frac {1}{972 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \left (\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4}{3} \int \frac {1}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \left (\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \int \frac {\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\int \frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4+216 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )+\int \frac {\left (864 \left (\frac {1}{x}-\frac {1}{6}\right )^2+24\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {1}{2} \int \frac {24 \left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1\right )}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )^2+\int \frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4+216 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (12 \int \frac {36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )^2+\int \frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4+216 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\int \frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4+216 \left (\frac {1}{x}-\frac {1}{6}\right )^2+1}{\left (1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613\right )^{3/2}}d\left (\frac {1}{x}-\frac {1}{6}\right )+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 2206 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (-\frac {\int \frac {20736 \left (88885-267984 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{24367309056}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (-\frac {\int \frac {88885-267984 \left (\frac {1}{x}-\frac {1}{6}\right )^2}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{1175121}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 1497 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {-\left (\left (88885-7444 \sqrt {613}\right ) \int \frac {1}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )\right )-7444 \sqrt {613} \int \frac {\sqrt {613}-36 \left (\frac {1}{x}-\frac {1}{6}\right )^2}{\sqrt {613} \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{1175121}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {-\left (\left (88885-7444 \sqrt {613}\right ) \int \frac {1}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )\right )-7444 \int \frac {\sqrt {613}-36 \left (\frac {1}{x}-\frac {1}{6}\right )^2}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{1175121}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 1409 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {-7444 \int \frac {\sqrt {613}-36 \left (\frac {1}{x}-\frac {1}{6}\right )^2}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )-\frac {\left (88885-7444 \sqrt {613}\right ) \left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right ) \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {6 \left (\frac {1}{x}-\frac {1}{6}\right )}{\sqrt [4]{613}}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}}{1175121}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
\(\Big \downarrow \) 1496 |
\(\displaystyle -\frac {4 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {-\frac {\left (88885-7444 \sqrt {613}\right ) \left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right ) \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {6 \left (\frac {1}{x}-\frac {1}{6}\right )}{\sqrt [4]{613}}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}-7444 \left (\frac {\sqrt [4]{613} \left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right ) \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right )^2}} E\left (2 \arctan \left (\frac {6 \left (\frac {1}{x}-\frac {1}{6}\right )}{\sqrt [4]{613}}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{6 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}-\frac {613 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \left (\frac {1}{x}-\frac {1}{6}\right )}{22068 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613 \sqrt {613}}\right )}{1175121}+\frac {4 \left (44-207 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right )}{5751 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}+\frac {2 \left (45401-133992 \left (\frac {1}{x}-\frac {1}{6}\right )^2\right ) \left (\frac {1}{x}-\frac {1}{6}\right )}{1175121 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}\right )}{3 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
(-4*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4]*((4*(44 - 207*(-1/6 + x^(-1))^2))/(5751*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1 /6 + x^(-1))^4]) + (2*(45401 - 133992*(-1/6 + x^(-1))^2)*(-1/6 + x^(-1)))/ (1175121*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4]) + (- 7444*((-613*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4]*(- 1/6 + x^(-1)))/(613*Sqrt[613] + 22068*(-1/6 + x^(-1))^2) + (613^(1/4)*(Sqr t[613] + 36*(-1/6 + x^(-1))^2)*Sqrt[(613 - 6552*(-1/6 + x^(-1))^2 + 1296*( -1/6 + x^(-1))^4)/(Sqrt[613] + 36*(-1/6 + x^(-1))^2)^2]*EllipticE[2*ArcTan [(6*(-1/6 + x^(-1)))/613^(1/4)], (613 + 91*Sqrt[613])/1226])/(6*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4])) - ((88885 - 7444*Sqrt[ 613])*(Sqrt[613] + 36*(-1/6 + x^(-1))^2)*Sqrt[(613 - 6552*(-1/6 + x^(-1))^ 2 + 1296*(-1/6 + x^(-1))^4)/(Sqrt[613] + 36*(-1/6 + x^(-1))^2)^2]*Elliptic F[2*ArcTan[(6*(-1/6 + x^(-1)))/613^(1/4)], (613 + 91*Sqrt[613])/1226])/(12 *613^(1/4)*Sqrt[613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4]))/1 175121))/(3*(1 + 6*(-1/6 + x^(-1)))^2*Sqrt[(613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4)/(1 + 6*(-1/6 + x^(-1)))^4])
3.9.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[ b/a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ [c/a, 0] && LtQ[b/a, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.34 (sec) , antiderivative size = 5421, normalized size of antiderivative = 12.21
method | result | size |
default | \(\text {Expression too large to display}\) | \(5421\) |
risch | \(\text {Expression too large to display}\) | \(5421\) |
elliptic | \(\text {Expression too large to display}\) | \(5421\) |
\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)/(9*x^8 + 90*x^7 - 39*x^6 - 1356*x^5 + 1810*x^4 + 798*x^3 - 756*x^2 - 108*x + 81), x)
\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4+15\,x^3-44\,x^2-6\,x+9\right )}^{3/2}} \,d x \]