\(\int \frac {1}{(9-6 x-44 x^2+15 x^3+3 x^4)^{3/2}} \, dx\) [93]

Optimal result

Integrand size = 24, antiderivative size = 434 \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right ) \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right )^2}} x^2 E\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (7444-145 \sqrt {613}\right ) \left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \] Output:

-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/(613^(1/2)+(-1+6/x) 
^2)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+3722/31728267*(613^(1/2)+(-1+6/x)^2) 
*((613-182*(1-6/x)^2+(-1+6/x)^4)/(613^(1/2)+(-1+6/x)^2)^2)^(1/2)*x^2*Ellip 
ticE(sin(2*arctan(1/613*(6-x)*613^(3/4)/x)),1/1226*(751538+111566*613^(1/2 
))^(1/2))*613^(1/4)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)-1/126913068*(7444-14 
5*613^(1/2))*(613^(1/2)+(-1+6/x)^2)*((613-182*(1-6/x)^2+(-1+6/x)^4)/(613^( 
1/2)+(-1+6/x)^2)^2)^(1/2)*x^2*InverseJacobiAM(2*arctan(1/613*(6-x)*613^(3/ 
4)/x),1/1226*(751538+111566*613^(1/2))^(1/2))*613^(1/4)/(3*x^4+15*x^3-44*x 
^2-6*x+9)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 16.16 (sec) , antiderivative size = 5428, normalized size of antiderivative = 12.51 \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.21, 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}}}\)

Input:

Int[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]
 

Output:

(-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])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1158

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]
 

rule 1409

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]
 

rule 1496

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]
 

rule 1497

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]
 

rule 1576

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]
 

rule 2202

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]
 

rule 2206

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]
 

rule 2504

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 
]
 

rule 7270

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5420\) vs. \(2(389)=778\).

Time = 1.39 (sec) , antiderivative size = 5421, normalized size of antiderivative = 12.49

method	result	size
default	\(\text {Expression too large to display}\)	\(5421\)
risch	\(\text {Expression too large to display}\)	\(5421\)
elliptic	\(\text {Expression too large to display}\)	\(5421\)

Input:

int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(3/2),x)
 

Output:

Integral((3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="maxima")
 

Output:

integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="giac")
 

Output:

integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4+15\,x^3-44\,x^2-6\,x+9\right )}^{3/2}} \,d x \] Input:

int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(3/2),x)
 

Output:

int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}+15 x^{3}-44 x^{2}-6 x +9}}{9 x^{8}+90 x^{7}-39 x^{6}-1356 x^{5}+1810 x^{4}+798 x^{3}-756 x^{2}-108 x +81}d x \] Input:

int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x)
 

Output:

int(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)