\(\int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx\) [47]

Optimal result

Integrand size = 27, antiderivative size = 1895 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx =\text {Too large to display} \] Output:

2/9*C*(3*x^3-4*x^2+2)^(1/2)+2/729*(9*B+8*C)*(256-16*(179-9*345^(1/2))^(2/3 
)+(179-9*345^(1/2))^(4/3)+(179-9*345^(1/2)+16*(179-9*345^(1/2))^(1/3))*(4- 
9*x)+(179-9*345^(1/2))^(2/3)*(4-9*x)^2)^(1/2)*(4-16/(179-9*345^(1/2))^(1/3 
)-(179-9*345^(1/2))^(1/3)-9*x)*(-16+256/(179-9*345^(1/2))^(2/3)+(179-9*345 
^(1/2))^(2/3)+(16+(179-9*345^(1/2))^(2/3))*(4-9*x)/(179-9*345^(1/2))^(1/3) 
+(-4+9*x)^2)^(1/2)/(179-9*345^(1/2))^(1/3)/(4-(179-9*345^(1/2))^(1/3)-(16+ 
(768+(537-27*345^(1/2))*(179-9*345^(1/2))^(1/3)+48*(179-9*345^(1/2))^(2/3) 
)^(1/2))/(179-9*345^(1/2))^(1/3)-9*x)/(3*x^3-4*x^2+2)^(1/2)-4/729*(1850681 
-99195*345^(1/2)+(165424-8592*345^(1/2))*(179-9*345^(1/2))^(1/3)+64*(179-9 
*345^(1/2))^(5/3))^(1/2)*(9*B+8*C)*((256-16*(179-9*345^(1/2))^(2/3)+(179-9 
*345^(1/2))^(4/3)+(179-9*345^(1/2)+16*(179-9*345^(1/2))^(1/3))*(4-9*x)+(17 
9-9*345^(1/2))^(2/3)*(4-9*x)^2)/(1-(179-9*345^(1/2))^(1/3)*(4-16/(179-9*34 
5^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(768+48*(179-9*345^(1/2))^(2/3 
)+3*(179-9*345^(1/2))^(4/3))^(1/2))^2)^(1/2)*(1-(179-9*345^(1/2))^(1/3)*(4 
-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(768+48*(179-9*34 
5^(1/2))^(2/3)+3*(179-9*345^(1/2))^(4/3))^(1/2))*(-4+16/(179-9*345^(1/2))^ 
(1/3)+(179-9*345^(1/2))^(1/3)+9*x)^(1/2)*(-16+256/(179-9*345^(1/2))^(2/3)+ 
(179-9*345^(1/2))^(2/3)+(16+(179-9*345^(1/2))^(2/3))*(4-9*x)/(179-9*345^(1 
/2))^(1/3)+(-4+9*x)^2)^(1/2)*EllipticE(sin(2*arctan((179-9*345^(1/2))^(1/6 
)*(-4+16/(179-9*345^(1/2))^(1/3)+(179-9*345^(1/2))^(1/3)+9*x)^(1/2)/(76...
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.69 (sec) , antiderivative size = 1597, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*x + C*x^2)/Sqrt[2 - 4*x^2 + 3*x^3],x]
 

Output:

(2*(C*(2 - 4*x^2 + 3*x^3) + (9*A*EllipticF[ArcSin[Sqrt[(-x + Root[2 - 4*#1 
^2 + 3*#1^3 & , 3, 0])/(-Root[2 - 4*#1^2 + 3*#1^3 & , 2, 0] + Root[2 - 4*# 
1^2 + 3*#1^3 & , 3, 0])]], (Root[2 - 4*#1^2 + 3*#1^3 & , 2, 0] - Root[2 - 
4*#1^2 + 3*#1^3 & , 3, 0])/(Root[2 - 4*#1^2 + 3*#1^3 & , 1, 0] - Root[2 - 
4*#1^2 + 3*#1^3 & , 3, 0])]*(x - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])*Sqrt[ 
(-x + Root[2 - 4*#1^2 + 3*#1^3 & , 1, 0])/(Root[2 - 4*#1^2 + 3*#1^3 & , 1, 
 0] - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])]*Sqrt[(-x + Root[2 - 4*#1^2 + 3* 
#1^3 & , 2, 0])/(Root[2 - 4*#1^2 + 3*#1^3 & , 2, 0] - Root[2 - 4*#1^2 + 3* 
#1^3 & , 3, 0])])/Sqrt[(x - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])/(Root[2 - 
4*#1^2 + 3*#1^3 & , 2, 0] - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])] - (9*B*(x 
 - Root[2 - 4*#1^2 + 3*#1^3 & , 2, 0])*Sqrt[(-x + Root[2 - 4*#1^2 + 3*#1^3 
 & , 1, 0])/(Root[2 - 4*#1^2 + 3*#1^3 & , 1, 0] - Root[2 - 4*#1^2 + 3*#1^3 
 & , 3, 0])]*Sqrt[(-x + Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])/(-Root[2 - 4*# 
1^2 + 3*#1^3 & , 2, 0] + Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])]*(EllipticF[A 
rcSin[Sqrt[(-x + Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])/(-Root[2 - 4*#1^2 + 3 
*#1^3 & , 2, 0] + Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])]], (Root[2 - 4*#1^2 
+ 3*#1^3 & , 2, 0] - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])/(Root[2 - 4*#1^2 
+ 3*#1^3 & , 1, 0] - Root[2 - 4*#1^2 + 3*#1^3 & , 3, 0])]*Root[2 - 4*#1^2 
+ 3*#1^3 & , 1, 0] + EllipticE[ArcSin[Sqrt[(-x + Root[2 - 4*#1^2 + 3*#1^3 
& , 3, 0])/(-Root[2 - 4*#1^2 + 3*#1^3 & , 2, 0] + Root[2 - 4*#1^2 + 3*#...
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.91 (sec) , antiderivative size = 665, normalized size of antiderivative = 0.35, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2526, 2490, 2486, 27, 1269, 1172, 321, 327}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {3 x^3-4 x^2+2}} \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{9} \int \frac {9 A+(9 B+8 C) x}{\sqrt {3 x^3-4 x^2+2}}dx+\frac {2}{9} C \sqrt {3 x^3-4 x^2+2}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{9} \int \frac {\frac {1}{9} (81 A+4 (9 B+8 C))+(9 B+8 C) \left (x-\frac {4}{9}\right )}{\sqrt {3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}}}d\left (x-\frac {4}{9}\right )+\frac {2}{9} C \sqrt {3 x^3-4 x^2+2}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \int \frac {81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )}{\sqrt {3} \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16}}d\left (x-\frac {4}{9}\right )}{3 \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}+\frac {2}{9} C \sqrt {3 x^3-4 x^2+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \int \frac {81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )}{\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16}}d\left (x-\frac {4}{9}\right )}{3 \sqrt {3} \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}+\frac {2}{9} C \sqrt {3 x^3-4 x^2+2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \left (\left (81 A-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}+36 B+32 C\right ) \int \frac {1}{\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16}}d\left (x-\frac {4}{9}\right )+(9 B+8 C) \int \frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}}}{\sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16}}d\left (x-\frac {4}{9}\right )\right )}{3 \sqrt {3} \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}+\frac {2}{9} C \sqrt {3 x^3-4 x^2+2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2}{9} \sqrt {3 x^3-4 x^2+2} C+\frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \left (-\frac {2 i \sqrt {2} \sqrt [6]{179-9 \sqrt {345}} \left (81 A+36 B+32 C-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}} \int \frac {1}{\sqrt {1-\frac {i \sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{2 \sqrt {3} \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}} \sqrt {\frac {\sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{\sqrt {3} \left (16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{179-9 \sqrt {345}} \sqrt {i \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}}}{9 \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}}}-\frac {i \sqrt {2} (9 B+8 C) \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \int \frac {\sqrt {\frac {\sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{\sqrt {3} \left (16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}+1}}{\sqrt {1-\frac {i \sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{2 \sqrt {3} \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}}}d\frac {\sqrt [6]{179-9 \sqrt {345}} \sqrt {i \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}}}{9 \sqrt [6]{179-9 \sqrt {345}} \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}}}\right )}{3 \sqrt {3} \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2}{9} \sqrt {3 x^3-4 x^2+2} C+\frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \left (\frac {2 i \sqrt {2} \sqrt [6]{179-9 \sqrt {345}} \left (81 A+36 B+32 C-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4}{9}-x\right ),\frac {2 i \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}{16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )}{9 \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}}}-\frac {i \sqrt {2} (9 B+8 C) \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \int \frac {\sqrt {\frac {\sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{\sqrt {3} \left (16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}+1}}{\sqrt {1-\frac {i \sqrt [3]{179-9 \sqrt {345}} \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{2 \sqrt {3} \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}}}d\frac {\sqrt [6]{179-9 \sqrt {345}} \sqrt {i \left (18 \left (x-\frac {4}{9}\right )+\frac {i \left (16 \left (i-\sqrt {3}\right )+\left (i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}\right )}{\sqrt [3]{179-9 \sqrt {345}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}}}{9 \sqrt [6]{179-9 \sqrt {345}} \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}}}\right )}{3 \sqrt {3} \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2}{9} C \sqrt {3 x^3-4 x^2+2}+\frac {\sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} \sqrt {81 \left (x-\frac {4}{9}\right )^2-\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\left (179-9 \sqrt {345}\right )^{2/3}+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}-16} \left (\frac {2 i \sqrt {2} \sqrt [6]{179-9 \sqrt {345}} \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (-\sqrt {3}+3 i\right )+\left (\sqrt {3}+3 i\right ) \left (179-9 \sqrt {345}\right )^{2/3}}} \left (81 A-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}+36 B+32 C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4}{9}-x\right ),\frac {2 i \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}{16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )}{9 \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}}}+\frac {i \sqrt {2} \sqrt {9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}} (9 B+8 C) E\left (\arcsin \left (\frac {4}{9}-x\right )|\frac {2 i \left (16-\left (179-9 \sqrt {345}\right )^{2/3}\right )}{16 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )}{9 \sqrt [6]{179-9 \sqrt {345}} \sqrt {\frac {i \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (-\sqrt {3}+3 i\right )+\left (\sqrt {3}+3 i\right ) \left (179-9 \sqrt {345}\right )^{2/3}}}}\right )}{3 \sqrt {3} \sqrt {729 \left (x-\frac {4}{9}\right )^3-432 \left (x-\frac {4}{9}\right )+358}}\)

Input:

Int[(A + B*x + C*x^2)/Sqrt[2 - 4*x^2 + 3*x^3],x]
 

Output:

(2*C*Sqrt[2 - 4*x^2 + 3*x^3])/9 + (Sqrt[(16 + (179 - 9*Sqrt[345])^(2/3))/( 
179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)]*Sqrt[-16 + 256/(179 - 9*Sqrt[345] 
)^(2/3) + (179 - 9*Sqrt[345])^(2/3) - (9*(16 + (179 - 9*Sqrt[345])^(2/3))* 
(-4/9 + x))/(179 - 9*Sqrt[345])^(1/3) + 81*(-4/9 + x)^2]*(((I/9)*Sqrt[2]*( 
9*B + 8*C)*Sqrt[(16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 9*Sqrt[345])^(1/3) 
 + 9*(-4/9 + x)]*EllipticE[ArcSin[4/9 - x], ((2*I)*(16 - (179 - 9*Sqrt[345 
])^(2/3)))/(16*(I + Sqrt[3]) - (I - Sqrt[3])*(179 - 9*Sqrt[345])^(2/3))])/ 
((179 - 9*Sqrt[345])^(1/6)*Sqrt[(I*((16 + (179 - 9*Sqrt[345])^(2/3))/(179 
- 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)))/(16*(3*I - Sqrt[3]) + (3*I + Sqrt[3] 
)*(179 - 9*Sqrt[345])^(2/3))]) + (((2*I)/9)*Sqrt[2]*(179 - 9*Sqrt[345])^(1 
/6)*(81*A + 36*B + 32*C - ((16 + (179 - 9*Sqrt[345])^(2/3))*(9*B + 8*C))/( 
179 - 9*Sqrt[345])^(1/3))*Sqrt[(I*((16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 
 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)))/(16*(3*I - Sqrt[3]) + (3*I + Sqrt[3]) 
*(179 - 9*Sqrt[345])^(2/3))]*EllipticF[ArcSin[4/9 - x], ((2*I)*(16 - (179 
- 9*Sqrt[345])^(2/3)))/(16*(I + Sqrt[3]) - (I - Sqrt[3])*(179 - 9*Sqrt[345 
])^(2/3))])/Sqrt[(16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 9*Sqrt[345])^(1/3 
) + 9*(-4/9 + x)]))/(3*Sqrt[3]*Sqrt[358 - 432*(-4/9 + x) + 729*(-4/9 + x)^ 
3])
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2486

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x 
]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 
3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p)   Int[(e + f*x)^m*Sim 
p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 
*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 
3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* 
b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.11 (sec) , antiderivative size = 1236, normalized size of antiderivative = 0.65

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1236\)
risch	\(\text {Expression too large to display}\)	\(1238\)
default	\(\text {Expression too large to display}\)	\(1968\)

Input:

int((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/9*C*(3*x^3-4*x^2+2)^(1/2)+2/3*I*A*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3)+ 
16/9/(179+9*345^(1/2))^(1/3))*(I*(x-1/18*(179+9*345^(1/2))^(1/3)-8/9/(179+ 
9*345^(1/2))^(1/3)-4/9+1/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3)+16/9/(1 
79+9*345^(1/2))^(1/3)))*3^(1/2)/(1/9*(179+9*345^(1/2))^(1/3)-16/9/(179+9*3 
45^(1/2))^(1/3)))^(1/2)*((x+1/9*(179+9*345^(1/2))^(1/3)+16/9/(179+9*345^(1 
/2))^(1/3)-4/9)/(1/6*(179+9*345^(1/2))^(1/3)+8/3/(179+9*345^(1/2))^(1/3)-1 
/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3)+16/9/(179+9*345^(1/2))^(1/3)))) 
^(1/2)*(-I*(x-1/18*(179+9*345^(1/2))^(1/3)-8/9/(179+9*345^(1/2))^(1/3)-4/9 
-1/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3)+16/9/(179+9*345^(1/2))^(1/3)) 
)*3^(1/2)/(1/9*(179+9*345^(1/2))^(1/3)-16/9/(179+9*345^(1/2))^(1/3)))^(1/2 
)/(3*x^3-4*x^2+2)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/18*(179+9*345^(1/2)) 
^(1/3)-8/9/(179+9*345^(1/2))^(1/3)-4/9+1/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2 
))^(1/3)+16/9/(179+9*345^(1/2))^(1/3)))*3^(1/2)/(1/9*(179+9*345^(1/2))^(1/ 
3)-16/9/(179+9*345^(1/2))^(1/3)))^(1/2),(I*3^(1/2)*(1/9*(179+9*345^(1/2))^ 
(1/3)-16/9/(179+9*345^(1/2))^(1/3))/(1/6*(179+9*345^(1/2))^(1/3)+8/3/(179+ 
9*345^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3)+16/9/(179+9 
*345^(1/2))^(1/3))))^(1/2))+2/3*I*(B+8/9*C)*3^(1/2)*(-1/9*(179+9*345^(1/2) 
)^(1/3)+16/9/(179+9*345^(1/2))^(1/3))*(I*(x-1/18*(179+9*345^(1/2))^(1/3)-8 
/9/(179+9*345^(1/2))^(1/3)-4/9+1/2*I*3^(1/2)*(-1/9*(179+9*345^(1/2))^(1/3) 
+16/9/(179+9*345^(1/2))^(1/3)))*3^(1/2)/(1/9*(179+9*345^(1/2))^(1/3)-16...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.03 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\frac {2}{243} \, \sqrt {3} {\left (81 \, A + 36 \, B + 32 \, C\right )} {\rm weierstrassPInverse}\left (\frac {64}{27}, -\frac {1432}{729}, x - \frac {4}{9}\right ) - \frac {2}{27} \, \sqrt {3} {\left (9 \, B + 8 \, C\right )} {\rm weierstrassZeta}\left (\frac {64}{27}, -\frac {1432}{729}, {\rm weierstrassPInverse}\left (\frac {64}{27}, -\frac {1432}{729}, x - \frac {4}{9}\right )\right ) + \frac {2}{9} \, \sqrt {3 \, x^{3} - 4 \, x^{2} + 2} C \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^(1/2),x, algorithm="fricas")
 

Output:

2/243*sqrt(3)*(81*A + 36*B + 32*C)*weierstrassPInverse(64/27, -1432/729, x 
 - 4/9) - 2/27*sqrt(3)*(9*B + 8*C)*weierstrassZeta(64/27, -1432/729, weier 
strassPInverse(64/27, -1432/729, x - 4/9)) + 2/9*sqrt(3*x^3 - 4*x^2 + 2)*C
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {3 x^{3} - 4 x^{2} + 2}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(3*x**3-4*x**2+2)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/sqrt(3*x**3 - 4*x**2 + 2), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {3 \, x^{3} - 4 \, x^{2} + 2}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^(1/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/sqrt(3*x^3 - 4*x^2 + 2), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {3 \, x^{3} - 4 \, x^{2} + 2}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^(1/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/sqrt(3*x^3 - 4*x^2 + 2), x)
 

Mupad [B] (verification not implemented)

Time = 13.13 (sec) , antiderivative size = 3476, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\text {Too large to display} \] Input:

int((A + B*x + C*x^2)/(3*x^3 - 4*x^2 + 2)^(1/2),x)
 

Output:

(C*((2*(x^3 - (4*x^2)/3 + 2/3)^(1/2))/3 - (16*(ellipticF(asin(((8/(81*(179 
/729 - 345^(1/2)/81)^(1/3)) - x + (179/729 - 345^(1/2)/81)^(1/3)/2 - 3^(1/ 
2)*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2)/81)^(1/3) 
/2)*1i + 4/9)/(8/(27*(179/729 - 345^(1/2)/81)^(1/3)) + (3*(179/729 - 345^( 
1/2)/81)^(1/3))/2 - 3^(1/2)*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) - (179/ 
729 - 345^(1/2)/81)^(1/3)/2)*1i))^(1/2)), (3^(1/2)*(8/(81*(179/729 - 345^( 
1/2)/81)^(1/3)) + (179/729 - 345^(1/2)/81)^(1/3)/2 - (3^(1/2)*(8/(81*(179/ 
729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2)/81)^(1/3)/2)*1i)/3)*1i)/ 
(16/(81*(179/729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2)/81)^(1/3))) 
*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) + (179/729 - 345^(1/2)/81)^(1/3)/2 
 + 3^(1/2)*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2)/8 
1)^(1/3)/2)*1i + 4/9) - 3^(1/2)*(16/(81*(179/729 - 345^(1/2)/81)^(1/3)) - 
(179/729 - 345^(1/2)/81)^(1/3))*ellipticE(asin(((8/(81*(179/729 - 345^(1/2 
)/81)^(1/3)) - x + (179/729 - 345^(1/2)/81)^(1/3)/2 - 3^(1/2)*(8/(81*(179/ 
729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2)/81)^(1/3)/2)*1i + 4/9)/( 
8/(27*(179/729 - 345^(1/2)/81)^(1/3)) + (3*(179/729 - 345^(1/2)/81)^(1/3)) 
/2 - 3^(1/2)*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) - (179/729 - 345^(1/2) 
/81)^(1/3)/2)*1i))^(1/2)), (3^(1/2)*(8/(81*(179/729 - 345^(1/2)/81)^(1/3)) 
 + (179/729 - 345^(1/2)/81)^(1/3)/2 - (3^(1/2)*(8/(81*(179/729 - 345^(1/2) 
/81)^(1/3)) - (179/729 - 345^(1/2)/81)^(1/3)/2)*1i)/3)*1i)/(16/(81*(179...
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x^2+3 x^3}} \, dx=\frac {2 \sqrt {3 x^{3}-4 x^{2}+2}\, c}{9}+\left (\int \frac {\sqrt {3 x^{3}-4 x^{2}+2}}{3 x^{3}-4 x^{2}+2}d x \right ) a +\left (\int \frac {\sqrt {3 x^{3}-4 x^{2}+2}\, x}{3 x^{3}-4 x^{2}+2}d x \right ) b +\frac {8 \left (\int \frac {\sqrt {3 x^{3}-4 x^{2}+2}\, x}{3 x^{3}-4 x^{2}+2}d x \right ) c}{9} \] Input:

int((C*x^2+B*x+A)/(3*x^3-4*x^2+2)^(1/2),x)
 

Output:

(2*sqrt(3*x**3 - 4*x**2 + 2)*c + 9*int(sqrt(3*x**3 - 4*x**2 + 2)/(3*x**3 - 
 4*x**2 + 2),x)*a + 9*int((sqrt(3*x**3 - 4*x**2 + 2)*x)/(3*x**3 - 4*x**2 + 
 2),x)*b + 8*int((sqrt(3*x**3 - 4*x**2 + 2)*x)/(3*x**3 - 4*x**2 + 2),x)*c) 
/9