\(\int \frac {A+B x+C x^2}{\sqrt {2-4 x+3 x^3}} \, dx\) [20]

Optimal result

Integrand size = 25, antiderivative size = 1731 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x+3 x^3}} \, dx =\text {Too large to display} \] Output:

2/9*B*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)*(-4+16/(9-17^(1/2))^ 
(2/3)+(9-17^(1/2))^(2/3)-3*(4+(9-17^(1/2))^(2/3))*x/(9-17^(1/2))^(1/3)+9*x 
^2)^(1/2)*(16-4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3)-3*(9-17^(1/2)+4*(9-1 
7^(1/2))^(1/3))*x+9*(9-17^(1/2))^(2/3)*x^2)^(1/2)/(9-17^(1/2))^(1/3)/((4+( 
9-17^(1/2))^(2/3)+(48+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(1/2))/( 
9-17^(1/2))^(1/3)+3*x)/(3*x^3-4*x+2)^(1/2)+2/9*C*(3*x^3-4*x+2)^(1/2)-4/3*( 
585-97*17^(1/2)+4*(65-9*17^(1/2))*(9-17^(1/2))^(1/3)+12*(9-17^(1/2))^(5/3) 
)^(1/2)*B*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)^(1/2)*(-4+16/(9- 
17^(1/2))^(2/3)+(9-17^(1/2))^(2/3)-3*(4+(9-17^(1/2))^(2/3))*x/(9-17^(1/2)) 
^(1/3)+9*x^2)^(1/2)*((16-4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3)-3*(9-17^( 
1/2)+4*(9-17^(1/2))^(1/3))*x+9*(9-17^(1/2))^(2/3)*x^2)/(1+(4+(9-17^(1/2))^ 
(2/3)+3*(9-17^(1/2))^(1/3)*x)/(48+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/ 
3))^(1/2))^2)^(1/2)*(1+(4+(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(1/3)*x)/(48+1 
2*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(1/2))*EllipticE(sin(2*arctan(( 
9-17^(1/2))^(1/6)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)^(1/2)/(4 
8+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(1/4))),1/2*(2+(4+(9-17^(1/2 
))^(2/3))*3^(1/2)/(16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3))^(1/2))^(1/2 
))/(9-17^(1/2))^(1/3)/(48+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(3/4 
)/(16-4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3)-3*(9-17^(1/2)+4*(9-17^(1/2)) 
^(1/3))*x+9*(9-17^(1/2))^(2/3)*x^2)^(1/2)/(3*x^3-4*x+2)^(1/2)+1/81*(9*(...
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.58 (sec) , antiderivative size = 1260, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x+3 x^3}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 803, normalized size of antiderivative = 0.46, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2526, 2486, 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.

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

\(\Big \downarrow \) 2526

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

\(\Big \downarrow \) 2486

\(\displaystyle \frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \int \frac {9 A+4 C+9 B x}{\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4}}dx}{9 \sqrt {3 x^3-4 x+2}}+\frac {2}{9} C \sqrt {3 x^3-4 x+2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \left (\left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \int \frac {1}{\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4}}dx+3 B \int \frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}}}{\sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4}}dx\right )}{9 \sqrt {3 x^3-4 x+2}}+\frac {2}{9} C \sqrt {3 x^3-4 x+2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2}{9} \sqrt {3 x^3-4 x+2} C+\frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \left (-\frac {2 i \sqrt {2} \sqrt [6]{9-\sqrt {17}} \left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}} \int \frac {1}{\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{2 \sqrt {3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+1} \sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{\sqrt {3} \left (4-4 i \sqrt {3}-\left (1+i \sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}}{3 \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}}}-\frac {i \sqrt {2} B \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{\sqrt {3} \left (4-4 i \sqrt {3}-\left (1+i \sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}+1}}{\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{2 \sqrt {3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}}{\sqrt [6]{9-\sqrt {17}} \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}}}\right )}{9 \sqrt {3 x^3-4 x+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2}{9} \sqrt {3 x^3-4 x+2} C+\frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \left (-\frac {2 i \sqrt {2} \sqrt [6]{9-\sqrt {17}} \left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}\right ),\frac {2 i \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}{4 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )}{3 \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}}}-\frac {i \sqrt {2} B \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{\sqrt {3} \left (4-4 i \sqrt {3}-\left (1+i \sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}+1}}{\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{2 \sqrt {3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}}{\sqrt [6]{9-\sqrt {17}} \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}}}\right )}{9 \sqrt {3 x^3-4 x+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2}{9} \sqrt {3 x^3-4 x+2} C+\frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \left (-\frac {i \sqrt {2} B \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} E\left (\arcsin \left (\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}\right )|\frac {2 i \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}{4 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )}{\sqrt [6]{9-\sqrt {17}} \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}}}-\frac {2 i \sqrt {2} \sqrt [6]{9-\sqrt {17}} \left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}\right ),\frac {2 i \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}{4 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )}{3 \sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}}}\right )}{9 \sqrt {3 x^3-4 x+2}}\)

Input:

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

Output:

(2*C*Sqrt[2 - 4*x + 3*x^3])/9 + (Sqrt[(4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt 
[17])^(1/3) + 3*x]*Sqrt[-4 + 16/(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(2/3 
) - (3*(4 + (9 - Sqrt[17])^(2/3))*x)/(9 - Sqrt[17])^(1/3) + 9*x^2]*(((-I)* 
Sqrt[2]*B*Sqrt[(4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x]*Elli 
pticE[ArcSin[((9 - Sqrt[17])^(1/6)*Sqrt[(-I)*((4 + I*(4*Sqrt[3] - (I + Sqr 
t[3])*(9 - Sqrt[17])^(2/3)))/(9 - Sqrt[17])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2 
*(4 - (9 - Sqrt[17])^(2/3))])], ((2*I)*(4 - (9 - Sqrt[17])^(2/3)))/(4*(I + 
 Sqrt[3]) - (I - Sqrt[3])*(9 - Sqrt[17])^(2/3))])/((9 - Sqrt[17])^(1/6)*Sq 
rt[(I*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x))/(4*(3*I - S 
qrt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3))]) - (((2*I)/3)*Sqrt[2]*(9 
- Sqrt[17])^(1/6)*(9*A - (3*(4 + (9 - Sqrt[17])^(2/3))*B)/(9 - Sqrt[17])^( 
1/3) + 4*C)*Sqrt[(I*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x 
))/(4*(3*I - Sqrt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3))]*EllipticF[A 
rcSin[((9 - Sqrt[17])^(1/6)*Sqrt[(-I)*((4 + I*(4*Sqrt[3] - (I + Sqrt[3])*( 
9 - Sqrt[17])^(2/3)))/(9 - Sqrt[17])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(4 - ( 
9 - Sqrt[17])^(2/3))])], ((2*I)*(4 - (9 - Sqrt[17])^(2/3)))/(4*(I + Sqrt[3 
]) - (I - Sqrt[3])*(9 - Sqrt[17])^(2/3))])/Sqrt[(4 + (9 - Sqrt[17])^(2/3)) 
/(9 - Sqrt[17])^(1/3) + 3*x]))/(9*Sqrt[2 - 4*x + 3*x^3])
 

Defintions of rubi rules used

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 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 = 1.28 (sec) , antiderivative size = 1056, normalized size of antiderivative = 0.61

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1056\)
risch	\(\text {Expression too large to display}\)	\(1460\)
default	\(\text {Expression too large to display}\)	\(1461\)

Input:

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

Output:

2/9*C*(3*x^3-4*x+2)^(1/2)+2/3*I*(A+4/9*C)*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3) 
+4/3/(9+17^(1/2))^(1/3))*(I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/(9+17^(1/2))^(1/ 
3)+1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3)))*3^(1/2) 
/(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^(1/2))^(1/3)))^(1/2)*((x+1/3*(9+17^(1/2 
))^(1/3)+4/3/(9+17^(1/2))^(1/3))/(1/2*(9+17^(1/2))^(1/3)+2/(9+17^(1/2))^(1 
/3)-1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))))^(1/2) 
*(-I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/(9+17^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3* 
(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3 
)-4/3/(9+17^(1/2))^(1/3)))^(1/2)/(3*x^3-4*x+2)^(1/2)*EllipticF(1/3*3^(1/2) 
*(I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/(9+17^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*( 
9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3) 
-4/3/(9+17^(1/2))^(1/3)))^(1/2),(I*3^(1/2)*(1/3*(9+17^(1/2))^(1/3)-4/3/(9+ 
17^(1/2))^(1/3))/(1/2*(9+17^(1/2))^(1/3)+2/(9+17^(1/2))^(1/3)-1/2*I*3^(1/2 
)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))))^(1/2))+2/3*I*B*3^(1/2 
)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))*(I*(x-1/6*(9+17^(1/2))^ 
(1/3)-2/3/(9+17^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9 
+17^(1/2))^(1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^(1/2))^(1/3)) 
)^(1/2)*((x+1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))/(1/2*(9+17^(1/2 
))^(1/3)+2/(9+17^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/( 
9+17^(1/2))^(1/3))))^(1/2)*(-I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/(9+17^(1/2...
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.03 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x+3 x^3}} \, dx=\frac {2}{27} \, \sqrt {3} {\left (9 \, A + 4 \, C\right )} {\rm weierstrassPInverse}\left (\frac {16}{3}, -\frac {8}{3}, x\right ) - \frac {2}{3} \, \sqrt {3} B {\rm weierstrassZeta}\left (\frac {16}{3}, -\frac {8}{3}, {\rm weierstrassPInverse}\left (\frac {16}{3}, -\frac {8}{3}, x\right )\right ) + \frac {2}{9} \, \sqrt {3 \, x^{3} - 4 \, x + 2} C \] Input:

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

Output:

2/27*sqrt(3)*(9*A + 4*C)*weierstrassPInverse(16/3, -8/3, x) - 2/3*sqrt(3)* 
B*weierstrassZeta(16/3, -8/3, weierstrassPInverse(16/3, -8/3, x)) + 2/9*sq 
rt(3*x^3 - 4*x + 2)*C
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 12.68 (sec) , antiderivative size = 3117, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x+C x^2}{\sqrt {2-4 x+3 x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

(C*((2*(x^3 - (4*x)/3 + 2/3)^(1/2))/3 - (8*(-(x - 2/(9*(1/3 - 17^(1/2)/27) 
^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)/2 + (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27) 
^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2)/(2/(3*(1/3 - 17^(1/2)/27)^(1/3 
)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^ 
(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2))^(1/2)*ellipticF(asin((-(x - 2/ 
(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)/2 + (3^(1/2)*(4/ 
(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2)/(2/(3*(1 
/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/( 
9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2))^(1/2)), 
(3^(1/2)*(2/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/ 
2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)) 
*1i)/2)*1i)/(3*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3 
))))*((x + 4/(9*(1/3 - 17^(1/2)/27)^(1/3)) + (1/3 - 17^(1/2)/27)^(1/3))/(2 
/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2 
)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2))^(1 
/2)*(2/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - ( 
3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/ 
2)*(-(3^(1/2)*(2/(9*(1/3 - 17^(1/2)/27)^(1/3)) - x + (1/3 - 17^(1/2)/27)^( 
1/3)/2 + (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^( 
1/3))*1i)/2)*1i)/(3*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/...
 

Reduce [F]

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

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

Output:

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