\(\int (d+e x+f x^2+g x^3) \sqrt {a+b x^2+c x^4} \, dx\) [101]

Optimal result

Integrand size = 32, antiderivative size = 505 \[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\frac {\left (5 b c d-2 b^2 f+6 a c f\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) (2 c e-b g) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {\sqrt [4]{a} \left (5 b c d-2 b^2 f+6 a c f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}} \] Output:

1/15*(6*a*c*f-2*b^2*f+5*b*c*d)*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(a^(1/2)+c^ 
(1/2)*x^2)+1/16*(-b*g+2*c*e)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^2+1/15*x* 
(3*c*f*x^2+b*f+5*c*d)*(c*x^4+b*x^2+a)^(1/2)/c+1/6*g*(c*x^4+b*x^2+a)^(3/2)/ 
c-1/32*(-4*a*c+b^2)*(-b*g+2*c*e)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b* 
x^2+a)^(1/2))/c^(5/2)-1/15*a^(1/4)*(6*a*c*f-2*b^2*f+5*b*c*d)*(a^(1/2)+c^(1 
/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*a 
rctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+ 
b*x^2+a)^(1/2)+1/30*a^(1/4)*(b+2*a^(1/2)*c^(1/2))*(5*c*d-2*b*f+3*a^(1/2)*c 
^(1/2)*f)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^ 
(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2) 
)^(1/2))/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.25 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.31 \[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\frac {2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-15 b^2 g+2 b c (15 e+x (8 f+5 g x))+4 c (10 a g+c x (20 d+x (15 e+2 x (6 f+5 g x))))\right )+\frac {-8 i \sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \left (-5 b c d+2 b^2 f-6 a c f\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+8 i \sqrt {2} \sqrt {c} \left (-2 b^3 f+b c \left (-5 \sqrt {b^2-4 a c} d+8 a f\right )+b^2 \left (5 c d+2 \sqrt {b^2-4 a c} f\right )-2 a c \left (10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-15 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} (-2 c e+b g) \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{480 c^{5/2} \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]
                                                                                    
                                                                                    
 

Output:

(2*Sqrt[c]*(a + b*x^2 + c*x^4)*(-15*b^2*g + 2*b*c*(15*e + x*(8*f + 5*g*x)) 
 + 4*c*(10*a*g + c*x*(20*d + x*(15*e + 2*x*(6*f + 5*g*x))))) + ((-8*I)*Sqr 
t[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*(-5*b*c*d + 2*b^2*f - 6*a*c*f)*Sqrt[ 
(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[ 
b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[ 
2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b 
^2 - 4*a*c])] + (8*I)*Sqrt[2]*Sqrt[c]*(-2*b^3*f + b*c*(-5*Sqrt[b^2 - 4*a*c 
]*d + 8*a*f) + b^2*(5*c*d + 2*Sqrt[b^2 - 4*a*c]*f) - 2*a*c*(10*c*d + 3*Sqr 
t[b^2 - 4*a*c]*f))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 
4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*E 
llipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b 
^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 15*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[ 
b^2 - 4*a*c])]*(-2*c*e + b*g)*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 - 2* 
Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(480*c^ 
(5/2)*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2202, 1490, 1511, 27, 1416, 1509, 1576, 1160, 1087, 1092, 219}

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.

\(\displaystyle \int \sqrt {a+b x^2+c x^4} \left (d+e x+f x^2+g x^3\right ) \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \left (f x^2+d\right ) \sqrt {c x^4+b x^2+a}dx+\int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {\int \frac {\left (-2 f b^2+5 c d b+6 a c f\right ) x^2+a (10 c d-b f)}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (6 a c f-2 b^2 f+5 b c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+\int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+\int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+\int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \int x \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {1}{2} \int \left (g x^2+e\right ) \sqrt {c x^4+b x^2+a}dx^2+\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \int \sqrt {c x^4+b x^2+a}dx^2}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c f-2 b^2 f+5 b c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+\frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}\)

Input:

Int[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

(x*(5*c*d + b*f + 3*c*f*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) + ((g*(a + b* 
x^2 + c*x^4)^(3/2))/(3*c) + ((2*c*e - b*g)*(((b + 2*c*x^2)*Sqrt[a + b*x^2 
+ c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + 
 b*x^2 + c*x^4])])/(8*c^(3/2))))/(2*c))/2 + (-(((5*b*c*d - 2*b^2*f + 6*a*c 
*f)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sq 
rt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*E 
llipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^( 
1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c] 
)*(5*c*d - 2*b*f + 3*Sqrt[a]*Sqrt[c]*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + 
b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a 
^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4]) 
)/(15*c)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1416

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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1509

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] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

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] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

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]
 

Maple [A] (verified)

Time = 3.86 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.23

method	result	size
elliptic	\(\frac {g \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{6}+\frac {f \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {\left (\frac {b g}{6}+c e \right ) x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (\frac {b f}{5}+c d \right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (\frac {a g}{3}+b e -\frac {3 b \left (\frac {b g}{6}+c e \right )}{4 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}+\frac {\left (a d -\frac {a \left (\frac {b f}{5}+c d \right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (a e -\frac {a \left (\frac {b g}{6}+c e \right )}{2 c}-\frac {b \left (\frac {a g}{3}+b e -\frac {3 b \left (\frac {b g}{6}+c e \right )}{4 c}\right )}{2 c}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {\left (\frac {2 a f}{5}+b d -\frac {2 b \left (\frac {b f}{5}+c d \right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(620\)
risch	\(\frac {\left (40 g \,x^{4} c^{2}+48 f \,x^{3} c^{2}+10 b c g \,x^{2}+60 c^{2} e \,x^{2}+16 b f x c +80 c^{2} x d +40 a c g -15 b^{2} g +30 c e b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{240 c^{2}}-\frac {\frac {8 c \left (6 a c f -2 b^{2} f +5 b c d \right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {\left (-60 a b c g +120 a \,c^{2} e +15 b^{3} g -30 b^{2} c e \right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {40 a \,c^{2} d \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {4 a b c f \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{240 c^{2}}\)	\(681\)
default	\(d \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+e \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\right )+f \left (\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {2 a}{5}-\frac {2 b^{2}}{15 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+g \left (\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {b \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )\)	\(974\)

Input:

int((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*g*x^4*(c*x^4+b*x^2+a)^(1/2)+1/5*f*x^3*(c*x^4+b*x^2+a)^(1/2)+1/4*(1/6*b 
*g+c*e)/c*x^2*(c*x^4+b*x^2+a)^(1/2)+1/3*(1/5*b*f+c*d)/c*x*(c*x^4+b*x^2+a)^ 
(1/2)+1/2*(1/3*a*g+b*e-3/4*b/c*(1/6*b*g+c*e))/c*(c*x^4+b*x^2+a)^(1/2)+1/4* 
(a*d-1/3*a/c*(1/5*b*f+c*d))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2 
*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^( 
1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2) 
)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/2*(a*e-1/2*a/c 
*(1/6*b*g+c*e)-1/2*b/c*(1/3*a*g+b*e-3/4*b/c*(1/6*b*g+c*e)))*ln((2*c*x^2+b) 
/c^(1/2)+2*(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-1/2*(2/5*a*f+b*d-2/3*b/c*(1/5*b* 
f+c*d))*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^ 
(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+ 
a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2) 
^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE( 
1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2 
)^(1/2))/a/c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.14 \[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\frac {32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, b c^{2} d - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, b^{2} c d - 2 \, {\left (b^{3} - 3 \, a b c\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, {\left (b c^{2} - 2 \, c^{3}\right )} d - {\left (2 \, b^{2} c - {\left (6 \, a + b\right )} c^{2}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, {\left (b^{2} c + 2 \, b c^{2}\right )} d - {\left (2 \, b^{3} - {\left (6 \, a b - b^{2}\right )} c\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 15 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e - {\left (b^{3} - 4 \, a b c\right )} g\right )} \sqrt {c} x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) + 4 \, {\left (40 \, c^{3} g x^{5} + 48 \, c^{3} f x^{4} + 80 \, b c^{2} d + 10 \, {\left (6 \, c^{3} e + b c^{2} g\right )} x^{3} + 16 \, {\left (5 \, c^{3} d + b c^{2} f\right )} x^{2} - 32 \, {\left (b^{2} c - 3 \, a c^{2}\right )} f + 5 \, {\left (6 \, b c^{2} e - {\left (3 \, b^{2} c - 8 \, a c^{2}\right )} g\right )} x\right )} \sqrt {c x^{4} + b x^{2} + a}}{960 \, c^{3} x} \] Input:

integrate((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

1/960*(32*sqrt(1/2)*((5*b*c^2*d - 2*(b^2*c - 3*a*c^2)*f)*x*sqrt((b^2 - 4*a 
*c)/c^2) - (5*b^2*c*d - 2*(b^3 - 3*a*b*c)*f)*x)*sqrt(c)*sqrt((c*sqrt((b^2 
- 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a* 
c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) 
 - 32*sqrt(1/2)*((5*(b*c^2 - 2*c^3)*d - (2*b^2*c - (6*a + b)*c^2)*f)*x*sqr 
t((b^2 - 4*a*c)/c^2) - (5*(b^2*c + 2*b*c^2)*d - (2*b^3 - (6*a*b - b^2)*c)* 
f)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sq 
rt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4 
*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 15*(2*(b^2*c - 4*a*c^2)*e - (b^3 - 4*a* 
b*c)*g)*sqrt(c)*x*log(8*c^2*x^4 + 8*b*c*x^2 + b^2 - 4*sqrt(c*x^4 + b*x^2 + 
 a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) + 4*(40*c^3*g*x^5 + 48*c^3*f*x^4 + 80*b 
*c^2*d + 10*(6*c^3*e + b*c^2*g)*x^3 + 16*(5*c^3*d + b*c^2*f)*x^2 - 32*(b^2 
*c - 3*a*c^2)*f + 5*(6*b*c^2*e - (3*b^2*c - 8*a*c^2)*g)*x)*sqrt(c*x^4 + b* 
x^2 + a))/(c^3*x)
 

Sympy [F]

\[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\int \sqrt {a + b x^{2} + c x^{4}} \left (d + e x + f x^{2} + g x^{3}\right )\, dx \] Input:

integrate((g*x**3+f*x**2+e*x+d)*(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral(sqrt(a + b*x**2 + c*x**4)*(d + e*x + f*x**2 + g*x**3), x)
 

Maxima [F]

\[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (g x^{3} + f x^{2} + e x + d\right )} \,d x } \] Input:

integrate((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d), x)
 

Giac [F]

\[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (g x^{3} + f x^{2} + e x + d\right )} \,d x } \] Input:

integrate((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(g*x^3 + f*x^2 + e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx=\int \sqrt {c\,x^4+b\,x^2+a}\,\left (g\,x^3+f\,x^2+e\,x+d\right ) \,d x \] Input:

int((a + b*x^2 + c*x^4)^(1/2)*(d + e*x + f*x^2 + g*x^3),x)
 

Output:

int((a + b*x^2 + c*x^4)^(1/2)*(d + e*x + f*x^2 + g*x^3), x)
 

Reduce [F]

\[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Input:

int((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

(80*sqrt(a + b*x**2 + c*x**4)*a*c**2*g - 30*sqrt(a + b*x**2 + c*x**4)*b**2 
*c*g + 60*sqrt(a + b*x**2 + c*x**4)*b*c**2*e + 32*sqrt(a + b*x**2 + c*x**4 
)*b*c**2*f*x + 20*sqrt(a + b*x**2 + c*x**4)*b*c**2*g*x**2 + 160*sqrt(a + b 
*x**2 + c*x**4)*c**3*d*x + 120*sqrt(a + b*x**2 + c*x**4)*c**3*e*x**2 + 96* 
sqrt(a + b*x**2 + c*x**4)*c**3*f*x**3 + 80*sqrt(a + b*x**2 + c*x**4)*c**3* 
g*x**4 + 60*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a*b*c*g 
- 120*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a*c**2*e - 15* 
sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*b**3*g + 30*sqrt(c)* 
log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*b**2*c*e - 60*sqrt(c)*log(sq 
rt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a*b*c*g + 120*sqrt(c)*log(sqrt(a + 
 b*x**2 + c*x**4) + sqrt(c)*x**2)*a*c**2*e + 15*sqrt(c)*log(sqrt(a + b*x** 
2 + c*x**4) + sqrt(c)*x**2)*b**3*g - 30*sqrt(c)*log(sqrt(a + b*x**2 + c*x* 
*4) + sqrt(c)*x**2)*b**2*c*e - 32*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2* 
a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**2*b*c**2*f + 320*int(sqr 
t(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x** 
6),x)*a**2*c**3*d + 192*int((sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b 
*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a*b*c**3*f - 64*int((sqrt(a + 
b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x** 
6),x)*b**3*c**2*f + 160*int((sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b 
*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*b**2*c**3*d + 192*int((sqrt...