\(\int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2} \, dx\) [252]

Optimal result

Integrand size = 24, antiderivative size = 855 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2} \, dx=\frac {2 \sqrt {c} x \sqrt {a+b x^2+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+b x^2+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+b x^2+c x^4}}{d^2-e^2 x^2}-\frac {d \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d^4+b d^2 e^2+a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 e^3 \sqrt {c d^4+b d^2 e^2+a e^4}}-\frac {\sqrt {c} d \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{e^3}+\frac {d \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 e^3 \sqrt {c d^4+b d^2 e^2+a e^4}}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \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 )}{e^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (4 c d^2+b e^2+2 \sqrt {a} \sqrt {c} e^2\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 )}{2 \sqrt [4]{c} e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\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 {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},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 )}{4 \sqrt [4]{a} \sqrt [4]{c} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}} \] Output:

2*c^(1/2)*x*(c*x^4+b*x^2+a)^(1/2)/e^2/(a^(1/2)+c^(1/2)*x^2)-d*(c*x^4+b*x^2 
+a)^(1/2)/e/(-e^2*x^2+d^2)+x*(c*x^4+b*x^2+a)^(1/2)/(-e^2*x^2+d^2)-1/2*d*(b 
*e^2+2*c*d^2)*arctanh((a*e^4+b*d^2*e^2+c*d^4)^(1/2)*x/d/e/(c*x^4+b*x^2+a)^ 
(1/2))/e^3/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)-c^(1/2)*d*arctanh(1/2*(2*c*x^2+b) 
/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/e^3+1/2*d*(b*e^2+2*c*d^2)*arctanh(1/2*(b*d 
^2+2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2 
+a)^(1/2))/e^3/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)-2*a^(1/4)*c^(1/4)*(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 
*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/e^2/(c*x^4+b* 
x^2+a)^(1/2)+1/2*a^(1/4)*(4*c*d^2+b*e^2+2*a^(1/2)*c^(1/2)*e^2)*(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^(1/4)/e^2/ 
(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)+1/4*(c^(1/2)*d^2-a^(1/2)*e 
^2)*(b*e^2+2*c*d^2)*(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)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(c^(1/2)*d 
^2+a^(1/2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2*(2-b/a^(1/2)/c^(1/2))^(1/2)) 
/a^(1/4)/c^(1/4)/e^4/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 19.07 (sec) , antiderivative size = 4221, normalized size of antiderivative = 4.94 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

-(Sqrt[a + b*x^2 + c*x^4]/(e*(d + e*x))) + ((I*(-b + Sqrt[b^2 - 4*a*c])*Sq 
rt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (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])] - EllipticF[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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a* 
c]))]*e*Sqrt[a + b*x^2 + c*x^4]) - (I*Sqrt[2]*c*d^2*Sqrt[1 - (2*c*x^2)/(-b 
 - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Ellipt 
icF[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])])/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]) 
)]*e^3*Sqrt[a + b*x^2 + c*x^4]) - (I*b*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 
 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*EllipticF[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])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*e* 
Sqrt[a + b*x^2 + c*x^4]) - (4*c*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2 
] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*d^3*(-(Sqrt[-(b/c) - Sqrt[ 
b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-( 
Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 
 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt 
[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^...
 

Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2}dx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 749, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{e \left (e x +d \right )}+\frac {\left (\frac {b \,e^{2}+3 c \,d^{2}}{e^{4}}-\frac {c \,d^{2}}{e^{4}}\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 {\sqrt {c}\, d \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{e^{3}}-\frac {c 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 )}{e^{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 )}-\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{5}}\)	\(749\)
elliptic	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{e \left (e x +d \right )}+\frac {\left (\frac {b \,e^{2}+3 c \,d^{2}}{e^{4}}-\frac {c \,d^{2}}{e^{4}}\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 {\sqrt {c}\, d \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{e^{3}}-\frac {c 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 )}{e^{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 )}-\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{5}}\)	\(749\)

Input:

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

Output:

-1/e*(c*x^4+b*x^2+a)^(1/2)/(e*x+d)+1/4*((b*e^2+3*c*d^2)/e^4-c*d^2/e^4)*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)*Elli 
pticF(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))-c^(1/2)*d/e^3*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+b 
*x^2+a)^(1/2))-c/e^2*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)))-d/e^5*(b*e^2+2*c*d^2)*(-1/2/(c*d^4/e^4+ 
b*d^2/e^2+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+b*d^2/e^2+b*x^2+2*a)/(c*d^ 
4/e^4+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/((-b+(-4*a*c+b^2)^ 
(1/2))/a)^(1/2)/d*e*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(1+1/2*(b+ 
(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^ 
(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),2/(-b+(-4*a*c+b^2)^(1/2))*a/d^2*e^ 
2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a 
)^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*x**2 + c*x**4)/(d + e*x)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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