\(\int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^3} \, dx\) [253]

Optimal result

Integrand size = 24, antiderivative size = 1140 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^3} \, dx =\text {Too large to display} \] Output:

-1/2*c^(1/2)*d*(b*e^2+2*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/e^2/(a*e^4+b*d^2*e^ 
2+c*d^4)/(a^(1/2)+c^(1/2)*x^2)+d*x*(c*x^4+b*x^2+a)^(1/2)/(-e^2*x^2+d^2)^2- 
1/2*d*(b*e^2+2*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)/(-e^ 
2*x^2+d^2)+1/2*(d^2*(-a*e^4+c*d^4)-e^2*(a*e^4+2*b*d^2*e^2+3*c*d^4)*x^2)*(c 
*x^4+b*x^2+a)^(1/2)/e/(a*e^4+b*d^2*e^2+c*d^4)/(-e^2*x^2+d^2)^2+1/4*(a*b*e^ 
6+6*a*c*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)*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)^(3/2)+1/2*c^ 
(1/2)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/e^3-1/4*(a*b* 
e^6+6*a*c*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)*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)^(3/2)+1/2*a^(1/4)*c^(1/4)*d*(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)*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/(a*e 
^4+b*d^2*e^2+c*d^4)/(c*x^4+b*x^2+a)^(1/2)-a^(1/4)*c^(3/4)*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)*InverseJacobiAM(2* 
arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/e^2/(c^(1/2)*d^ 
2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)-1/8*(c^(1/4)*d-a^(1/4)*e)*(c^(1/4)*d+ 
a^(1/4)*e)*(a*b*e^6+6*a*c*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)*(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*ar 
ctan(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...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.10 (sec) , antiderivative size = 8346, normalized size of antiderivative = 7.32 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

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.

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

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{(d+e x)^3}dx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 917, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e \left (e x +d \right )^{2}}+\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (e x +d \right )}+\frac {\left (-\frac {3 c d}{e^{4}}+\frac {c d \left (2 e^{4} a +3 b \,d^{2} e^{2}+4 c \,d^{4}\right )}{2 e^{4} \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )}\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}\, \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e^{3}}+\frac {c d \left (b \,e^{2}+2 c \,d^{2}\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 )}{4 e^{2} \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\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 (a b \,e^{6}+6 a c \,d^{2} e^{4}+3 b c \,d^{4} e^{2}+2 c^{2} d^{6}\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 )}{2 \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) e^{5}}\)	\(917\)
elliptic	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e \left (e x +d \right )^{2}}+\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 e \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (e x +d \right )}+\frac {\left (-\frac {3 c d}{e^{4}}+\frac {c d \left (2 e^{4} a +3 b \,d^{2} e^{2}+4 c \,d^{4}\right )}{2 e^{4} \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )}\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}\, \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e^{3}}+\frac {c d \left (b \,e^{2}+2 c \,d^{2}\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 )}{4 e^{2} \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\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 (a b \,e^{6}+6 a c \,d^{2} e^{4}+3 b c \,d^{4} e^{2}+2 c^{2} d^{6}\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 )}{2 \left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) e^{5}}\)	\(917\)

Input:

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

Output:

-1/2/e*(c*x^4+b*x^2+a)^(1/2)/(e*x+d)^2+1/2*d*(b*e^2+2*c*d^2)/e/(a*e^4+b*d^ 
2*e^2+c*d^4)*(c*x^4+b*x^2+a)^(1/2)/(e*x+d)+1/4*(-3*c*d/e^4+1/2*c*d/e^4*(2* 
a*e^4+3*b*d^2*e^2+4*c*d^4)/(a*e^4+b*d^2*e^2+c*d^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)*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*c^(1/2)/e^3*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+b*x^2+a)^(1/2))+1/ 
4*c*d*(b*e^2+2*c*d^2)/e^2/(a*e^4+b*d^2*e^2+c*d^4)*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)))+1/2*(a*b*e 
^6+6*a*c*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)/(a*e^4+b*d^2*e^2+c*d^4)/e^5*(-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)*Ellipt 
icPi(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...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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