\(\int \frac {1}{(d+e x)^2 (a+b x^2+c x^4)^{3/2}} \, dx\) [271]

Optimal result

Integrand size = 24, antiderivative size = 1490 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

e^4*x/(a*e^4+b*d^2*e^2+c*d^4)/(-e^2*x^2+d^2)/(c*x^4+b*x^2+a)^(1/2)+2*d*e*( 
b*c*d^2+b^2*e^2-2*a*c*e^2+c*(b*e^2+2*c*d^2)*x^2)/(-4*a*c+b^2)/(a*e^4+b*d^2 
*e^2+c*d^4)/(-e^2*x^2+d^2)/(c*x^4+b*x^2+a)^(1/2)-x*(a*e^2*(-b*e^2+2*c*d^2) 
*(b*c*d^2+(-2*a*c+b^2)*e^2)-(c*(-2*a*c+b^2)*d^2+b*(-3*a*c+b^2)*e^2)*(-3*a* 
e^4+b*d^2*e^2+c*d^4)-c*((b*c*d^2+(-2*a*c+b^2)*e^2)*(-3*a*e^4+b*d^2*e^2+c*d 
^4)-a*e^2*(-b^2*e^4+4*c^2*d^4))*x^2)/a/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2+c*d^4 
)^2/(c*x^4+b*x^2+a)^(1/2)-c^(1/2)*(b^3*d^2*e^4+b*c*d^2*(-5*a*e^4+c*d^4)-6* 
a*c*e^2*(-a*e^4+c*d^4)+2*b^2*(-a*e^6+c*d^4*e^2))*x*(c*x^4+b*x^2+a)^(1/2)/a 
/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2+c*d^4)^2/(a^(1/2)+c^(1/2)*x^2)-d*e^3*(-8*a* 
c*e^4+3*b^2*e^4+4*b*c*d^2*e^2+4*c^2*d^4)*(c*x^4+b*x^2+a)^(1/2)/(-4*a*c+b^2 
)/(a*e^4+b*d^2*e^2+c*d^4)^2/(-e^2*x^2+d^2)+3/2*d*e^5*(b*e^2+2*c*d^2)*arcta 
nh((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))/(a*e^4+b*d^2 
*e^2+c*d^4)^(5/2)-3/2*d*e^5*(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))/(a* 
e^4+b*d^2*e^2+c*d^4)^(5/2)+c^(1/4)*(b^3*d^2*e^4+b*c*d^2*(-5*a*e^4+c*d^4)-6 
*a*c*e^2*(-a*e^4+c*d^4)+2*b^2*(-a*e^6+c*d^4*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)*EllipticE(sin(2*arctan(c^(1/ 
4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(-4*a*c+b^2)/(a*e^ 
4+b*d^2*e^2+c*d^4)^2/(c*x^4+b*x^2+a)^(1/2)-1/2*c^(1/4)*(c^(3/2)*d^4-2*a^(1 
/2)*c*d^2*e^2-2*a^(1/2)*b*e^4+c^(1/2)*(3*a*e^4+b*d^2*e^2))*(a^(1/2)+c^(...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.72 (sec) , antiderivative size = 10488, normalized size of antiderivative = 7.04 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}}dx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 1545, normalized size of antiderivative = 1.04

method	result	size
default	\(\text {Expression too large to display}\)	\(1545\)
elliptic	\(\text {Expression too large to display}\)	\(1545\)

Input:

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

Output:

-e^7/(a*e^4+b*d^2*e^2+c*d^4)^2*(c*x^4+b*x^2+a)^(1/2)/(e*x+d)-2*c*(1/2*(2*a 
^2*c*e^6-a*b^2*e^6-5*a*b*c*d^2*e^4-6*a*c^2*d^4*e^2+b^3*d^2*e^4+2*b^2*c*d^4 
*e^2+b*c^2*d^6)/a/(4*a*c-b^2)/(a*e^4+b*d^2*e^2+c*d^4)^2*x^3-d*e*(2*a*c*e^4 
-b^2*e^4-2*b*c*d^2*e^2-2*c^2*d^4)/(4*a*c-b^2)/(a*e^4+b*d^2*e^2+c*d^4)^2*x^ 
2+1/2*(3*a^2*b*c*e^6+6*a^2*c^2*d^2*e^4-a*b^3*e^6-6*a*b^2*c*d^2*e^4-7*a*b*c 
^2*d^4*e^2-2*a*c^3*d^6+b^4*d^2*e^4+2*b^3*c*d^4*e^2+b^2*c^2*d^6)/a/(4*a*c-b 
^2)/c/(a*e^4+b*d^2*e^2+c*d^4)^2*x-d*e*(3*a*b*c*e^4+4*a*c^2*d^2*e^2-b^3*e^4 
-2*b^2*c*d^2*e^2-b*c^2*d^4)/(4*a*c-b^2)/(a*e^4+b*d^2*e^2+c*d^4)^2/c)/((x^4 
+b/c*x^2+a/c)*c)^(1/2)+1/4*(-c*d^2*e^4/(a*e^4+b*d^2*e^2+c*d^4)^2-(a*b*e^6+ 
3*a*c*d^2*e^4-b^2*d^2*e^4-2*b*c*d^4*e^2-c^2*d^6)/a/(a*e^4+b*d^2*e^2+c*d^4) 
^2+(3*a^2*b*c*e^6+6*a^2*c^2*d^2*e^4-a*b^3*e^6-6*a*b^2*c*d^2*e^4-7*a*b*c^2* 
d^4*e^2-2*a*c^3*d^6+b^4*d^2*e^4+2*b^3*c*d^4*e^2+b^2*c^2*d^6)/a/(4*a*c-b^2) 
/(a*e^4+b*d^2*e^2+c*d^4)^2)*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*(e^6*c/(a*e^ 
4+b*d^2*e^2+c*d^4)^2+c*(2*a^2*c*e^6-a*b^2*e^6-5*a*b*c*d^2*e^4-6*a*c^2*d^4* 
e^2+b^3*d^2*e^4+2*b^2*c*d^4*e^2+b*c^2*d^6)/a/(4*a*c-b^2)/(a*e^4+b*d^2*e^2+ 
c*d^4)^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...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \] Input:

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

Output:

sage0*x
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:

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

Output:

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