\(\int \frac {1}{(a+b x)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [6]

Optimal result

Integrand size = 30, antiderivative size = 595 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{\left (b^2 c+a^2 d\right ) \left (a^2-b^2 x^2\right ) \sqrt {e+f x^2}}-\frac {a b^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (b^2 c+a^2 d\right ) \left (b^2 e+a^2 f\right ) \left (a^2-b^2 x^2\right )}-\frac {a b \left (2 a^2 d f+b^2 (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b^2 e+a^2 f} \sqrt {c+d x^2}}{\sqrt {b^2 c+a^2 d} \sqrt {e+f x^2}}\right )}{\left (b^2 c+a^2 d\right )^{3/2} \left (b^2 e+a^2 f\right )^{3/2}}-\frac {b^2 \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\left (b^2 c+a^2 d\right ) \left (b^2 e+a^2 f\right ) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {a^2 \sqrt {e} f^{3/2} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \left (b^2 e+a^2 f\right )^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 e^{3/2} \left (2 a^2 d f+b^2 (d e+c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1+\frac {b^2 e}{a^2 f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \left (b^2 c+a^2 d\right ) \sqrt {f} \left (b^2 e+a^2 f\right )^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \] Output:

b^2*x*(d*x^2+c)^(1/2)/(a^2*d+b^2*c)/(-b^2*x^2+a^2)/(f*x^2+e)^(1/2)-a*b^3*( 
d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(a^2*d+b^2*c)/(a^2*f+b^2*e)/(-b^2*x^2+a^2)- 
a*b*(2*a^2*d*f+b^2*(c*f+d*e))*arctanh((a^2*f+b^2*e)^(1/2)*(d*x^2+c)^(1/2)/ 
(a^2*d+b^2*c)^(1/2)/(f*x^2+e)^(1/2))/(a^2*d+b^2*c)^(3/2)/(a^2*f+b^2*e)^(3/ 
2)-b^2*e^(1/2)*f^(1/2)*(d*x^2+c)^(1/2)*EllipticE(f^(1/2)*x/e^(1/2)/(1+f*x^ 
2/e)^(1/2),(1-d*e/c/f)^(1/2))/(a^2*d+b^2*c)/(a^2*f+b^2*e)/(e*(d*x^2+c)/c/( 
f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+a^2*e^(1/2)*f^(3/2)*(d*x^2+c)^(1/2)*Invers 
eJacobiAM(arctan(f^(1/2)*x/e^(1/2)),(1-d*e/c/f)^(1/2))/c/(a^2*f+b^2*e)^2/( 
e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+b^2*e^(3/2)*(2*a^2*d*f+b^2* 
(c*f+d*e))*(d*x^2+c)^(1/2)*EllipticPi(f^(1/2)*x/e^(1/2)/(1+f*x^2/e)^(1/2), 
1+b^2*e/a^2/f,(1-d*e/c/f)^(1/2))/c/(a^2*d+b^2*c)/f^(1/2)/(a^2*f+b^2*e)^2/( 
e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.91 (sec) , antiderivative size = 2848, normalized size of antiderivative = 4.79 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((a + b*x)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
 

Output:

-((b^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/((b^2*c + a^2*d)*(b^2*e + a^2*f)*( 
a + b*x))) + (Sqrt[(c + d*x^2)*(e + f*x^2)]*(((-I)*b^2*d*e*Sqrt[1 + (d*x^2 
)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - 
 EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x 
^2)*(e + f*x^2)]) + (I*a^2*d*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ell 
ipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*( 
e + f*x^2)]) - ((2*I)*a*b^2*d^(3/2)*e*((I*Sqrt[c])/Sqrt[d] + (I*Sqrt[e])/S 
qrt[f])*(((-I)*Sqrt[c])/Sqrt[d] + x)^2*Sqrt[(I*Sqrt[c]*(((-I)*Sqrt[e])/Sqr 
t[f] + x))/(Sqrt[d]*((I*Sqrt[c])/Sqrt[d] + (I*Sqrt[e])/Sqrt[f])*(((-I)*Sqr 
t[c])/Sqrt[d] + x))]*Sqrt[(I*Sqrt[c]*((I*Sqrt[e])/Sqrt[f] + x))/(Sqrt[d]*( 
(I*Sqrt[c])/Sqrt[d] - (I*Sqrt[e])/Sqrt[f])*(((-I)*Sqrt[c])/Sqrt[d] + x))]* 
Sqrt[-(((-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/(( 
Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x)))]*((-a + (I 
*b*Sqrt[c])/Sqrt[d])*EllipticF[ArcSin[Sqrt[-(((-(Sqrt[d]*Sqrt[e]) + Sqrt[c 
]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*( 
(-I)*Sqrt[c] + Sqrt[d]*x)))]], (Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])^2/(Sqrt 
[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2] - ((2*I)*b*Sqrt[c]*EllipticPi[((a + (I*b 
*Sqrt[c])/Sqrt[d])*((I*Sqrt[c])/Sqrt[d] + (I*Sqrt[e])/Sqrt[f]))/((a - (I*b 
*Sqrt[c])/Sqrt[d])*(((-I)*Sqrt[c])/Sqrt[d] + (I*Sqrt[e])/Sqrt[f])), ArcSin 
[Sqrt[-(((-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x)...
 

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.

\(\displaystyle \int \frac {1}{(a+b x)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{(a+b x)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}dx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [A] (verified)

Time = 10.24 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.14

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {b^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{\left (a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e \right ) \left (b x +a \right )}-\frac {a^{2} d f \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\left (a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b^{2} d e \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\left (a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {a \left (2 a^{2} d f +b^{2} c f +d e \,b^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+2 c e +\left (c f +d e \right ) x^{2}+\frac {2 d f \,x^{2} a^{2}}{b^{2}}}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}}+\frac {b \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, -\frac {c \,b^{2}}{d \,a^{2}}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{\sqrt {-\frac {d}{c}}\, a \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\left (a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e \right ) b}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(679\)
default	\(\text {Expression too large to display}\)	\(2422\)

Input:

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

Output:

((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(-b^3/(a^4*d*f 
+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(b*x 
+a)-a^2*d*f/(a^4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/(-d/c)^(1/2)*(1+1/c* 
x^2*d)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Ellipti 
cF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-b^2*d/(a^4*d*f+a^2*b^2*c*f+a^2 
*b^2*d*e+b^4*c*e)*e/(-d/c)^(1/2)*(1+1/c*x^2*d)^(1/2)*(1+f*x^2/e)^(1/2)/(d* 
f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e 
/d)^(1/2))-EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2)))+a*(2*a^2*d* 
f+b^2*c*f+b^2*d*e)/(a^4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/b*(-1/2/(d*f* 
a^4/b^4+(c*f+d*e)*a^2/b^2+c*e)^(1/2)*arctanh(1/2*((c*f+d*e)*a^2/b^2+2*c*e+ 
(c*f+d*e)*x^2+2*d*f*x^2*a^2/b^2)/(d*f*a^4/b^4+(c*f+d*e)*a^2/b^2+c*e)^(1/2) 
/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2))+1/(-d/c)^(1/2)/a*b*(1+1/c*x^2*d)^(1/ 
2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-d/ 
c)^(1/2),-1/d*c/a^2*b^2,(-f/e)^(1/2)/(-d/c)^(1/2))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(1/((a + b*x)**2*sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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