\(\int \frac {A+B x^2}{\sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)} \, dx\) [22]

Optimal result

Integrand size = 39, antiderivative size = 377 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {d} (B c-A d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \left (a B d^2 e-b B c^2 f-A d (b d e-2 b c f+a d f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

d^(1/2)*(-A*d+B*c)*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c) 
^(1/2),(1-b*c/a/d)^(1/2))/c^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(c*(b*x^2+a)/a/(d* 
x^2+c))^(1/2)/(d*x^2+c)^(1/2)-c^(1/2)*(a*B*d^2*e-b*B*c^2*f-A*d*(a*d*f-2*b* 
c*f+b*d*e))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b 
*c/a/d)^(1/2))/a/d^(1/2)/(-a*d+b*c)/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c)) 
^(1/2)/(d*x^2+c)^(1/2)-c^(3/2)*f*(-A*f+B*e)*(b*x^2+a)^(1/2)*EllipticPi(d^( 
1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/ 
(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d (B c-A d) e x \left (a+b x^2\right )+i b c (B c-A d) e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) (B e-A f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c (-b c+a d) e (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*(B*c - A*d)*e*x*(a + b*x^2) + I*b*c*(B*c - A*d)*e*Sqrt[1 + (b 
*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c) 
] + I*c*(-(b*c) + a*d)*(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c] 
*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]* 
c*(-(b*c) + a*d)*e*(-(d*e) + c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {A f-B e}{f \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {B}{f \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c^{3/2} f \sqrt {a+b x^2} (B e-A f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d^{3/2} \sqrt {a+b x^2} (B e-A f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} f \sqrt {c+d x^2} (b c-a d) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} (B e-A f) (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a f \sqrt {c+d x^2} (b c-a d) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b B \sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} f \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {B \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} f \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\)

Input:

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

Output:

-((B*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b 
*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*S 
qrt[c + d*x^2])) + (d^(3/2)*(B*e - A*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[( 
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*f*(d*e - c*f)* 
Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*B*Sqrt[c]*Sqrt 
[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sq 
rt[d]*(b*c - a*d)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) 
 - (Sqrt[c]*Sqrt[d]*(B*e - A*f)*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[a + b*x^2]* 
EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*f* 
(d*e - c*f)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^ 
(3/2)*f*(B*e - A*f)*Sqrt[a + b*x^2]*EllipticPi[1 - (c*f)/(d*e), ArcTan[(Sq 
rt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*e*(d*e - c*f)^2*Sqrt[(c*(a 
 + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 8.57 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.42

method	result	size
default	\(\frac {\left (-A \sqrt {-\frac {b}{a}}\, b \,d^{2} e \,x^{3}+B \sqrt {-\frac {b}{a}}\, b c d e \,x^{3}+A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e +A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c d f -A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,c^{2} f -B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} e -B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c d e +B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,c^{2} e -A \sqrt {-\frac {b}{a}}\, a \,d^{2} e x +B \sqrt {-\frac {b}{a}}\, a c d e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{c \left (a d -b c \right ) e \sqrt {-\frac {b}{a}}\, \left (c f -d e \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(537\)
elliptic	\(\text {Expression too large to display}\)	\(1180\)

Input:

int((B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVE 
RBOSE)
 

Output:

(-A*(-b/a)^(1/2)*b*d^2*e*x^3+B*(-b/a)^(1/2)*b*c*d*e*x^3+A*((b*x^2+a)/a)^(1 
/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*d*e+ 
A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/ 
e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c*d*f-A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c) 
^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c^ 
2*f-B*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a* 
d/b/c)^(1/2))*b*c^2*e-B*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi 
(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c*d*e+B*((b*x^2+a)/ 
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1 
/2)/(-b/a)^(1/2))*b*c^2*e-A*(-b/a)^(1/2)*a*d^2*e*x+B*(-b/a)^(1/2)*a*c*d*e* 
x)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/c/(a*d-b*c)/e/(-b/a)^(1/2)/(c*f-d*e)/(b 
*d*x^4+a*d*x^2+b*c*x^2+a*c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{2} f \,x^{6}+2 c d f \,x^{4}+d^{2} e \,x^{4}+c^{2} f \,x^{2}+2 c d e \,x^{2}+c^{2} e}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**2*e + c**2*f*x**2 + 2*c*d*e*x* 
*2 + 2*c*d*f*x**4 + d**2*e*x**4 + d**2*f*x**6),x)