\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4)}{(c+d x^2)^{3/2} (e+f x^2)} \, dx\) [11]

Optimal result

Integrand size = 44, antiderivative size = 481 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\left (c^2 C-B c d+A d^2\right ) x \sqrt {a+b x^2}}{c d (d e-c f) \sqrt {c+d x^2}}-\frac {b \left (2 c^2 C f+A d^2 f-c d (C e+B f)\right ) x \sqrt {c+d x^2}}{c d^2 f (d e-c f) \sqrt {a+b x^2}}+\frac {\sqrt {a} \sqrt {b} \left (2 c^2 C f+A d^2 f-c d (C e+B f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{c d^2 f (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} C \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c d f \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (C e^2-B e f+A f^2\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.69 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-i b c e f \left (2 c^2 C f+A d^2 f-c d (C e+B f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c e (-d e+c f) (b B d f+a C d f-b C (d e+2 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} \left (c^2 C-B c d+A d^2\right ) e f^2 x \left (a+b x^2\right )+i c d (-b e+a f) \left (C e^2+f (-B e+A f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c d^2 e f^2 (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. \(1299\) vs. \(2(459)=918\).

Time = 6.56 (sec) , antiderivative size = 1300, normalized size of antiderivative = 2.70

method	result	size
default	\(\text {Expression too large to display}\)	\(1300\)
elliptic	\(\text {Expression too large to display}\)	\(1542\)

Input:

int((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RE 
TURNVERBOSE)
 

Output:

(-A*(-b/a)^(1/2)*b*d^3*e*f^2*x^3+B*(-b/a)^(1/2)*b*c*d^2*e*f^2*x^3-C*(-b/a) 
^(1/2)*b*c^2*d*e*f^2*x^3+A*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti 
cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*d^2*e*f^2+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^2*f^3-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*d^2*e*f^2-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*d*e*f^2-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^2*e*f^2+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*d^2*e^2*f+B*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ell 
ipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2*d*e*f^2-B*((b*x^2+a)/a)^(1/2) 
*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*d^2*e^2 
*f+2*C*((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^3*e*f^2-C*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellip 
ticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2*d*e^2*f+C*((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^2*e^2*f-C*((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*d^2*e^3+C*((b*x^2 
+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/...
 

Fricas [F(-1)]

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

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(3/2)/(f*x^2+e),x, alg 
orithm="giac")
 

Output:

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*x + sqrt(c + d*x**2)*sqrt(a + b*x** 
2)*b**2*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a*c**3*e*f + 
 2*a*c**3*f**2*x**2 + a*c**2*d*e**2 + 5*a*c**2*d*e*f*x**2 + 4*a*c**2*d*f** 
2*x**4 + 2*a*c*d**2*e**2*x**2 + 4*a*c*d**2*e*f*x**4 + 2*a*c*d**2*f**2*x**6 
 + a*d**3*e**2*x**4 + a*d**3*e*f*x**6 + 2*b*c**3*e*f*x**2 + 2*b*c**3*f**2* 
x**4 + b*c**2*d*e**2*x**2 + 5*b*c**2*d*e*f*x**4 + 4*b*c**2*d*f**2*x**6 + 2 
*b*c*d**2*e**2*x**4 + 4*b*c*d**2*e*f*x**6 + 2*b*c*d**2*f**2*x**8 + b*d**3* 
e**2*x**6 + b*d**3*e*f*x**8),x)*a*b*c**3*d*f**2 - int((sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*x**6)/(2*a*c**3*e*f + 2*a*c**3*f**2*x**2 + a*c**2*d*e**2 + 
5*a*c**2*d*e*f*x**2 + 4*a*c**2*d*f**2*x**4 + 2*a*c*d**2*e**2*x**2 + 4*a*c* 
d**2*e*f*x**4 + 2*a*c*d**2*f**2*x**6 + a*d**3*e**2*x**4 + a*d**3*e*f*x**6 
+ 2*b*c**3*e*f*x**2 + 2*b*c**3*f**2*x**4 + b*c**2*d*e**2*x**2 + 5*b*c**2*d 
*e*f*x**4 + 4*b*c**2*d*f**2*x**6 + 2*b*c*d**2*e**2*x**4 + 4*b*c*d**2*e*f*x 
**6 + 2*b*c*d**2*f**2*x**8 + b*d**3*e**2*x**6 + b*d**3*e*f*x**8),x)*a*b*c* 
*2*d**2*e*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a*c**3*e*f 
 + 2*a*c**3*f**2*x**2 + a*c**2*d*e**2 + 5*a*c**2*d*e*f*x**2 + 4*a*c**2*d*f 
**2*x**4 + 2*a*c*d**2*e**2*x**2 + 4*a*c*d**2*e*f*x**4 + 2*a*c*d**2*f**2*x* 
*6 + a*d**3*e**2*x**4 + a*d**3*e*f*x**6 + 2*b*c**3*e*f*x**2 + 2*b*c**3*f** 
2*x**4 + b*c**2*d*e**2*x**2 + 5*b*c**2*d*e*f*x**4 + 4*b*c**2*d*f**2*x**6 + 
 2*b*c*d**2*e**2*x**4 + 4*b*c*d**2*e*f*x**6 + 2*b*c*d**2*f**2*x**8 + b*...