\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^{3/2} (A+B x^2+C x^4)}{e+f x^2} \, dx\) [8]

Optimal result

Integrand size = 44, antiderivative size = 1070 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{e+f x^2} \, dx =\text {Too large to display} \] Output:

1/105*(8*a^3*C*d^3*f^3+a^2*b*d^2*f^2*(-14*B*d*f-19*C*c*f+14*C*d*e)-a*b^2*d 
*f*(7*d*f*(-5*A*d*f-7*B*c*f+5*B*d*e)-C*(9*c^2*f^2-49*c*d*e*f+35*d^2*e^2))- 
b^3*(C*(6*c^3*f^3+21*c^2*d*e*f^2-140*c*d^2*e^2*f+105*d^3*e^3)+7*d*f*(5*A*d 
*f*(-4*c*f+3*d*e)-B*(3*c^2*f^2-20*c*d*e*f+15*d^2*e^2))))*x*(d*x^2+c)^(1/2) 
/b^2/d^2/f^4/(b*x^2+a)^(1/2)-1/105*(4*a^2*C*d^2*f^2+a*b*d*f*(-7*B*d*f-9*C* 
c*f+7*C*d*e)+b^2*(7*d*f*(-5*A*d*f-6*B*c*f+5*B*d*e)-C*(3*c^2*f^2-42*c*d*e*f 
+35*d^2*e^2)))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d/f^3+1/35*(a*C*d*f-b 
*(-7*B*d*f-8*C*c*f+7*C*d*e))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/f^2+1/7 
*C*d*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f-1/105*a^(1/2)*(8*a^3*C*d^3*f^3+ 
a^2*b*d^2*f^2*(-14*B*d*f-19*C*c*f+14*C*d*e)-a*b^2*d*f*(7*d*f*(-5*A*d*f-7*B 
*c*f+5*B*d*e)-C*(9*c^2*f^2-49*c*d*e*f+35*d^2*e^2))-b^3*(C*(6*c^3*f^3+21*c^ 
2*d*e*f^2-140*c*d^2*e^2*f+105*d^3*e^3)+7*d*f*(5*A*d*f*(-4*c*f+3*d*e)-B*(3* 
c^2*f^2-20*c*d*e*f+15*d^2*e^2))))*(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))/b^(5/2)/d^2/f^4/(b*x^2+a)^(1/2)/( 
a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/105*a^(3/2)*(4*a^2*c*C*d^2*f^3+a*b*c*d*f^ 
2*(-7*B*d*f-9*C*c*f+7*C*d*e)-b^2*(C*(3*c^3*f^3+63*c^2*d*e*f^2-175*c*d^2*e^ 
2*f+105*d^3*e^3)+7*d*f*(5*A*d*f*(-5*c*f+3*d*e)-B*(9*c^2*f^2-25*c*d*e*f+15* 
d^2*e^2))))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a 
*d/b/c)^(1/2))/b^(5/2)/c/d/f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^( 
1/2)+a^(3/2)*(-c*f+d*e)^2*(A*f^2-B*e*f+C*e^2)*(d*x^2+c)^(1/2)*EllipticP...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.16 (sec) , antiderivative size = 6720, normalized size of antiderivative = 6.28 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{e+f x^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.45 (sec) , antiderivative size = 1321, normalized size of antiderivative = 1.23, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x^3}{7 f}+\frac {C (3 b c+a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b f}-\frac {d (C e-B f) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x}{5 b f^2}-\frac {2 (3 b c-a d) (C e-B f) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b f^2}+\frac {d \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 f^3}+\frac {C \left (-\frac {4 d a^2}{b}+9 c a+\frac {3 b c^2}{d}\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 b f}-\frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) (C e-B f) \sqrt {b x^2+a} x}{15 b^2 f^2 \sqrt {d x^2+c}}-\frac {d (3 b d e-4 b c f-a d f) \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} x}{3 b f^4 \sqrt {d x^2+c}}-\frac {C (2 b c-a d) \left (3 b^2 c^2-3 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} x}{105 b^3 d f \sqrt {d x^2+c}}+\frac {\sqrt {c} \left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) (C e-B f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 \sqrt {d} f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} \sqrt {d} (3 b d e-4 b c f-a d f) \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b f^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} C (2 b c-a d) \left (3 b^2 c^2-3 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b^3 d^{3/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} (9 b c-a d) (C e-B f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b \sqrt {d} f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {d} (3 d e-5 c f) \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 f^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} C \left (3 b^2 c^2+9 a b d c-4 a^2 d^2\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 b^2 d^{3/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {a^{3/2} (d e-c f)^2 \left (C e^2-B f e+A f^2\right ) \sqrt {d x^2+c} \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^4 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\)

Input:

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

Output:

-1/105*(C*(2*b*c - a*d)*(3*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*x*Sqrt[a + b*x 
^2])/(b^3*d*f*Sqrt[c + d*x^2]) - ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*(C*e 
 - B*f)*x*Sqrt[a + b*x^2])/(15*b^2*f^2*Sqrt[c + d*x^2]) - (d*(3*b*d*e - 4* 
b*c*f - a*d*f)*(C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2])/(3*b*f^4*Sqrt[c 
+ d*x^2]) + (C*(9*a*c + (3*b*c^2)/d - (4*a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[ 
c + d*x^2])/(105*b*f) - (2*(3*b*c - a*d)*(C*e - B*f)*x*Sqrt[a + b*x^2]*Sqr 
t[c + d*x^2])/(15*b*f^2) + (d*(C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2]*Sq 
rt[c + d*x^2])/(3*f^3) + (C*(3*b*c + a*d)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x 
^2])/(35*b*f) - (d*(C*e - B*f)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b*f 
^2) + (C*x^3*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(7*f) + (Sqrt[c]*C*(2*b*c 
- a*d)*(3*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTa 
n[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b^3*d^(3/2)*f*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(3*b^2*c^2 + 7*a*b*c* 
d - 2*a^2*d^2)*(C*e - B*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq 
rt[c]], 1 - (b*c)/(a*d)])/(15*b^2*Sqrt[d]*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + 
 d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[d]*(3*b*d*e - 4*b*c*f - a*d*f)* 
(C*e^2 - B*e*f + A*f^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[ 
c]], 1 - (b*c)/(a*d)])/(3*b*f^4*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt 
[c + d*x^2]) - (c^(3/2)*C*(3*b^2*c^2 + 9*a*b*c*d - 4*a^2*d^2)*Sqrt[a + b*x 
^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b^2*d...
 

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 = 19.54 (sec) , antiderivative size = 1360, normalized size of antiderivative = 1.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(1360\)
default	\(\text {Expression too large to display}\)	\(5349\)
elliptic	\(\text {Expression too large to display}\)	\(6489\)

Input:

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

Output:

1/105*x/d*(15*C*b^2*d^2*f^2*x^4+21*B*b^2*d^2*f^2*x^2+3*C*a*b*d^2*f^2*x^2+2 
4*C*b^2*c*d*f^2*x^2-21*C*b^2*d^2*e*f*x^2+35*A*b^2*d^2*f^2+7*B*a*b*d^2*f^2+ 
42*B*b^2*c*d*f^2-35*B*b^2*d^2*e*f-4*C*a^2*d^2*f^2+9*C*a*b*c*d*f^2-7*C*a*b* 
d^2*e*f+3*C*b^2*c^2*f^2-42*C*b^2*c*d*e*f+35*C*b^2*d^2*e^2)*(b*x^2+a)^(1/2) 
*(d*x^2+c)^(1/2)/b^2/f^3+1/105/f^3/d/b^2*((175*A*a*b^2*c*d^2*f^4-105*A*a*b 
^2*d^3*e*f^3+105*A*b^3*c^2*d*f^4-210*A*b^3*c*d^2*e*f^3+105*A*b^3*d^3*e^2*f 
^2-7*B*a^2*b*c*d^2*f^4+63*B*a*b^2*c^2*d*f^4-175*B*a*b^2*c*d^2*e*f^3+105*B* 
a*b^2*d^3*e^2*f^2-105*B*b^3*c^2*d*e*f^3+210*B*b^3*c*d^2*e^2*f^2-105*B*b^3* 
d^3*e^3*f+4*C*a^3*c*d^2*f^4-9*C*a^2*b*c^2*d*f^4+7*C*a^2*b*c*d^2*e*f^3-3*C* 
a*b^2*c^3*f^4-63*C*a*b^2*c^2*d*e*f^3+175*C*a*b^2*c*d^2*e^2*f^2-105*C*a*b^2 
*d^3*e^3*f+105*C*b^3*c^2*d*e^2*f^2-210*C*b^3*c*d^2*e^3*f+105*C*b^3*d^3*e^4 
)/f^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b* 
c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/f*(3 
5*A*a*b^2*d^3*f^3+140*A*b^3*c*d^2*f^3-105*A*b^3*d^3*e*f^2-14*B*a^2*b*d^3*f 
^3+49*B*a*b^2*c*d^2*f^3-35*B*a*b^2*d^3*e*f^2+21*B*b^3*c^2*d*f^3-140*B*b^3* 
c*d^2*e*f^2+105*B*b^3*d^3*e^2*f+8*C*a^3*d^3*f^3-19*C*a^2*b*c*d^2*f^3+14*C* 
a^2*b*d^3*e*f^2+9*C*a*b^2*c^2*d*f^3-49*C*a*b^2*c*d^2*e*f^2+35*C*a*b^2*d^3* 
e^2*f-6*C*b^3*c^3*f^3-21*C*b^3*c^2*d*e*f^2+140*C*b^3*c*d^2*e^2*f-105*C*b^3 
*d^3*e^3)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d* 
x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(C*x^4+B*x^2+A)/(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]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(C*x^4+B*x^2+A)/(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. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{e+f x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (C\,x^4+B\,x^2+A\right )}{f\,x^2+e} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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