\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4)}{(c+d x^2)^{7/2} (e+f x^2)} \, dx\) [13]

Optimal result

Integrand size = 44, antiderivative size = 1093 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Output:

1/5*(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)^(5/2)-1 
/15*(a*d*(4*A*d^3*e+c^3*C*f+c*d^2*(-9*A*f+B*e)-c^2*(-4*B*d*f+6*C*d*e))-b*c 
*(3*A*d^3*e+2*c^3*C*f+2*c*d^2*(-4*A*f+B*e)-c^2*(-3*B*d*f+7*C*d*e)))*x*(b*x 
^2+a)^(1/2)/c^2/d/(-a*d+b*c)/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*(a^2*d^2*(8 
*A*d^4*e^2-2*c^4*C*f^2+2*c*d^3*e*(-13*A*f+B*e)+2*c^3*d*f*(-4*B*f+7*C*e)+3* 
c^2*d^2*(11*A*f^2-3*B*e*f+C*e^2))+b^2*c^2*(3*A*d^4*e^2-2*c^4*C*f^2+c*d^3*e 
*(-11*A*f+2*B*e)+3*c^3*d*f*(-B*f+3*C*e)+c^2*d^2*(23*A*f^2-14*B*e*f+8*C*e^2 
))-a*b*c*d*(13*A*d^4*e^2-2*c^4*C*f^2+c*d^3*e*(-41*A*f+2*B*e)+c^3*d*f*(-13* 
B*f+19*C*e)+c^2*d^2*(58*A*f^2-19*B*e*f+13*C*e^2)))*(b*x^2+a)^(1/2)*Ellipti 
cE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/c^(5/2)/d^(3/2)/ 
(-a*d+b*c)^2/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+ 
1/15*(15*b^3*c^5*d*f*(C*e^2-f*(-A*f+B*e))-15*a^3*c^2*d^4*f*(C*e^2-f*(-A*f+ 
B*e))-a^2*b*d*(4*A*d^5*e^3+c^5*C*f^3+c*d^4*e^2*(-17*A*f+B*e)-4*c^4*d*f^2*( 
-B*f+2*C*e)-2*c^2*d^3*e*(-11*A*f^2-B*e*f+3*C*e^2)-2*c^3*d^2*f*(27*A*f^2-19 
*B*e*f+16*C*e^2))+a*b^2*c*(6*A*d^5*e^3-c^5*C*f^3-c*d^4*e^2*(23*A*f+B*e)-2* 
c^4*d*f^2*(-3*B*f+C*e)-2*c^3*d^2*f*(19*C*e^2-4*f*(-7*A*f+4*B*e))-4*c^2*d^3 
*e*(C*e^2-f*(7*A*f+2*B*e))))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2 
)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/c^(3/2)/d^(3/2)/(-a*d+b*c)^2/(-c*f+d*e)^ 
4/(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)*(A* 
f^2-B*e*f+C*e^2)*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.76 (sec) , antiderivative size = 1031, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d e x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 \left (c^2 C-B c d+A d^2\right ) (d e-c f)^2-c (b c-a d) (d e-c f) \left (-b c \left (3 A d^3 e+2 c^3 C f+2 c d^2 (B e-4 A f)+c^2 (-7 C d e+3 B d f)\right )+a d \left (4 A d^3 e+c^3 C f+c d^2 (B e-9 A f)+c^2 (-6 C d e+4 B d f)\right )\right ) \left (c+d x^2\right )+\left (a b c d \left (-13 A d^4 e^2+2 c^4 C f^2+c d^3 e (-2 B e+41 A f)+c^3 d f (-19 C e+13 B f)+c^2 d^2 \left (-13 C e^2+19 B e f-58 A f^2\right )\right )+a^2 d^2 \left (8 A d^4 e^2-2 c^4 C f^2+2 c d^3 e (B e-13 A f)+2 c^3 d f (7 C e-4 B f)+3 c^2 d^2 \left (C e^2-3 B e f+11 A f^2\right )\right )+b^2 c^2 \left (3 A d^4 e^2-2 c^4 C f^2+c d^3 e (2 B e-11 A f)-3 c^3 d f (-3 C e+B f)+c^2 d^2 \left (8 C e^2-14 B e f+23 A f^2\right )\right )\right ) \left (c+d x^2\right )^2\right )+i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (b e \left (a b c d \left (-13 A d^4 e^2+2 c^4 C f^2+c d^3 e (-2 B e+41 A f)+c^3 d f (-19 C e+13 B f)+c^2 d^2 \left (-13 C e^2+19 B e f-58 A f^2\right )\right )+a^2 d^2 \left (8 A d^4 e^2-2 c^4 C f^2+2 c d^3 e (B e-13 A f)+2 c^3 d f (7 C e-4 B f)+3 c^2 d^2 \left (C e^2-3 B e f+11 A f^2\right )\right )+b^2 c^2 \left (3 A d^4 e^2-2 c^4 C f^2+c d^3 e (2 B e-11 A f)-3 c^3 d f (-3 C e+B f)+c^2 d^2 \left (8 C e^2-14 B e f+23 A f^2\right )\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (b e (-d e+c f) \left (b c \left (3 A d^3 e+2 c^3 C f+2 c d^2 (B e-4 A f)+c^2 (-7 C d e+3 B d f)\right )-a d \left (4 A d^3 e+c^3 C f+c d^2 (B e-9 A f)+c^2 (-6 C d e+4 B d f)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-15 c^2 d^2 (b c-a d) (b e-a f) \left (C e^2+f (-B e+A f)\right ) \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 )\right )}{15 \sqrt {\frac {b}{a}} c^3 d^2 (b c-a d)^2 e (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4))/((c + d*x^2)^(7/2)*(e + f* 
x^2)),x]
 

Output:

(Sqrt[b/a]*d*e*x*(a + b*x^2)*(3*c^2*(b*c - a*d)^2*(c^2*C - B*c*d + A*d^2)* 
(d*e - c*f)^2 - c*(b*c - a*d)*(d*e - c*f)*(-(b*c*(3*A*d^3*e + 2*c^3*C*f + 
2*c*d^2*(B*e - 4*A*f) + c^2*(-7*C*d*e + 3*B*d*f))) + a*d*(4*A*d^3*e + c^3* 
C*f + c*d^2*(B*e - 9*A*f) + c^2*(-6*C*d*e + 4*B*d*f)))*(c + d*x^2) + (a*b* 
c*d*(-13*A*d^4*e^2 + 2*c^4*C*f^2 + c*d^3*e*(-2*B*e + 41*A*f) + c^3*d*f*(-1 
9*C*e + 13*B*f) + c^2*d^2*(-13*C*e^2 + 19*B*e*f - 58*A*f^2)) + a^2*d^2*(8* 
A*d^4*e^2 - 2*c^4*C*f^2 + 2*c*d^3*e*(B*e - 13*A*f) + 2*c^3*d*f*(7*C*e - 4* 
B*f) + 3*c^2*d^2*(C*e^2 - 3*B*e*f + 11*A*f^2)) + b^2*c^2*(3*A*d^4*e^2 - 2* 
c^4*C*f^2 + c*d^3*e*(2*B*e - 11*A*f) - 3*c^3*d*f*(-3*C*e + B*f) + c^2*d^2* 
(8*C*e^2 - 14*B*e*f + 23*A*f^2)))*(c + d*x^2)^2) + I*c*Sqrt[1 + (b*x^2)/a] 
*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*(b*e*(a*b*c*d*(-13*A*d^4*e^2 + 2*c^4*C* 
f^2 + c*d^3*e*(-2*B*e + 41*A*f) + c^3*d*f*(-19*C*e + 13*B*f) + c^2*d^2*(-1 
3*C*e^2 + 19*B*e*f - 58*A*f^2)) + a^2*d^2*(8*A*d^4*e^2 - 2*c^4*C*f^2 + 2*c 
*d^3*e*(B*e - 13*A*f) + 2*c^3*d*f*(7*C*e - 4*B*f) + 3*c^2*d^2*(C*e^2 - 3*B 
*e*f + 11*A*f^2)) + b^2*c^2*(3*A*d^4*e^2 - 2*c^4*C*f^2 + c*d^3*e*(2*B*e - 
11*A*f) - 3*c^3*d*f*(-3*C*e + B*f) + c^2*d^2*(8*C*e^2 - 14*B*e*f + 23*A*f^ 
2)))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b*c - a*d)*(b*e*(-( 
d*e) + c*f)*(b*c*(3*A*d^3*e + 2*c^3*C*f + 2*c*d^2*(B*e - 4*A*f) + c^2*(-7* 
C*d*e + 3*B*d*f)) - a*d*(4*A*d^3*e + c^3*C*f + c*d^2*(B*e - 9*A*f) + c^2*( 
-6*C*d*e + 4*B*d*f)))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - ...
 

Rubi [A] (verified)

Time = 2.74 (sec) , antiderivative size = 1396, normalized size of antiderivative = 1.28, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {\sqrt {a+b x^2} \left (A f^2-B e f+C e^2\right )}{f^2 \left (c+d x^2\right )^{7/2} \left (e+f x^2\right )}-\frac {\sqrt {a+b x^2} (C e-B f)}{f^2 \left (c+d x^2\right )^{7/2}}+\frac {C x^2 \sqrt {a+b x^2}}{f \left (c+d x^2\right )^{7/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {f \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 ) a^{3/2}}{\sqrt {b} c e (d e-c f)^3 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\left (3 b^2 c^2-13 a b d c+8 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 c^{5/2} \sqrt {d} (b c-a d)^2 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 {d} \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 )}{\sqrt {c} (d e-c f)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (C e^2-B f e+A f^2\right ) \left (-b^2 (3 d e-8 c f) c^2+a b d (13 d e-28 c f) c-2 a^2 d^2 (4 d e-9 c f)\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} (b c-a d)^2 f^2 (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 C \left (b^2 c^2-a b d c+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 )}{15 c^{3/2} d^{3/2} (b c-a d)^2 f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 b (3 b c-2 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 c^{3/2} \sqrt {d} (b c-a d)^2 f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {d} \left (C e^2-B f e+A f^2\right ) (b c (6 d e-11 c f)-a d (4 d e-9 c 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 c^{3/2} (b c-a d)^2 f^2 (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b C (b c+a d) \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 \sqrt {c} d^{3/2} (b c-a d)^2 f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(3 b c-4 a d) (C e-B f) x \sqrt {b x^2+a}}{15 c^2 (b c-a d) f^2 \left (d x^2+c\right )^{3/2}}-\frac {d \left (C e^2-B f e+A f^2\right ) (a d (4 d e-9 c f)-b c (3 d e-8 c f)) x \sqrt {b x^2+a}}{15 c^2 (b c-a d) f^2 (d e-c f)^2 \left (d x^2+c\right )^{3/2}}+\frac {C (2 b c-a d) x \sqrt {b x^2+a}}{15 c d (b c-a d) f \left (d x^2+c\right )^{3/2}}-\frac {(C e-B f) x \sqrt {b x^2+a}}{5 c f^2 \left (d x^2+c\right )^{5/2}}+\frac {d \left (C e^2-B f e+A f^2\right ) x \sqrt {b x^2+a}}{5 c f^2 (d e-c f) \left (d x^2+c\right )^{5/2}}-\frac {C x \sqrt {b x^2+a}}{5 d f \left (d x^2+c\right )^{5/2}}\)

Input:

Int[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4))/((c + d*x^2)^(7/2)*(e + f*x^2)), 
x]
 

Output:

-1/5*(C*x*Sqrt[a + b*x^2])/(d*f*(c + d*x^2)^(5/2)) - ((C*e - B*f)*x*Sqrt[a 
 + b*x^2])/(5*c*f^2*(c + d*x^2)^(5/2)) + (d*(C*e^2 - B*e*f + A*f^2)*x*Sqrt 
[a + b*x^2])/(5*c*f^2*(d*e - c*f)*(c + d*x^2)^(5/2)) + (C*(2*b*c - a*d)*x* 
Sqrt[a + b*x^2])/(15*c*d*(b*c - a*d)*f*(c + d*x^2)^(3/2)) - ((3*b*c - 4*a* 
d)*(C*e - B*f)*x*Sqrt[a + b*x^2])/(15*c^2*(b*c - a*d)*f^2*(c + d*x^2)^(3/2 
)) - (d*(C*e^2 - B*e*f + A*f^2)*(a*d*(4*d*e - 9*c*f) - b*c*(3*d*e - 8*c*f) 
)*x*Sqrt[a + b*x^2])/(15*c^2*(b*c - a*d)*f^2*(d*e - c*f)^2*(c + d*x^2)^(3/ 
2)) + (2*C*(b^2*c^2 - a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[ 
(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*c^(3/2)*d^(3/2)*(b*c - a*d)^2* 
f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - ((3*b^2*c^2 - 1 
3*a*b*c*d + 8*a^2*d^2)*(C*e - B*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[ 
d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*c^(5/2)*Sqrt[d]*(b*c - a*d)^2*f^2*Sq 
rt[(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)/Sqrt[c]], 1 - ( 
b*c)/(a*d)])/(Sqrt[c]*(d*e - c*f)^3*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)*(a*b*c*d*(13*d*e - 28* 
c*f) - 2*a^2*d^2*(4*d*e - 9*c*f) - b^2*c^2*(3*d*e - 8*c*f))*Sqrt[a + b*x^2 
]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*c^(5/2)*(b* 
c - a*d)^2*f^2*(d*e - c*f)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c 
+ d*x^2]) - (b*C*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(13343\) vs. \(2(1063)=2126\).

Time = 11.50 (sec) , antiderivative size = 13344, normalized size of antiderivative = 12.21

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(13344\)
default	\(\text {Expression too large to display}\)	\(18396\)

Input:

int((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(7/2)/(f*x^2+e),x,method=_RE 
TURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(7/2)/(f*x^2+e),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(1/2)*(C*x**4+B*x**2+A)/(d*x**2+c)**(7/2)/(f*x**2+e) 
,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(7/2)/(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)^(7/2)*(f*x^2 + 
e)), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(7/2)/(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)^(7/2)*(f*x^2 + 
e)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (d\,x^2+c\right )}^{7/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4))/((c + d*x^2)^(7/2)*(e + f*x^2) 
),x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4))/((c + d*x^2)^(7/2)*(e + f*x^2) 
), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(7/2)/(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 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(4*a**2*c**4 
*d*e*f + 4*a**2*c**4*d*f**2*x**2 + 16*a**2*c**3*d**2*e*f*x**2 + 16*a**2*c* 
*3*d**2*f**2*x**4 + 24*a**2*c**2*d**3*e*f*x**4 + 24*a**2*c**2*d**3*f**2*x* 
*6 + 16*a**2*c*d**4*e*f*x**6 + 16*a**2*c*d**4*f**2*x**8 + 4*a**2*d**5*e*f* 
x**8 + 4*a**2*d**5*f**2*x**10 - 2*a*b*c**5*e*f - 2*a*b*c**5*f**2*x**2 + 3* 
a*b*c**4*d*e**2 - a*b*c**4*d*e*f*x**2 - 4*a*b*c**4*d*f**2*x**4 + 12*a*b*c* 
*3*d**2*e**2*x**2 + 16*a*b*c**3*d**2*e*f*x**4 + 4*a*b*c**3*d**2*f**2*x**6 
+ 18*a*b*c**2*d**3*e**2*x**4 + 34*a*b*c**2*d**3*e*f*x**6 + 16*a*b*c**2*d** 
3*f**2*x**8 + 12*a*b*c*d**4*e**2*x**6 + 26*a*b*c*d**4*e*f*x**8 + 14*a*b*c* 
d**4*f**2*x**10 + 3*a*b*d**5*e**2*x**8 + 7*a*b*d**5*e*f*x**10 + 4*a*b*d**5 
*f**2*x**12 - 2*b**2*c**5*e*f*x**2 - 2*b**2*c**5*f**2*x**4 + 3*b**2*c**4*d 
*e**2*x**2 - 5*b**2*c**4*d*e*f*x**4 - 8*b**2*c**4*d*f**2*x**6 + 12*b**2*c* 
*3*d**2*e**2*x**4 - 12*b**2*c**3*d**2*f**2*x**8 + 18*b**2*c**2*d**3*e**2*x 
**6 + 10*b**2*c**2*d**3*e*f*x**8 - 8*b**2*c**2*d**3*f**2*x**10 + 12*b**2*c 
*d**4*e**2*x**8 + 10*b**2*c*d**4*e*f*x**10 - 2*b**2*c*d**4*f**2*x**12 + 3* 
b**2*d**5*e**2*x**10 + 3*b**2*d**5*e*f*x**12),x)*a**2*b*c**4*d**2*f**2 + 1 
2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(4*a**2*c**4*d*e*f + 4*a**2 
*c**4*d*f**2*x**2 + 16*a**2*c**3*d**2*e*f*x**2 + 16*a**2*c**3*d**2*f**2*x* 
*4 + 24*a**2*c**2*d**3*e*f*x**4 + 24*a**2*c**2*d**3*f**2*x**6 + 16*a**2...