\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^{3/2}}{x^6 (e+f x^2)^2} \, dx\) [108]

Optimal result

Integrand size = 35, antiderivative size = 1412 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^6 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Output:

-1/30*b*(4*b^2*c^2*e^2-2*a*b*c*e*(-10*c*f+7*d*e)-a^2*(105*c^2*f^2-95*c*d*e 
*f+6*d^2*e^2))*x*(d*x^2+c)^(1/2)/a^2/c/e^4/(b*x^2+a)^(1/2)+1/30*(2*b^3*c*d 
*e^3*(-c*f+d*e)+5*a^3*d*f^2*(42*c^2*f^2-59*c*d*e*f+20*d^2*e^2)-a^2*b*f*(-1 
05*c^3*f^3+380*c^2*d*e*f^2-397*c*d^2*e^2*f+122*d^3*e^3)+2*a*b^2*e*(-45*c^3 
*f^3+86*c^2*d*e*f^2-52*c*d^2*e^2*f+11*d^3*e^3))*x*(b*x^2+a)^(1/2)*(d*x^2+c 
)^(1/2)/a^2/c/e^4/(-a*f+b*e)/(-c*f+d*e)+1/30*d*(16*b^3*c*d*e^3*(-c*f+d*e)+ 
5*a^3*d*f^2*(21*c^2*f^2-26*c*d*e*f+8*d^2*e^2)+2*a*b^2*e*(-90*c^3*f^3+148*c 
^2*d*e*f^2-63*c*d^2*e^2*f+5*d^3*e^3)-5*a^2*b*f*(-42*c^3*f^3+77*c^2*d*e*f^2 
-48*c*d^2*e^2*f+10*d^3*e^3))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e 
^4/(-a*f+b*e)/(-c*f+d*e)+1/6*b*d^2*(2*b^2*d*e^3*(-c*f+d*e)-2*a*b*e*f*(9*c^ 
2*f^2-14*c*d*e*f+5*d^2*e^2)+a^2*f^2*(21*c^2*f^2-26*c*d*e*f+8*d^2*e^2))*x^5 
*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^4/(-a*f+b*e)/(-c*f+d*e)-1/5*(b* 
x^2+a)^(3/2)*(d*x^2+c)^(5/2)/a/c/e/x^5/(f*x^2+e)+1/15*(7*a*f+2*b*e)*(b*x^2 
+a)^(3/2)*(d*x^2+c)^(5/2)/a^2/c/e^2/x^3/(f*x^2+e)+1/15*(5*a*f*(-7*c*f+4*d* 
e)-b*e*(-2*c*f+5*d*e))*(b*x^2+a)^(3/2)*(d*x^2+c)^(5/2)/a^2/c^2/e^3/x/(f*x^ 
2+e)-1/6*f*(2*b^2*d*e^3*(-c*f+d*e)-2*a*b*e*f*(9*c^2*f^2-14*c*d*e*f+5*d^2*e 
^2)+a^2*f^2*(21*c^2*f^2-26*c*d*e*f+8*d^2*e^2))*x*(b*x^2+a)^(3/2)*(d*x^2+c) 
^(5/2)/a^2/c^2/e^4/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/30*b^(1/2)*(4*b^2*c^2 
*e^2-2*a*b*c*e*(-10*c*f+7*d*e)-a^2*(105*c^2*f^2-95*c*d*e*f+6*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)^...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.83 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^6 \left (e+f x^2\right )^2} \, dx=\frac {\frac {e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (15 a^2 c f^2 (d e-c f) x^6-6 a^2 c^2 e^2 \left (e+f x^2\right )-2 a c e (b c e+6 a d e-10 a c f) x^2 \left (e+f x^2\right )+2 \left (2 b^2 c^2 e^2+a b c e (-7 d e+10 c f)+a^2 \left (-3 d^2 e^2+40 c d e f-45 c^2 f^2\right )\right ) x^4 \left (e+f x^2\right )\right )}{c x^5 \left (e+f x^2\right )}+\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (b e \left (4 b^2 c^2 e^2+2 a b c e (-7 d e+10 c f)+a^2 \left (-6 d^2 e^2+95 c d e f-105 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e \left (4 b^2 c^2 e^2+4 a b c e (-4 d e+5 c f)+a^2 \left (-33 d^2 e^2+130 c d e f-105 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+15 a^2 (d e-c f) (a f (4 d e-7 c f)-3 b e (d e-2 c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}}}}{30 a^2 e^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(x^6*(e + f*x^2)^2),x]
 

Output:

((e*(a + b*x^2)*(c + d*x^2)*(15*a^2*c*f^2*(d*e - c*f)*x^6 - 6*a^2*c^2*e^2* 
(e + f*x^2) - 2*a*c*e*(b*c*e + 6*a*d*e - 10*a*c*f)*x^2*(e + f*x^2) + 2*(2* 
b^2*c^2*e^2 + a*b*c*e*(-7*d*e + 10*c*f) + a^2*(-3*d^2*e^2 + 40*c*d*e*f - 4 
5*c^2*f^2))*x^4*(e + f*x^2)))/(c*x^5*(e + f*x^2)) + (I*Sqrt[1 + (b*x^2)/a] 
*Sqrt[1 + (d*x^2)/c]*(b*e*(4*b^2*c^2*e^2 + 2*a*b*c*e*(-7*d*e + 10*c*f) + a 
^2*(-6*d^2*e^2 + 95*c*d*e*f - 105*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]* 
x], (a*d)/(b*c)] - b*e*(4*b^2*c^2*e^2 + 4*a*b*c*e*(-4*d*e + 5*c*f) + a^2*( 
-33*d^2*e^2 + 130*c*d*e*f - 105*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x] 
, (a*d)/(b*c)] + 15*a^2*(d*e - c*f)*(a*f*(4*d*e - 7*c*f) - 3*b*e*(d*e - 2* 
c*f))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/Sqrt[ 
b/a])/(30*a^2*e^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^6 \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^6 \left (e+f x^2\right )^2}dx\)

Input:

Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(x^6*(e + f*x^2)^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [A] (verified)

Time = 23.15 (sec) , antiderivative size = 1826, normalized size of antiderivative = 1.29

method	result	size
risch	\(\text {Expression too large to display}\)	\(1826\)
elliptic	\(\text {Expression too large to display}\)	\(2056\)
default	\(\text {Expression too large to display}\)	\(3497\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x,method=_RETURNVERBOS 
E)
 

Output:

-1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(45*a^2*c^2*f^2*x^4-40*a^2*c*d*e*f*x 
^4+3*a^2*d^2*e^2*x^4-10*a*b*c^2*e*f*x^4+7*a*b*c*d*e^2*x^4-2*b^2*c^2*e^2*x^ 
4-10*a^2*c^2*e*f*x^2+6*a^2*c*d*e^2*x^2+a*b*c^2*e^2*x^2+3*a^2*c^2*e^2)/a^2/ 
c/e^4/x^5+1/15/a^2/c/e^4*(-b*(45*a^2*c^2*f^2-40*a^2*c*d*e*f+3*a^2*d^2*e^2- 
10*a*b*c^2*e*f+7*a*b*c*d*e^2-2*b^2*c^2*e^2)*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)*(EllipticF(x*(-b/ 
a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/ 
c/b)^(1/2)))-a*b^2*c^2*d*e^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))-6*a^2*b*c*d^2*e^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))-15*a^2*c*e*(a*c^2*f^3-2*a*c*d*e*f^2+a*d^2*e^2*f-b*c 
^2*e*f^2+2*b*c*d*e^2*f-b*d^2*e^3)*(1/2*f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^ 
2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-1/2*d*b/(a*c*f^2-a*d* 
e*f-b*c*e*f+b*d*e^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))+1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*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)*Elliptic 
F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e* 
f+b*d*e^2)/e*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^...
 

Fricas [F(-1)]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/x**6/(f*x**2+e)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^6), x)
 

Giac [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^6), x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^6*(e + f*x^2)^2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^6*(e + f*x^2)^2), x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^6/(f*x^2+e)^2,x)