\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)^2}{(c+d x^2)^{9/2}} \, dx\) [67]

Optimal result

Integrand size = 32, antiderivative size = 737 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\left (a d (d e-c f)^2-b c \left (d^2 e^2-2 c d e f-6 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{7 c d^3 \left (c+d x^2\right )^{7/2}}-\frac {b f^2 x^5 \sqrt {a+b x^2}}{d \left (c+d x^2\right )^{7/2}}+\frac {2 \left (b c \left (d^2 e^2-9 c d e f-27 c^2 f^2\right )+a d \left (3 d^2 e^2+c d e f-4 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{35 c^2 d^3 \left (c+d x^2\right )^{5/2}}+\frac {\left (a b c d \left (15 d^2 e^2-2 c d e f-48 c^2 f^2\right )-a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+2 b^2 c^2 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{105 c^3 d^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {2 \left (a b^2 c^2 d \left (6 d^2 e^2-5 c d e f-36 c^2 f^2\right )-a^2 b c d^2 \left (36 d^2 e^2+5 c d e f-6 c^2 f^2\right )+a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+b^3 c^3 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{7/2} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \left (a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )-a b c d \left (33 d^2 e^2+4 c d e f+33 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 c^{5/2} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/7*(a*d*(-c*f+d*e)^2-b*c*(-6*c^2*f^2-2*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2 
)/c/d^3/(d*x^2+c)^(7/2)-b*f^2*x^5*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(7/2)+2/35*( 
b*c*(-27*c^2*f^2-9*c*d*e*f+d^2*e^2)+a*d*(-4*c^2*f^2+c*d*e*f+3*d^2*e^2))*x* 
(b*x^2+a)^(1/2)/c^2/d^3/(d*x^2+c)^(5/2)+1/105*(a*b*c*d*(-48*c^2*f^2-2*c*d* 
e*f+15*d^2*e^2)-a^2*d^2*(3*c^2*f^2+8*c*d*e*f+24*d^2*e^2)+2*b^2*c^2*(24*c^2 
*f^2+8*c*d*e*f+3*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^3/d^3/(-a*d+b*c)/(d*x^2+c)^ 
(3/2)+2/105*(a*b^2*c^2*d*(-36*c^2*f^2-5*c*d*e*f+6*d^2*e^2)-a^2*b*c*d^2*(-6 
*c^2*f^2+5*c*d*e*f+36*d^2*e^2)+a^3*d^3*(3*c^2*f^2+8*c*d*e*f+24*d^2*e^2)+b^ 
3*c^3*(24*c^2*f^2+8*c*d*e*f+3*d^2*e^2))*(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^(7/2)/d^(7/2)/(-a*d+b*c)^ 
2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/105*b*(a^2*d^2*(3*c^2* 
f^2+8*c*d*e*f+24*d^2*e^2)+b^2*c^2*(24*c^2*f^2+8*c*d*e*f+3*d^2*e^2)-a*b*c*d 
*(33*c^2*f^2+4*c*d*e*f+33*d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan 
(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c^(5/2)/d^(7/2)/(-a*d+b*c)^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 = 8.42 (sec) , antiderivative size = 720, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 c^3 (b c-a d)^3 (d e-c f)^2-6 c^2 (b c-a d)^2 (d e-c f) (b c (d e-8 c f)+a d (3 d e+4 c f)) \left (c+d x^2\right )+c (b c-a d) \left (a b c d \left (-15 d^2 e^2+2 c d e f-57 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (-6 d^2 e^2-16 c d e f+57 c^2 f^2\right )\right ) \left (c+d x^2\right )^2-2 \left (a b^2 c^2 d \left (6 d^2 e^2-5 c d e f-36 c^2 f^2\right )+a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+a^2 b c d^2 \left (-36 d^2 e^2-5 c d e f+6 c^2 f^2\right )+b^3 c^3 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )\right ) \left (c+d x^2\right )^3\right )+i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \left (2 \left (a b^2 c^2 d \left (6 d^2 e^2-5 c d e f-36 c^2 f^2\right )+a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+a^2 b c d^2 \left (-36 d^2 e^2-5 c d e f+6 c^2 f^2\right )+b^3 c^3 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (a b c d \left (15 d^2 e^2-2 c d e f-48 c^2 f^2\right )-a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+2 b^2 c^2 \left (3 d^2 e^2+8 c d e f+24 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 \sqrt {\frac {b}{a}} c^4 d^4 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \] Input:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2),x]
 

Output:

(-(Sqrt[b/a]*d*x*(a + b*x^2)*(15*c^3*(b*c - a*d)^3*(d*e - c*f)^2 - 6*c^2*( 
b*c - a*d)^2*(d*e - c*f)*(b*c*(d*e - 8*c*f) + a*d*(3*d*e + 4*c*f))*(c + d* 
x^2) + c*(b*c - a*d)*(a*b*c*d*(-15*d^2*e^2 + 2*c*d*e*f - 57*c^2*f^2) + a^2 
*d^2*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(-6*d^2*e^2 - 16*c*d*e 
*f + 57*c^2*f^2))*(c + d*x^2)^2 - 2*(a*b^2*c^2*d*(6*d^2*e^2 - 5*c*d*e*f - 
36*c^2*f^2) + a^3*d^3*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + a^2*b*c*d^2*( 
-36*d^2*e^2 - 5*c*d*e*f + 6*c^2*f^2) + b^3*c^3*(3*d^2*e^2 + 8*c*d*e*f + 24 
*c^2*f^2))*(c + d*x^2)^3)) + I*b*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^3*Sqrt[ 
1 + (d*x^2)/c]*(2*(a*b^2*c^2*d*(6*d^2*e^2 - 5*c*d*e*f - 36*c^2*f^2) + a^3* 
d^3*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + a^2*b*c*d^2*(-36*d^2*e^2 - 5*c* 
d*e*f + 6*c^2*f^2) + b^3*c^3*(3*d^2*e^2 + 8*c*d*e*f + 24*c^2*f^2))*Ellipti 
cE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(a*b*c*d*(15*d^2*e^2 
 - 2*c*d*e*f - 48*c^2*f^2) - a^2*d^2*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) 
+ 2*b^2*c^2*(3*d^2*e^2 + 8*c*d*e*f + 24*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt 
[b/a]*x], (a*d)/(b*c)]))/(105*Sqrt[b/a]*c^4*d^4*(b*c - a*d)^2*Sqrt[a + b*x 
^2]*(c + d*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 1213, normalized size of antiderivative = 1.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 433

\(\displaystyle \int \left (\frac {e^2 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}}+\frac {2 e f x^2 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}}+\frac {f^2 x^4 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 (2 b c-a d) f^2 \sqrt {b x^2+a} x^3}{35 c d^2 \left (d x^2+c\right )^{5/2}}-\frac {f^2 \left (b x^2+a\right )^{3/2} x^3}{7 d \left (d x^2+c\right )^{7/2}}+\frac {\left (2 b^2 c^2+5 a b d c-8 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{35 c^3 d (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\left (8 b^2 c^2-5 a b d c-2 a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{35 c d^3 (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {2 \left (8 b^2 c^2-a b d c-4 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{105 c^2 d^2 (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {2 (b c+3 a d) e^2 \sqrt {b x^2+a} x}{35 c^2 d \left (d x^2+c\right )^{5/2}}-\frac {2 (4 b c-a d) e f \sqrt {b x^2+a} x}{35 c d^2 \left (d x^2+c\right )^{5/2}}-\frac {2 e f \left (b x^2+a\right )^{3/2} x}{7 d \left (d x^2+c\right )^{7/2}}-\frac {(b c-a d) e^2 \sqrt {b x^2+a} x}{7 c d \left (d x^2+c\right )^{7/2}}+\frac {2 (b c-2 a d) \left (b^2 c^2+4 a b d c-4 a^2 d^2\right ) e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 c^{7/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 (2 b c-a d) \left (4 b^2 c^2-4 a b d c-a^2 d^2\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 c^{3/2} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 (b c+a d) \left (8 b^2 c^2-13 a b d c+8 a^2 d^2\right ) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{5/2} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \left (b^2 c^2-11 a b d c+8 a^2 d^2\right ) e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 c^{5/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 \sqrt {c} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {4 b \left (2 b^2 c^2-a b d c+2 a^2 d^2\right ) e 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 )}{105 c^{3/2} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

Int[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2),x]
 

Output:

-1/7*((b*c - a*d)*e^2*x*Sqrt[a + b*x^2])/(c*d*(c + d*x^2)^(7/2)) - (2*e*f* 
x*(a + b*x^2)^(3/2))/(7*d*(c + d*x^2)^(7/2)) - (f^2*x^3*(a + b*x^2)^(3/2)) 
/(7*d*(c + d*x^2)^(7/2)) + (2*(b*c + 3*a*d)*e^2*x*Sqrt[a + b*x^2])/(35*c^2 
*d*(c + d*x^2)^(5/2)) - (2*(4*b*c - a*d)*e*f*x*Sqrt[a + b*x^2])/(35*c*d^2* 
(c + d*x^2)^(5/2)) - (3*(2*b*c - a*d)*f^2*x^3*Sqrt[a + b*x^2])/(35*c*d^2*( 
c + d*x^2)^(5/2)) + ((2*b^2*c^2 + 5*a*b*c*d - 8*a^2*d^2)*e^2*x*Sqrt[a + b* 
x^2])/(35*c^3*d*(b*c - a*d)*(c + d*x^2)^(3/2)) + (2*(8*b^2*c^2 - a*b*c*d - 
 4*a^2*d^2)*e*f*x*Sqrt[a + b*x^2])/(105*c^2*d^2*(b*c - a*d)*(c + d*x^2)^(3 
/2)) - ((8*b^2*c^2 - 5*a*b*c*d - 2*a^2*d^2)*f^2*x*Sqrt[a + b*x^2])/(35*c*d 
^3*(b*c - a*d)*(c + d*x^2)^(3/2)) + (2*(b*c - 2*a*d)*(b^2*c^2 + 4*a*b*c*d 
- 4*a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 
- (b*c)/(a*d)])/(35*c^(7/2)*d^(3/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a* 
(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*(b*c + a*d)*(8*b^2*c^2 - 13*a*b*c*d + 
8*a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
(b*c)/(a*d)])/(105*c^(5/2)*d^(5/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*( 
c + d*x^2))]*Sqrt[c + d*x^2]) + (2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - 
a^2*d^2)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b 
*c)/(a*d)])/(35*c^(3/2)*d^(7/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + 
 d*x^2))]*Sqrt[c + d*x^2]) - (b*(b^2*c^2 - 11*a*b*c*d + 8*a^2*d^2)*e^2*Sqr 
t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(...
 

Defintions of rubi rules used

rule 433

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) 
^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, 
 q, r}, x]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1473\) vs. \(2(698)=1396\).

Time = 10.53 (sec) , antiderivative size = 1474, normalized size of antiderivative = 2.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1474\)
default	\(\text {Expression too large to display}\)	\(7943\)

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/7*(a*c^2*d* 
f^2-2*a*c*d^2*e*f+a*d^3*e^2-b*c^3*f^2+2*b*c^2*d*e*f-b*c*d^2*e^2)/c/d^7*x*( 
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^4-2/35*(4*a*c^2*d*f^2-a*c*d^2 
*e*f-3*a*d^3*e^2-8*b*c^3*f^2+9*b*c^2*d*e*f-b*c*d^2*e^2)/c^2/d^6*x*(b*d*x^4 
+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^3+1/105*(3*a^2*c^2*d^2*f^2+8*a^2*c*d 
^3*e*f+24*a^2*d^4*e^2-57*a*b*c^3*d*f^2+2*a*b*c^2*d^2*e*f-15*a*b*c*d^3*e^2+ 
57*b^2*c^4*f^2-16*b^2*c^3*d*e*f-6*b^2*c^2*d^2*e^2)/d^5/(a*d-b*c)/c^3*x*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^2+2/105*(b*d*x^2+a*d)/c^4/d^4/( 
a*d-b*c)^2*x*(3*a^3*c^2*d^3*f^2+8*a^3*c*d^4*e*f+24*a^3*d^5*e^2+6*a^2*b*c^3 
*d^2*f^2-5*a^2*b*c^2*d^3*e*f-36*a^2*b*c*d^4*e^2-36*a*b^2*c^4*d*f^2-5*a*b^2 
*c^3*d^2*e*f+6*a*b^2*c^2*d^3*e^2+24*b^3*c^5*f^2+8*b^3*c^4*d*e*f+3*b^3*c^3* 
d^2*e^2)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(b^2*f^2/d^4+1/105*b*(3*a^2*c^2*d 
^2*f^2+8*a^2*c*d^3*e*f+24*a^2*d^4*e^2-57*a*b*c^3*d*f^2+2*a*b*c^2*d^2*e*f-1 
5*a*b*c*d^3*e^2+57*b^2*c^4*f^2-16*b^2*c^3*d*e*f-6*b^2*c^2*d^2*e^2)/d^4/(a* 
d-b*c)/c^3+2/105/d^4/(a*d-b*c)*(3*a^3*c^2*d^3*f^2+8*a^3*c*d^4*e*f+24*a^3*d 
^5*e^2+6*a^2*b*c^3*d^2*f^2-5*a^2*b*c^2*d^3*e*f-36*a^2*b*c*d^4*e^2-36*a*b^2 
*c^4*d*f^2-5*a*b^2*c^3*d^2*e*f+6*a*b^2*c^2*d^3*e^2+24*b^3*c^5*f^2+8*b^3*c^ 
4*d*e*f+3*b^3*c^3*d^2*e^2)/c^4-2/105*a/d^3/c^4/(a*d-b*c)^2*(3*a^3*c^2*d^3* 
f^2+8*a^3*c*d^4*e*f+24*a^3*d^5*e^2+6*a^2*b*c^3*d^2*f^2-5*a^2*b*c^2*d^3*e*f 
-36*a^2*b*c*d^4*e^2-36*a*b^2*c^4*d*f^2-5*a*b^2*c^3*d^2*e*f+6*a*b^2*c^2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2915 vs. \(2 (698) = 1396\).

Time = 0.21 (sec) , antiderivative size = 2915, normalized size of antiderivative = 3.96 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="fricas 
")
 

Output:

-1/105*(2*((3*(b^4*c^3*d^6 + 2*a*b^3*c^2*d^7 - 12*a^2*b^2*c*d^8 + 8*a^3*b* 
d^9)*e^2 + (8*b^4*c^4*d^5 - 5*a*b^3*c^3*d^6 - 5*a^2*b^2*c^2*d^7 + 8*a^3*b* 
c*d^8)*e*f + 3*(8*b^4*c^5*d^4 - 12*a*b^3*c^4*d^5 + 2*a^2*b^2*c^3*d^6 + a^3 
*b*c^2*d^7)*f^2)*x^8 + 4*(3*(b^4*c^4*d^5 + 2*a*b^3*c^3*d^6 - 12*a^2*b^2*c^ 
2*d^7 + 8*a^3*b*c*d^8)*e^2 + (8*b^4*c^5*d^4 - 5*a*b^3*c^4*d^5 - 5*a^2*b^2* 
c^3*d^6 + 8*a^3*b*c^2*d^7)*e*f + 3*(8*b^4*c^6*d^3 - 12*a*b^3*c^5*d^4 + 2*a 
^2*b^2*c^4*d^5 + a^3*b*c^3*d^6)*f^2)*x^6 + 6*(3*(b^4*c^5*d^4 + 2*a*b^3*c^4 
*d^5 - 12*a^2*b^2*c^3*d^6 + 8*a^3*b*c^2*d^7)*e^2 + (8*b^4*c^6*d^3 - 5*a*b^ 
3*c^5*d^4 - 5*a^2*b^2*c^4*d^5 + 8*a^3*b*c^3*d^6)*e*f + 3*(8*b^4*c^7*d^2 - 
12*a*b^3*c^6*d^3 + 2*a^2*b^2*c^5*d^4 + a^3*b*c^4*d^5)*f^2)*x^4 + 3*(b^4*c^ 
7*d^2 + 2*a*b^3*c^6*d^3 - 12*a^2*b^2*c^5*d^4 + 8*a^3*b*c^4*d^5)*e^2 + (8*b 
^4*c^8*d - 5*a*b^3*c^7*d^2 - 5*a^2*b^2*c^6*d^3 + 8*a^3*b*c^5*d^4)*e*f + 3* 
(8*b^4*c^9 - 12*a*b^3*c^8*d + 2*a^2*b^2*c^7*d^2 + a^3*b*c^6*d^3)*f^2 + 4*( 
3*(b^4*c^6*d^3 + 2*a*b^3*c^5*d^4 - 12*a^2*b^2*c^4*d^5 + 8*a^3*b*c^3*d^6)*e 
^2 + (8*b^4*c^7*d^2 - 5*a*b^3*c^6*d^3 - 5*a^2*b^2*c^5*d^4 + 8*a^3*b*c^4*d^ 
5)*e*f + 3*(8*b^4*c^8*d - 12*a*b^3*c^7*d^2 + 2*a^2*b^2*c^6*d^3 + a^3*b*c^5 
*d^4)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/ 
(b*c)) - ((3*(2*b^4*c^3*d^6 + (a^2*b^2 + 4*a*b^3)*c^2*d^7 - (11*a^3*b + 24 
*a^2*b^2)*c*d^8 + 8*(a^4 + 2*a^3*b)*d^9)*e^2 + 2*(8*b^4*c^4*d^5 + (4*a^2*b 
^2 - 5*a*b^3)*c^3*d^6 - (2*a^3*b + 5*a^2*b^2)*c^2*d^7 + 4*(a^4 + 2*a^3*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)**2/(d*x**2+c)**(9/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="maxima 
")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*e*f*x - 9*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*a*b*c**2*f**2*x - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
a*b*c*d*e*f*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**3 - 3 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*e**2*x - 6*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b*d**2*e*f*x**3 - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b 
*d**2*f**2*x**5 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x**3 
+ 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e*f*x**3 + 3*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*b**2*c*d*f**2*x**5 + 27*int((sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*x**4)/(3*a**2*c**5*d + 15*a**2*c**4*d**2*x**2 + 30*a**2*c**3*d**3 
*x**4 + 30*a**2*c**2*d**4*x**6 + 15*a**2*c*d**5*x**8 + 3*a**2*d**6*x**10 - 
 a*b*c**6 - 2*a*b*c**5*d*x**2 + 5*a*b*c**4*d**2*x**4 + 20*a*b*c**3*d**3*x* 
*6 + 25*a*b*c**2*d**4*x**8 + 14*a*b*c*d**5*x**10 + 3*a*b*d**6*x**12 - b**2 
*c**6*x**2 - 5*b**2*c**5*d*x**4 - 10*b**2*c**4*d**2*x**6 - 10*b**2*c**3*d* 
*3*x**8 - 5*b**2*c**2*d**4*x**10 - b**2*c*d**5*x**12),x)*a**4*c**4*d**4*f* 
*2 + 108*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(3*a**2*c**5*d + 15* 
a**2*c**4*d**2*x**2 + 30*a**2*c**3*d**3*x**4 + 30*a**2*c**2*d**4*x**6 + 15 
*a**2*c*d**5*x**8 + 3*a**2*d**6*x**10 - a*b*c**6 - 2*a*b*c**5*d*x**2 + 5*a 
*b*c**4*d**2*x**4 + 20*a*b*c**3*d**3*x**6 + 25*a*b*c**2*d**4*x**8 + 14*a*b 
*c*d**5*x**10 + 3*a*b*d**6*x**12 - b**2*c**6*x**2 - 5*b**2*c**5*d*x**4 - 1 
0*b**2*c**4*d**2*x**6 - 10*b**2*c**3*d**3*x**8 - 5*b**2*c**2*d**4*x**10...