Integrand size = 35, antiderivative size = 1080 \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=-\frac {\left (16 a^3 d f^3 (d e-c f)^2+8 a^2 b f^2 (d e-c f)^2 (5 d e+2 c f)+b^3 e^2 \left (105 d^3 e^3-170 c d^2 e^2 f+40 c^2 d e f^2+16 c^3 f^3\right )-a b^2 e f \left (170 d^3 e^3-275 c d^2 e^2 f+64 c^2 d e f^2+32 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{24 b d^2 f^4 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d f^3}-\frac {e^4 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 f^3 (b e-a f) (d e-c f) \left (e+f x^2\right )^2}-\frac {e^3 (a f (14 d e-17 c f)-b e (11 d e-14 c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 f^3 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {\sqrt {a} \left (16 a^3 d f^3 (d e-c f)^2+8 a^2 b f^2 (d e-c f)^2 (5 d e+2 c f)+b^3 e^2 \left (105 d^3 e^3-170 c d^2 e^2 f+40 c^2 d e f^2+16 c^3 f^3\right )-a b^2 e f \left (170 d^3 e^3-275 c d^2 e^2 f+64 c^2 d e f^2+32 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{24 b^{3/2} d^2 f^4 (b e-a f)^2 (d e-c f)^2 \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 (8 a^3 c f^4 (d e-c f)-b^3 e^3 \left (105 d^2 e^2-100 c d e f-8 c^2 f^2\right )+3 a b^2 e^2 f \left (80 d^2 e^2-73 c d e f-8 c^2 f^2\right )-24 a^2 b e f^2 \left (6 d^2 e^2-5 c d e f-c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{24 b^{3/2} c d f^4 (b e-a f)^3 (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} e^2 \left (3 a^2 f^2 \left (16 d^2 e^2-36 c d e f+21 c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-80 c d e f+48 c^2 f^2\right )-2 a b e f \left (40 d^2 e^2-91 c d e f+54 c^2 f^2\right )\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 )}{8 \sqrt {b} c f^4 (b e-a f)^3 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/24*(16*a^3*d*f^3*(-c*f+d*e)^2+8*a^2*b*f^2*(-c*f+d*e)^2*(2*c*f+5*d*e)+b^ 3*e^2*(16*c^3*f^3+40*c^2*d*e*f^2-170*c*d^2*e^2*f+105*d^3*e^3)-a*b^2*e*f*(3 2*c^3*f^3+64*c^2*d*e*f^2-275*c*d^2*e^2*f+170*d^3*e^3))*x*(d*x^2+c)^(1/2)/b /d^2/f^4/(-a*f+b*e)^2/(-c*f+d*e)^2/(b*x^2+a)^(1/2)+1/3*x*(b*x^2+a)^(1/2)*( d*x^2+c)^(1/2)/b/d/f^3-1/4*e^4*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^3/(-a*f +b*e)/(-c*f+d*e)/(f*x^2+e)^2-1/8*e^3*(a*f*(-17*c*f+14*d*e)-b*e*(-14*c*f+11 *d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^3/(-a*f+b*e)^2/(-c*f+d*e)^2/(f* x^2+e)+1/24*a^(1/2)*(16*a^3*d*f^3*(-c*f+d*e)^2+8*a^2*b*f^2*(-c*f+d*e)^2*(2 *c*f+5*d*e)+b^3*e^2*(16*c^3*f^3+40*c^2*d*e*f^2-170*c*d^2*e^2*f+105*d^3*e^3 )-a*b^2*e*f*(32*c^3*f^3+64*c^2*d*e*f^2-275*c*d^2*e^2*f+170*d^3*e^3))*(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^(3/2)/d^2/f^4/(-a*f+b*e)^2/(-c*f+d*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c /(b*x^2+a))^(1/2)+1/24*a^(3/2)*(8*a^3*c*f^4*(-c*f+d*e)-b^3*e^3*(-8*c^2*f^2 -100*c*d*e*f+105*d^2*e^2)+3*a*b^2*e^2*f*(-8*c^2*f^2-73*c*d*e*f+80*d^2*e^2) -24*a^2*b*e*f^2*(-c^2*f^2-5*c*d*e*f+6*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJac obiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/c/d/f^4/(-a*f+b *e)^3/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/8*a^(3/ 2)*e^2*(3*a^2*f^2*(21*c^2*f^2-36*c*d*e*f+16*d^2*e^2)+b^2*e^2*(48*c^2*f^2-8 0*c*d*e*f+35*d^2*e^2)-2*a*b*e*f*(54*c^2*f^2-91*c*d*e*f+40*d^2*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...
Result contains complex when optimal does not.
Time = 6.10 (sec) , antiderivative size = 730, normalized size of antiderivative = 0.68 \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {-\sqrt {\frac {b}{a}} d f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 b d e^4 (b e-a f) (d e-c f)-3 b d e^3 (b e (11 d e-14 c f)+a f (-14 d e+17 c f)) \left (e+f x^2\right )-8 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )^2\right )+i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (c f \left (16 a^3 d f^3 (d e-c f)^2+8 a^2 b f^2 (d e-c f)^2 (5 d e+2 c f)+b^3 e^2 \left (105 d^3 e^3-170 c d^2 e^2 f+40 c^2 d e f^2+16 c^3 f^3\right )-a b^2 e f \left (170 d^3 e^3-275 c d^2 e^2 f+64 c^2 d e f^2+32 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(d e-c f) \left (8 a^3 c d f^4 (d e-c f)+b^3 e^2 \left (105 d^3 e^3-30 c d^2 e^2 f-56 c^2 d e f^2-16 c^3 f^3\right )-8 a^2 b f^2 \left (-18 d^3 e^3+11 c d^2 e^2 f+5 c^2 d e f^2+2 c^3 f^3\right )+a b^2 e f \left (-240 d^3 e^3+101 c d^2 e^2 f+104 c^2 d e f^2+32 c^3 f^3\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 b d^2 e^2 \left (3 a^2 f^2 \left (16 d^2 e^2-36 c d e f+21 c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-80 c d e f+48 c^2 f^2\right )-2 a b e f \left (40 d^2 e^2-91 c d e f+54 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{24 b \sqrt {\frac {b}{a}} d^2 f^5 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:
Integrate[x^10/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
Output:
(-(Sqrt[b/a]*d*f^2*x*(a + b*x^2)*(c + d*x^2)*(6*b*d*e^4*(b*e - a*f)*(d*e - c*f) - 3*b*d*e^3*(b*e*(11*d*e - 14*c*f) + a*f*(-14*d*e + 17*c*f))*(e + f* x^2) - 8*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x^2)^2)) + I*Sqrt[1 + (b*x^2)/ a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(c*f*(16*a^3*d*f^3*(d*e - c*f)^2 + 8* a^2*b*f^2*(d*e - c*f)^2*(5*d*e + 2*c*f) + b^3*e^2*(105*d^3*e^3 - 170*c*d^2 *e^2*f + 40*c^2*d*e*f^2 + 16*c^3*f^3) - a*b^2*e*f*(170*d^3*e^3 - 275*c*d^2 *e^2*f + 64*c^2*d*e*f^2 + 32*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], ( a*d)/(b*c)] - (d*e - c*f)*(8*a^3*c*d*f^4*(d*e - c*f) + b^3*e^2*(105*d^3*e^ 3 - 30*c*d^2*e^2*f - 56*c^2*d*e*f^2 - 16*c^3*f^3) - 8*a^2*b*f^2*(-18*d^3*e ^3 + 11*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 2*c^3*f^3) + a*b^2*e*f*(-240*d^3*e^3 + 101*c*d^2*e^2*f + 104*c^2*d*e*f^2 + 32*c^3*f^3))*EllipticF[I*ArcSinh[Sq rt[b/a]*x], (a*d)/(b*c)] + 3*b*d^2*e^2*(3*a^2*f^2*(16*d^2*e^2 - 36*c*d*e*f + 21*c^2*f^2) + b^2*e^2*(35*d^2*e^2 - 80*c*d*e*f + 48*c^2*f^2) - 2*a*b*e* f*(40*d^2*e^2 - 91*c*d*e*f + 54*c^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcSin h[Sqrt[b/a]*x], (a*d)/(b*c)]))/(24*b*Sqrt[b/a]*d^2*f^5*(b*e - a*f)^2*(d*e - c*f)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)
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.
| \(\displaystyle \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3}dx\) |
Input:
Int[x^10/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
Output:
$Aborted
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]
Leaf count of result is larger than twice the leaf count of optimal. \(3360\) vs. \(2(1036)=2072\).
Time = 30.41 (sec) , antiderivative size = 3361, normalized size of antiderivative = 3.11
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3361\) |
| risch | \(\text {Expression too large to display}\) | \(4045\) |
| default | \(\text {Expression too large to display}\) | \(10282\) |
Input:
int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x,method=_RETURNVERBO SE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(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*e/f^4*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-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*e/f^4*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-33/8/(-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))*b*d*e^4/f^3/(a*c*f^2 -a*d*e*f-b*c*e*f+b*d*e^2)^2*a*c-6*e^4/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/ f^3/(-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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/ 2))*a^2*d^2-6*e^4/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/f^3/(-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)*Ellip ticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^2*c^2-35/8*e^6 /(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/f^5/(-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)*EllipticPi(x*(-b/a)^(1 /2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^2*d^2+2/3*c^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)/d^2/f^ 3*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3*c^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)/d^ 2/f^3*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/4/f^3/(a*c*f...
Timed out. \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="f ricas")
Output:
Timed out
Timed out. \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**10/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="m axima")
Output:
integrate(x^10/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="g iac")
Output:
integrate(x^10/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {x^{10}}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(x^10/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3),x)
Output:
int(x^10/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3), x)
\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {x^{10}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)
Output:
int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)