Integrand size = 36, antiderivative size = 529 \[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=-\frac {(5 b d e-3 b c f+a d f) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 b f^2}+\frac {x \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{5 f}+\frac {\sqrt {a} \left (2 a^2 d^2 f^2-a b d f (5 d e-7 c f)-b^2 \left (15 d^2 e^2-20 c d e f+3 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 b^{3/2} d f^3 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} \left (a^2 c d^2 f^3-a b d f \left (15 d^2 e^2-20 c d e f+2 c^2 f^2\right )-b^2 \left (15 d^3 e^3-15 c d^2 e^2 f-5 c^2 d e f^2+3 c^3 f^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 b^{3/2} d f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} (b e+a f) (d e-c f)^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-1/15*(a*d*f-3*b*c*f+5*b*d*e)*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/f^2+1/5 *x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/f+1/15*a^(1/2)*(2*a^2*d^2*f^2-a*b*d*f* (-7*c*f+5*d*e)-b^2*(3*c^2*f^2-20*c*d*e*f+15*d^2*e^2))*(1-b*x^2/a)^(1/2)*(d *x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(3/2)/d/f^3/ (-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)-1/15*a^(1/2)*(a^2*c*d^2*f^3-a*b*d*f*(2* c^2*f^2-20*c*d*e*f+15*d^2*e^2)-b^2*(3*c^3*f^3-5*c^2*d*e*f^2-15*c*d^2*e^2*f +15*d^3*e^3))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1 /2),(-a*d/b/c)^(1/2))/b^(3/2)/d/f^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-a^(1/ 2)*(a*f+b*e)*(-c*f+d*e)^2*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b ^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/f^4/(-b*x^2+a)^(1/2)/( d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\frac {-\sqrt {-\frac {b}{a}} d f^2 x \left (a-b x^2\right ) \left (c+d x^2\right ) \left (a d f+b \left (5 d e-6 c f-3 d f x^2\right )\right )-i c f \left (2 a^2 d^2 f^2+a b d f (-5 d e+7 c f)+b^2 \left (-15 d^2 e^2+20 c d e f-3 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i \left (a^2 c d^2 f^3+a b d f \left (-15 d^2 e^2+20 c d e f-2 c^2 f^2\right )+b^2 \left (-15 d^3 e^3+15 c d^2 e^2 f+5 c^2 d e f^2-3 c^3 f^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+15 i b d (b e+a f) (d e-c f)^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{15 b \sqrt {-\frac {b}{a}} d f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^2*Sqrt[a - b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),x]
Output:
(-(Sqrt[-(b/a)]*d*f^2*x*(a - b*x^2)*(c + d*x^2)*(a*d*f + b*(5*d*e - 6*c*f - 3*d*f*x^2))) - I*c*f*(2*a^2*d^2*f^2 + a*b*d*f*(-5*d*e + 7*c*f) + b^2*(-1 5*d^2*e^2 + 20*c*d*e*f - 3*c^2*f^2))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/ c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I*(a^2*c*d^2*f^3 + a*b*d*f*(-15*d^2*e^2 + 20*c*d*e*f - 2*c^2*f^2) + b^2*(-15*d^3*e^3 + 15* c*d^2*e^2*f + 5*c^2*d*e*f^2 - 3*c^3*f^3))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d* x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + (15*I)*b*d* (b*e + a*f)*(d*e - c*f)^2*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic Pi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(15*b*Sqrt[ -(b/a)]*d*f^4*Sqrt[a - b*x^2]*Sqrt[c + d*x^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.
| \(\displaystyle \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2}dx\) |
Input:
Int[(x^2*Sqrt[a - b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),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]
Time = 9.38 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {x \left (-3 b d f \,x^{2}+a d f -6 b c f +5 b d e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b \,f^{2}}+\frac {\left (\frac {\left (a^{2} c d \,f^{3}+9 a b \,c^{2} f^{3}-25 a b c d e \,f^{2}+15 a b \,d^{2} e^{2} f +15 b^{2} c^{2} e \,f^{2}-30 b^{2} c d \,e^{2} f +15 b^{2} d^{2} e^{3}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (2 a^{2} d^{2} f^{2}+7 a b c d \,f^{2}-5 a b \,d^{2} e f -3 b^{2} c^{2} f^{2}+20 b^{2} c d e f -15 b^{2} d^{2} e^{2}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}-\frac {15 \left (a \,c^{2} f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f +e^{3} b \,d^{2}\right ) b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 f^{2} b \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(610\) |
| default | \(\text {Expression too large to display}\) | \(1615\) |
| elliptic | \(\text {Expression too large to display}\) | \(2154\) |
Input:
int(x^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE )
Output:
-1/15/b*x*(-3*b*d*f*x^2+a*d*f-6*b*c*f+5*b*d*e)*(-b*x^2+a)^(1/2)*(d*x^2+c)^ (1/2)/f^2+1/15/f^2/b*((a^2*c*d*f^3+9*a*b*c^2*f^3-25*a*b*c*d*e*f^2+15*a*b*d ^2*e^2*f+15*b^2*c^2*e*f^2-30*b^2*c*d*e^2*f+15*b^2*d^2*e^3)/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*(2*a^2*d^2*f^2+7*a*b *c*d*f^2-5*a*b*d^2*e*f-3*b^2*c^2*f^2+20*b^2*c*d*e*f-15*b^2*d^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)/d*(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)))-15*(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)*b/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)*EllipticPi(x*(b/a )^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2)))*((-b*x^2+a)*(d*x^2+c))^(1/2) /(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\text {Timed out} \] Input:
integrate(x^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fri cas")
Output:
Timed out
\[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int \frac {x^{2} \sqrt {a - b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}{e + f x^{2}}\, dx \] Input:
integrate(x**2*(-b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/(f*x**2+e),x)
Output:
Integral(x**2*sqrt(a - b*x**2)*(c + d*x**2)**(3/2)/(e + f*x**2), x)
\[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{f x^{2} + e} \,d x } \] Input:
integrate(x^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="max ima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)*x^2/(f*x^2 + e), x)
\[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{f x^{2} + e} \,d x } \] Input:
integrate(x^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="gia c")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)*x^2/(f*x^2 + e), x)
Timed out. \[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int \frac {x^2\,\sqrt {a-b\,x^2}\,{\left (d\,x^2+c\right )}^{3/2}}{f\,x^2+e} \,d x \] Input:
int((x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2),x)
Output:
int((x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2), x)
\[ \int \frac {x^2 \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx =\text {Too large to display} \] Input:
int(x^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*d*f*x + 6*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c*f*x - 5*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d*e*x + 3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d*f*x**3 + 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*a**2*d**2*f**2 + 7*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x **4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*a*b*c*d*f**2 - 5*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d* e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x )*a*b*d**2*e*f - 3*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a *c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*b**2*c**2*f**2 + 20*int((sqrt(c + d*x**2)*sqrt(a - b*x** 2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c* f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*b**2*c*d*e*f - 15*int((sqrt(c + d*x** 2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*b**2*d**2*e**2 + int ((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*a**2 *c*d*f**2 + 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c*f* x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 -...