Integrand size = 37, antiderivative size = 908 \[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\left (15 a^2 d^2 f^2-2 a b d f (11 d e-7 c f)+b^2 \left (3 d^2 e^2-22 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{48 b^2 d^3 f \sqrt {a+b x^2}}+\frac {(7 b d e-5 b c f-5 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{24 b^2 d^2}+\frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{6 b d}-\frac {\sqrt {b c-a d} e \left (15 a^2 d^2 f^2-2 a b d f (11 d e-7 c f)+b^2 \left (3 d^2 e^2-22 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^3 \sqrt {c} d^3 f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \left (15 a^3 d^3 f^2-3 a^2 b d^2 f (14 d e-3 c f)+a b^2 d \left (31 d^2 e^2-22 c d e f+9 c^2 f^2\right )+b^3 c \left (17 d^2 e^2-32 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^4 \sqrt {c} d^3 \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (5 a^3 d^3 f^3+3 a b^2 d f (d e-c f)^2-3 a^2 b d^2 f^2 (3 d e-c f)+b^3 (d e-c f)^2 (d e+5 c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{16 b^4 \sqrt {c} d^3 \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/48*(15*a^2*d^2*f^2-2*a*b*d*f*(-7*c*f+11*d*e)+b^2*(15*c^2*f^2-22*c*d*e*f+ 3*d^2*e^2))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2/d^3/f/(b*x^2+a)^(1/2)+1/ 24*(-5*a*d*f-5*b*c*f+7*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^ (1/2)/b^2/d^2+1/6*f*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/ d-1/48*(-a*d+b*c)^(1/2)*e*(15*a^2*d^2*f^2-2*a*b*d*f*(-7*c*f+11*d*e)+b^2*(1 5*c^2*f^2-22*c*d*e*f+3*d^2*e^2))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a)) ^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/ (-a*d+b*c)/e)^(1/2))/b^3/c^(1/2)/d^3/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f* x^2+e)^(1/2)+1/48*a*(15*a^3*d^3*f^2-3*a^2*b*d^2*f*(-3*c*f+14*d*e)+a*b^2*d* (9*c^2*f^2-22*c*d*e*f+31*d^2*e^2)+b^3*c*(15*c^2*f^2-32*c*d*e*f+17*d^2*e^2) )*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*d+b*c)^(1/ 2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^4/c^(1/2 )/d^3/(-a*d+b*c)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/1 6*a*(5*a^3*d^3*f^3+3*a*b^2*d*f*(-c*f+d*e)^2-3*a^2*b*d^2*f^2*(-c*f+3*d*e)+b ^3*(-c*f+d*e)^2*(5*c*f+d*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/ 2)*EllipticPi((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),b*c/(-a*d+b*c),(c *(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^4/c^(1/2)/d^3/(-a*d+b*c)^(1/2)/f/(a*(d* x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx \] Input:
Integrate[(x^4*(e + f*x^2)^(3/2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
Integrate[(x^4*(e + f*x^2)^(3/2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x]
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.
| \(\displaystyle \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}}dx\) |
Input:
Int[(x^4*(e + f*x^2)^(3/2))/(Sqrt[a + b*x^2]*Sqrt[c + d*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]
\[\int \frac {x^{4} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}d x\]
Input:
int(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
int(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}} x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm ="fricas")
Output:
integral((f*x^6 + e*x^4)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/( b*d*x^4 + (b*c + a*d)*x^2 + a*c), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (e + f x^{2}\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**4*(f*x**2+e)**(3/2)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**4*(e + f*x**2)**(3/2)/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}} x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm ="maxima")
Output:
integrate((f*x^2 + e)^(3/2)*x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}} x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm ="giac")
Output:
integrate((f*x^2 + e)^(3/2)*x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^4*(e + f*x^2)^(3/2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int((x^4*(e + f*x^2)^(3/2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\text {too large to display} \] Input:
int(x^4*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
( - 5*sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x - 5*sqrt( e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f*x + 7*sqrt(e + f*x**2) *sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 4*sqrt(e + f*x**2)*sqrt(c + d *x**2)*sqrt(a + b*x**2)*b*d*f*x**3 + 15*int((sqrt(e + f*x**2)*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 + 1 4*int((sqrt(e + f*x**2)*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 - 22*int((sqrt(e + f*x**2)*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 + 15*int( (sqrt(e + f*x**2)*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 - 22*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqr t(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 + 3*int((sqrt (e + f*x**2)*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 + 10*int((sqrt(e + f*x**2)*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*...