Integrand size = 38, antiderivative size = 805 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\frac {(B c d+b B e+4 A c e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{8 c e x}+\frac {1}{4} B x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}-\frac {\sqrt {b^2-4 a c} (B c d+b B e+12 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{8 \sqrt {2} c e \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} (8 A b c d+5 a B c d-a b B e-4 a A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{4 \sqrt {2} a c \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} \left (4 A c e (c d+b e)-B \left (c^2 d^2+b^2 e^2-2 c e (b d+2 a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} c \left (b+\sqrt {b^2-4 a c}\right ) e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:
1/8*(4*A*c*e+B*b*e+B*c*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e/x+1/4* B*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)-1/16*(-4*a*c+b^2)^(1/2)*(12*A*c* e+B*b*e+B*c*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*E llipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4 *a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/c/e/(-a *(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1 /8*(-4*a*c+b^2)^(1/2)*(-4*A*a*c*e+8*A*b*c*d-B*a*b*e+5*B*a*c*d)*(-a*(c+a/x^ 4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e ))^(1/2)*x^3*EllipticF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2 ),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2 ^(1/2)/a/c/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/4*(-4*a*c+b^2)^(1/2)*(4 *A*c*e*(b*e+c*d)-B*(c^2*d^2+b^2*e^2-2*c*e*(2*a*e+b*d)))*(-a*(c+a/x^4+b/x^2 )/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2 )*x^3*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(- 4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d +(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/c/(b+(-4*a*c+b^2)^(1/2))/e/(e *x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^2,x]
Output:
Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx\) |
| \(\Big \downarrow \) 2250 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^2,x]
Output:
$Aborted
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol yQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x)
Output:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x, algorithm ="fricas")
Output:
integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {d + e x^{2}} \sqrt {a + b x^{2} + c x^{4}}}{x^{2}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(c*x**4+b*x**2+a)**(1/2)/x**2,x)
Output:
Integral((A + B*x**2)*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)/x**2, x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x, algorithm ="maxima")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x, algorithm ="giac")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^2, x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}\,\sqrt {c\,x^4+b\,x^2+a}}{x^2} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^2,x)
Output:
int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^2,x)
Output:
(6*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*e + 4*sqrt(d + e*x**2)*s qrt(a + b*x**2 + c*x**4)*a*c*d + 2*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x* *4)*b**2*d + sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*e*x**2 + sqrt (d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c*d*x**2 - 8*int((sqrt(d + e*x**2 )*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a* c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x* *4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*a*b**2*c*e**3*x - 12*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e* *2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d **2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6 ),x)*a*b*c**2*d*e**2*x - 4*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4) *x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x**6 + c**2*d **2*x**4 + c**2*d*e*x**6),x)*a*c**3*d**2*e*x + int((sqrt(d + e*x**2)*sqrt( a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x **2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b* c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*b**4*e**3*x - int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c* d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b *c*d*e*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*b**3*c...