Integrand size = 38, antiderivative size = 588 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (A \left (b^2-2 a c\right )-a (b B-2 a C)+(A b c-2 a B c+a b C) x^2\right ) \sqrt {d+e x^2}}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (A c (b d-4 a e)-a (2 B c d-b C d-2 b B e+4 a C e)+\frac {A c \left (b^2 d-12 a c d+4 a b e\right )-a \left (4 a c C d+b^2 (C d+2 B e)-4 b (B c d+a C e)\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (A c \left (b^2 d-12 a c d-b \sqrt {b^2-4 a c} d+4 a b e+4 a \sqrt {b^2-4 a c} e\right )+a \left (2 B c \sqrt {b^2-4 a c} d-4 a c C d+4 a \sqrt {b^2-4 a c} C e-b^2 (C d+2 B e)+b \left (4 B c d-\sqrt {b^2-4 a c} C d-2 B \sqrt {b^2-4 a c} e+4 a C e\right )\right )\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:
1/2*x*(A*(-2*a*c+b^2)-a*(B*b-2*C*a)+(A*b*c-2*B*a*c+C*a*b)*x^2)*(e*x^2+d)^( 1/2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*(A*c*(-4*a*e+b*d)-a*(-2*B*b*e+2*B* c*d+4*C*a*e-C*b*d)+(A*c*(4*a*b*e-12*a*c*d+b^2*d)-a*(4*a*c*C*d+b^2*(2*B*e+C *d)-4*b*(B*c*d+C*a*e)))/(-4*a*c+b^2)^(1/2))*arctan((2*c*d-(b-(-4*a*c+b^2)^ (1/2))*e)^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(e*x^2+d)^(1/2))/a/(-4*a*c+ b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-1 /2*(A*c*(b^2*d-12*a*c*d-b*(-4*a*c+b^2)^(1/2)*d+4*a*b*e+4*a*(-4*a*c+b^2)^(1 /2)*e)+a*(2*B*c*(-4*a*c+b^2)^(1/2)*d-4*a*c*C*d+4*a*(-4*a*c+b^2)^(1/2)*C*e- b^2*(2*B*e+C*d)+b*(4*B*c*d-(-4*a*c+b^2)^(1/2)*C*d-2*B*(-4*a*c+b^2)^(1/2)*e +4*C*a*e)))*arctan((2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)*x/(b+(-4*a*c+b^2 )^(1/2))^(1/2)/(e*x^2+d)^(1/2))/a/(-4*a*c+b^2)^(3/2)/(b+(-4*a*c+b^2)^(1/2) )^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(21061\) vs. \(2(588)=1176\).
Time = 17.50 (sec) , antiderivative size = 21061, normalized size of antiderivative = 35.82 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[(Sqrt[d + e*x^2]*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x]
Output:
Result too large to show
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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \int \left (\frac {\sqrt {d+e x^2} \left (-a C+A c+x^2 (B c-b C)\right )}{c \left (a+b x^2+c x^4\right )^2}+\frac {C \sqrt {d+e x^2}}{c \left (a+b x^2+c x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(A c-a C) \int \frac {\sqrt {e x^2+d}}{\left (c x^4+b x^2+a\right )^2}dx}{c}+\frac {(B c-b C) \int \frac {x^2 \sqrt {e x^2+d}}{\left (c x^4+b x^2+a\right )^2}dx}{c}+\frac {C \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {C \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{c \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\) |
Input:
Int[(Sqrt[d + e*x^2]*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Time = 1.75 (sec) , antiderivative size = 579, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {\frac {3 \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (c \,x^{4}+b \,x^{2}+a \right ) \left (\left (\frac {C \,a^{2}}{3}+\left (A c -\frac {B b}{6}\right ) a -\frac {A \,b^{2}}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+\frac {4 C \,a^{3} e}{3}+\frac {2 \left (\left (B c -b C \right ) d +2 e \left (A c -\frac {B b}{2}\right )\right ) a^{2}}{3}-\frac {4 b \left (A c -\frac {B b}{8}\right ) d a}{3}+\frac {A d \,b^{3}}{6}\right ) d \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (-\frac {3 \left (c \,x^{4}+b \,x^{2}+a \right ) d \left (\left (\frac {C \,a^{2}}{3}+\left (A c -\frac {B b}{6}\right ) a -\frac {A \,b^{2}}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-\frac {4 C \,a^{3} e}{3}+\frac {2 \left (\left (-B c +b C \right ) d -2 e \left (A c -\frac {B b}{2}\right )\right ) a^{2}}{3}+\frac {4 b \left (A c -\frac {B b}{8}\right ) d a}{3}-\frac {A d \,b^{3}}{6}\right ) \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+\sqrt {e \,x^{2}+d}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \left (-C \,a^{2}+\left (\frac {\left (-C \,x^{2}+B \right ) b}{2}+c \left (B \,x^{2}+A \right )\right ) a -\frac {A b \left (c \,x^{2}+b \right )}{2}\right ) x \right )}{\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, a \left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}\) | \(579\) |
| default | \(-\frac {C d \sqrt {2}\, \left (-\frac {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}+\frac {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 c \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}+\frac {\frac {3 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (\left (-a^{2} c C +\left (A \,c^{2}-\frac {1}{6} b B c +\frac {1}{3} b^{2} C \right ) a -\frac {A \,b^{2} c}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-\frac {4 C \,a^{3} c e}{3}+\frac {2 \left (c \left (B c +b C \right ) d +2 e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a^{2}}{3}-\frac {4 b \left (A \,c^{2}-\frac {1}{8} b B c +\frac {1}{4} b^{2} C \right ) d a}{3}+\frac {A \,b^{3} c d}{6}\right ) d \sqrt {2}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (-\frac {3 \left (c \,x^{4}+b \,x^{2}+a \right ) d \sqrt {2}\, \left (\left (-a^{2} c C +\left (A \,c^{2}-\frac {1}{6} b B c +\frac {1}{3} b^{2} C \right ) a -\frac {A \,b^{2} c}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+\frac {4 C \,a^{3} c e}{3}+\frac {2 \left (-c \left (B c +b C \right ) d -2 e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a^{2}}{3}+\frac {4 b \left (A \,c^{2}-\frac {1}{8} b B c +\frac {1}{4} b^{2} C \right ) d a}{3}-\frac {A \,b^{3} c d}{6}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+c \sqrt {e \,x^{2}+d}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \left (-C \,a^{2}+\left (\frac {\left (-C \,x^{2}+B \right ) b}{2}+c \left (B \,x^{2}+A \right )\right ) a -\frac {A b \left (c \,x^{2}+b \right )}{2}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, x \right )}{c \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (4 a c -b^{2}\right ) a \left (c \,x^{4}+b \,x^{2}+a \right )}\) | \(883\) |
Input:
int((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERB OSE)
Output:
1/(-4*(a*c-1/4*b^2)*d^2)^(1/2)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))* a)^(1/2)*(3/2*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)*(c*x^4+b *x^2+a)*((1/3*C*a^2+(A*c-1/6*B*b)*a-1/6*A*b^2)*(-4*(a*c-1/4*b^2)*d^2)^(1/2 )+4/3*C*a^3*e+2/3*((B*c-C*b)*d+2*e*(A*c-1/2*B*b))*a^2-4/3*b*(A*c-1/8*B*b)* d*a+1/6*A*d*b^3)*d*2^(1/2)*arctanh(a*(e*x^2+d)^(1/2)/x*2^(1/2)/((2*a*e-b*d +(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))+((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2 )^(1/2))*a)^(1/2)*(-3/2*(c*x^4+b*x^2+a)*d*((1/3*C*a^2+(A*c-1/6*B*b)*a-1/6* A*b^2)*(-4*(a*c-1/4*b^2)*d^2)^(1/2)-4/3*C*a^3*e+2/3*((-B*c+C*b)*d-2*e*(A*c -1/2*B*b))*a^2+4/3*b*(A*c-1/8*B*b)*d*a-1/6*A*d*b^3)*2^(1/2)*arctan(a*(e*x^ 2+d)^(1/2)/x*2^(1/2)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))+ (e*x^2+d)^(1/2)*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)*(-4*(a *c-1/4*b^2)*d^2)^(1/2)*(-C*a^2+(1/2*(-C*x^2+B)*b+c*(B*x^2+A))*a-1/2*A*b*(c *x^2+b))*x))/((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)/a/(4*a*c-b ^2)/(c*x^4+b*x^2+a)
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm=" fricas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(1/2)*(C*x**4+B*x**2+A)/(c*x**4+b*x**2+a)**2,x)
Output:
Timed out
\[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm=" maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a)^2, x)
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm=" giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2+a\right )}^2} \,d x \] Input:
int(((d + e*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x)
Output:
int(((d + e*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2, x)
\[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int \frac {\sqrt {e \,x^{2}+d}}{c \,x^{4}+b \,x^{2}+a}d x \] Input:
int((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x)
Output:
int(sqrt(d + e*x**2)/(a + b*x**2 + c*x**4),x)