Integrand size = 43, antiderivative size = 333 \[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {A \sqrt {a-b x^2} \sqrt {c+d x^2}}{a c e x}-\frac {A \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} c e \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {A \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} e \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-A*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x-A*b^(1/2)*(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))/a^(1/2)/c/e/(- b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+A*b^(1/2)*(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))/a^(1/2)/e/(-b*x^2+a)^(1 /2)/(d*x^2+c)^(1/2)+a^(1/2)*(-A*f+B*e)*(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)/e^2/(-b*x ^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 10.75 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-A \sqrt {-\frac {b}{a}} e \left (a-b x^2\right ) \left (c+d x^2\right )+i A b c e x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i A b c e x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i a c (B e-A f) x \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{a \sqrt {-\frac {b}{a}} c e^2 x \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(A + B*x^2)/(x^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(-(A*Sqrt[-(b/a)]*e*(a - b*x^2)*(c + d*x^2)) + I*A*b*c*e*x*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*b*c*e*x*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcS inh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*a*c*(B*e - A*f)*x*Sqrt[1 - (b*x^2 )/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)] *x], -((a*d)/(b*c))])/(a*Sqrt[-(b/a)]*c*e^2*x*Sqrt[a - b*x^2]*Sqrt[c + d*x ^2])
Time = 2.09 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {B e-A f}{e \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {A}{e x^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {A \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} e \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {A \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} c e \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x^2}}{a c e x}\) |
Input:
Int[(A + B*x^2)/(x^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
-((A*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(a*c*e*x)) - (A*Sqrt[b]*Sqrt[1 - (b* x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b* c))])/(Sqrt[a]*c*e*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (A*Sqrt[b]*Sqrt[ 1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[a]*e*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[a]*(B *e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e) ), ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*e^2*Sqrt[a - b*x ^2]*Sqrt[c + d*x^2])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 10.07 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {A \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a c e x}-\frac {\left (-\frac {A b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {a c \left (A f -B e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{a c e \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(319\) |
| default | \(\frac {\left (A \sqrt {\frac {b}{a}}\, b d e \,x^{4}+A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e x -A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e x -A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c f x +B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c e x -A \sqrt {\frac {b}{a}}\, a d e \,x^{2}+A \sqrt {\frac {b}{a}}\, b c e \,x^{2}-A \sqrt {\frac {b}{a}}\, a c e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{\sqrt {\frac {b}{a}}\, x \,e^{2} c a \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(376\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {A \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{a c e x}+\frac {b A \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b A \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) A}{e^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) B}{e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(476\) |
Input:
int((B*x^2+A)/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RET URNVERBOSE)
Output:
-A*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x-1/a/c/e*(-A*b*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)*(El lipticF(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)))+a*c*(A*f-B*e)/e/(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. \[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {A + B x^{2}}{x^{2} \sqrt {a - b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**2/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e), x)
Output:
Integral((A + B*x**2)/(x**2*sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2) ), x)
\[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="maxima")
Output:
integrate((B*x^2 + A)/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^2), x)
\[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="giac")
Output:
integrate((B*x^2 + A)/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^2), x)
Timed out. \[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {B\,x^2+A}{x^2\,\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((A + B*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int((A + B*x^2)/(x^2*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {A+B x^2}{x^2 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{8}+a d f \,x^{6}-b c f \,x^{6}-b d e \,x^{6}+a c f \,x^{4}+a d e \,x^{4}-b c e \,x^{4}+a c e \,x^{2}}d x \right ) a +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b \] Input:
int((B*x^2+A)/x^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e*x**2 + a*c*f*x**4 + a*d*e*x **4 + a*d*f*x**6 - b*c*e*x**4 - b*c*f*x**6 - b*d*e*x**6 - b*d*f*x**8),x)*a + int((sqrt(c + d*x**2)*sqrt(a - b*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)*b