Integrand size = 31, antiderivative size = 268 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {A \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {a} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {B \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
A*c^(1/2)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x ^2)^(1/2)*EllipticF(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/( d+a^(1/2)*e/c^(1/2)))^(1/2))/a^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+B*(1 -a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*El lipticPi(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2) *e/c^(1/2)))^(1/2))/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx \] Input:
Integrate[(A + B*x^2)/(Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),x]
Output:
Integrate[(A + B*x^2)/(Sqrt[d + e*x^2]*Sqrt[a - c*x^4]), 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.
\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 2261 |
\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \sqrt {d+e x^2}}dx\) |
Input:
Int[(A + B*x^2)/(Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {B \,x^{2}+A}{\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]
Input:
int((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
Output:
int((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorithm="fricas" )
Output:
integral(-sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c*e*x^6 + c*d*x^4 - a*e*x^2 - a*d), x)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2}}{\sqrt {a - c x^{4}} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate((B*x**2+A)/(e*x**2+d)**(1/2)/(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2)/(sqrt(a - c*x**4)*sqrt(d + e*x**2)), x)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorithm="maxima" )
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)), x)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)), x)
Timed out. \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}} \,d x \] Input:
int((A + B*x^2)/((a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)),x)
Output:
int((A + B*x^2)/((a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)), x)
\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \] Input:
int((B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b + int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a