Integrand size = 31, antiderivative size = 374 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A+B x^2\right ) \sqrt {d+e x^2}}{2 a \sqrt {a-c x^4}}+\frac {B \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c x}+\frac {B \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(A c d-a B e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/2*x*(B*x^2+A)*(e*x^2+d)^(1/2)/a/(-c*x^4+a)^(1/2)+1/2*B*(e*x^2+d)^(1/2)*( -c*x^4+a)^(1/2)/a/c/x+1/2*B*(d+a^(1/2)*e/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)*EllipticE(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/(e* x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/2*(A*c*d-B*a*e)*(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^(3/2) /c^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[d + e*x^2])/(a - c*x^4)^(3/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[d + e*x^2])/(a - c*x^4)^(3/2), 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[d + e*x^2])/(a - c*x^4)^(3/2),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}}{\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas" )
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c^2*x^8 - 2*a*c*x^4 + a^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {d + e x^{2}}}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(1/2)/(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*sqrt(d + e*x**2)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((B*x^2 + A)*sqrt(e*x^2 + d)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*sqrt(e*x^2 + d)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^(1/2))/(a - c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^(1/2))/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a e x +\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b d x +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{2} b \,e^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{2} c d e -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a b c \,d^{2}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a b c \,e^{2} x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a \,c^{2} d e \,x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) b \,c^{2} d^{2} x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{3} d e -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{2} b \,d^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{2} c d e \,x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a b c \,d^{2} x^{4}}{2 a e \left (-c \,x^{4}+a \right )} \] Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e*x + sqrt(d + e*x**2)*sqrt(a - c*x** 4)*b*d*x + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e *x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2* b*e**2 - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x** 2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*d* e - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x**2 - 2 *a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*b*c*d**2 - 2 *int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x**2 - 2*a* c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*b*c*e**2*x**4 + int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x**2 - 2*a* c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*c**2*d*e*x**4 + int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x**2 - 2*a* c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*b*c**2*d**2*x**4 + int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2*a*c*d* x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**3*d*e - int((sqrt( d + e*x**2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c *e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*b*d**2 - int((sqrt(d + e*x** 2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*d*e*x**4 + int((sqrt(d + e*x**2)*sq rt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c...