Integrand size = 34, antiderivative size = 620 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\frac {\left (3 B c d^2+30 A c d e-8 a B e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c e x}+\frac {1}{24} (7 B d+6 A e) x \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {1}{6} B e x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 B c d^2+78 A c d e-8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (31 B c d^2+6 A c d e+8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (B c d^3-6 A c d^2 e+12 a B d e^2+8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/48*(30*A*c*d*e-8*B*a*e^2+3*B*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e /x+1/24*(6*A*e+7*B*d)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/6*B*e*x^3*(e*x^ 2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/48*(d+a^(1/2)*e/c^(1/2))*(78*A*c*d*e-8*B*a*e ^2+3*B*c*d^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))/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/48*a ^(1/2)*(6*A*c*d*e+8*B*a*e^2+31*B*c*d^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))/c^(1/2)/(e*x^2+ d)^(1/2)/(-c*x^4+a)^(1/2)+1/16*(8*A*a*e^3-6*A*c*d^2*e+12*B*a*d*e^2+B*c*d^3 )*(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 )*EllipticPi(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/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^2,x]
Output:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^2}dx\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^2,x]
Output:
$Aborted
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p , x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}{x^{2}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x, algorithm="fri cas")
Output:
integral((B*e*x^4 + (B*d + A*e)*x^2 + A*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d )/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2)/x**2,x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2)/x**2, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x, algorithm="max ima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x, algorithm="gia c")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^2, x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}}{x^2} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^2,x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^2, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^2} \, dx=\frac {-12 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a^{2} e^{2}-22 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b d e +24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,d^{2}+6 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d e \,x^{2}+7 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x^{2}+4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d e \,x^{4}-24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,e^{3} x -36 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c d \,e^{2} x +18 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{2} e x -3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{3} x -12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{8}-c d \,x^{6}+a e \,x^{4}+a d \,x^{2}}d x \right ) a^{3} d \,e^{2} x -22 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{8}-c d \,x^{6}+a e \,x^{4}+a d \,x^{2}}d x \right ) a^{2} b \,d^{2} e x +48 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{8}-c d \,x^{6}+a e \,x^{4}+a d \,x^{2}}d x \right ) a^{2} c \,d^{3} x +42 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,d^{2} e x +17 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{3} x}{24 c d x} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^2,x)
Output:
( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**2 - 22*sqrt(d + e*x**2)*s qrt(a - c*x**4)*a*b*d*e + 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2 + 6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e*x**2 + 7*sqrt(d + e*x**2)*sqrt (a - c*x**4)*b*c*d**2*x**2 + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d*e*x **4 - 24*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c* d*x**4 - c*e*x**6),x)*a**2*c*e**3*x - 36*int((sqrt(d + e*x**2)*sqrt(a - c* x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d*e**2*x + 18* int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c**2*d**2*e*x - 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x* *4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b*c**2*d**3*x - 12*int((sqrt (d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**2 + a*e*x**4 - c*d*x**6 - c*e*x**8) ,x)*a**3*d*e**2*x - 22*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**2 + a*e*x**4 - c*d*x**6 - c*e*x**8),x)*a**2*b*d**2*e*x + 48*int((sqrt(d + e*x **2)*sqrt(a - c*x**4))/(a*d*x**2 + a*e*x**4 - c*d*x**6 - c*e*x**8),x)*a**2 *c*d**3*x + 42*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**2*c*d**2*e*x + 17*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*b*c*d**3*x)/(24*c*d* x)