Integrand size = 24, antiderivative size = 597 \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\left (a-\frac {c d^2}{e^2}\right ) x \sqrt {a-c x^4}}{d \sqrt {d+e x^2}}+\frac {\left (15 c d^2-8 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 d e^3 x}-\frac {c x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 e^2}+\frac {\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) \left (15 c d^2-8 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 d e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \sqrt {c} \left (5 c d^2-8 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 d e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {3 c \left (5 c d^2-4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
(a-c*d^2/e^2)*x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(1/2)+1/8*(-8*a*e^2+15*c*d^2) *(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d/e^3/x-1/4*c*x*(e*x^2+d)^(1/2)*(-c*x^4+ a)^(1/2)/e^2+1/8*c^(1/2)*(c^(1/2)*d+a^(1/2)*e)*(-8*a*e^2+15*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)*Ellipti cE(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))/d/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/8*a^(1/2)*c^(1/2)*(-8 *a*e^2+5*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))/d/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+ 3/8*c*(-4*a*e^2+5*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)*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^3/(e*x^2+d)^(1/2)/(-c*x^4 +a)^(1/2)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx \] Input:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(3/2),x]
Output:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[(a - c*x^4)^(3/2)/(d + e*x^2)^(3/2),x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
\[\int \frac {\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x)
Output:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (a - c x^{4}\right )^{\frac {3}{2}}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-c*x**4+a)**(3/2)/(e*x**2+d)**(3/2),x)
Output:
Integral((a - c*x**4)**(3/2)/(d + e*x**2)**(3/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(3/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(3/2),x)
Output:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(3/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a e x -\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c d \,x^{3}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c d \,e^{2}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c \,e^{3} x^{2}-5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) c^{2} d^{3}-5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) c^{2} d^{2} e \,x^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c \,d^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c \,d^{2} e \,x^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a^{2} d^{2} e +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a^{2} d \,e^{2} x^{2}}{4 d e \left (e \,x^{2}+d \right )} \] Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(3/2),x)
Output:
(2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e*x - sqrt(d + e*x**2)*sqrt(a - c*x **4)*c*d*x**3 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2 *a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a *c*d*e**2 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d *e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c*e **3*x**2 - 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d* e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*c**2*d **3 - 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x** 2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*c**2*d**2*e *x**2 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d**2 + 2*a*d*e*x **2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c*d**3 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c*d**2*e*x**2 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2 + 2*a*d*e*x**2 + a*e* *2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2*d**2*e + 2*int ((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2*d*e**2*x**2)/(4*d*e*(d + e*x**2))