Integrand size = 24, antiderivative size = 369 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )^{3/2}}{2 a \sqrt {a-c x^4}}+\frac {e \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c x}+\frac {e \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (c d^2-a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/2*x*(e*x^2+d)^(3/2)/a/(-c*x^4+a)^(1/2)+1/2*e*(e*x^2+d)^(1/2)*(-c*x^4+a)^ (1/2)/a/c/x+1/2*e*(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*e^2+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))/a^(3/2)/c^(1/2)/ (e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx \] Input:
Integrate[(d + e*x^2)^(3/2)/(a - c*x^4)^(3/2),x]
Output:
Integrate[(d + e*x^2)^(3/2)/(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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}}dx\) |
Input:
Int[(d + e*x^2)^(3/2)/(a - c*x^4)^(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]
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
Output:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)/(c^2*x^8 - 2*a*c*x^4 + a^2), x )
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**(3/2)/(-c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x**2)**(3/2)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(3/2)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^(3/2)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)^(3/2)/(a - c*x^4)^(3/2),x)
Output:
int((d + e*x^2)^(3/2)/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, d x +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a^{2} e^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a c \,d^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a c \,e^{2} x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) c^{2} d^{2} x^{4}}{a \left (-c \,x^{4}+a \right )} \] Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*d*x + 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*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*c*d**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*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)*c**2*d**2*x**4)/(a*(a - c*x**4))