Integrand size = 24, antiderivative size = 490 \[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \sqrt {a-c x^4}}{d \sqrt {d+e x^2}}-\frac {\sqrt {d+e x^2} \sqrt {a-c x^4}}{d e x}-\frac {\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{d e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{d \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {c \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(1/2)-(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d/e/ x-c^(1/2)*(c^(1/2)*d+a^(1/2)*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)*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))/d/e/(e*x^2+d)^(1/2)/(-c *x^4+a)^(1/2)+a^(1/2)*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))/d/(e*x^2+d)^(1/2)/(-c*x^4 +a)^(1/2)-c*(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)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx \] Input:
Integrate[Sqrt[a - c*x^4]/(d + e*x^2)^(3/2),x]
Output:
Integrate[Sqrt[a - c*x^4]/(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.
\(\displaystyle \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[Sqrt[a - c*x^4]/(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]
\[\int \frac {\sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
Output:
int((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a - c x^{4}}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-c*x**4+a)**(1/2)/(e*x**2+d)**(3/2),x)
Output:
Integral(sqrt(a - c*x**4)/(d + e*x**2)**(3/2), x)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)/(e*x^2 + d)^(3/2), x)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)/(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a - c*x^4)^(1/2)/(d + e*x^2)^(3/2),x)
Output:
int((a - c*x^4)^(1/2)/(d + e*x^2)^(3/2), x)
\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \] Input:
int((-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(d**2 + 2*d*e*x**2 + e**2*x**4),x)