Integrand size = 24, antiderivative size = 390 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=-\frac {e \sqrt {d+e x^2}}{2 \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}+\frac {c d x \sqrt {d+e x^2}}{2 a \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt {c} e \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/2*e*(e*x^2+d)^(1/2)/(-a*e^2+c*d^2)/x/(-c*x^4+a)^(1/2)+1/2*c*d*x*(e*x^2+ d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)-1/2*c^(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) )/a/(c^(1/2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+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)*Ell ipticF(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)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx \] Input:
Integrate[1/(Sqrt[d + e*x^2]*(a - c*x^4)^(3/2)),x]
Output:
Integrate[1/(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.
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \sqrt {d+e x^2}}dx\) |
Input:
Int[1/(Sqrt[d + e*x^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]
\[\int \frac {1}{\sqrt {e \,x^{2}+d}\, \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]
Input:
int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
Output:
int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(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), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a - c x^{4}\right )^{\frac {3}{2}} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate(1/(e*x**2+d)**(1/2)/(-c*x**4+a)**(3/2),x)
Output:
Integral(1/((a - c*x**4)**(3/2)*sqrt(d + e*x**2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*sqrt(e*x^2 + d)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-c*x^4 + a)^(3/2)*sqrt(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,\sqrt {e\,x^2+d}} \,d x \] Input:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^(1/2)),x)
Output:
int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^(1/2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \] Input:
int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2),x)
Output:
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)