Integrand size = 24, antiderivative size = 465 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=-\frac {e \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 c x}-\frac {e \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (2 c d^2+a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {3 d e \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/2*e*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/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)*El lipticE(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*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/2*(a*e^2+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)*Elli pticF(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^(1/2)/c^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+3/2*d*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)*El lipticPi(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*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx \] Input:
Integrate[(d + e*x^2)^(3/2)/Sqrt[a - c*x^4],x]
Output:
Integrate[(d + e*x^2)^(3/2)/Sqrt[a - c*x^4], 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}}dx\) |
Input:
Int[(d + e*x^2)^(3/2)/Sqrt[a - c*x^4],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]
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\sqrt {-c \,x^{4}+a}}d x\]
Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
Output:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)/(c*x^4 - a), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{\sqrt {a - c x^{4}}}\, dx \] Input:
integrate((e*x**2+d)**(3/2)/(-c*x**4+a)**(1/2),x)
Output:
Integral((d + e*x**2)**(3/2)/sqrt(a - c*x**4), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(3/2)/sqrt(-c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^(3/2)/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{\sqrt {a-c\,x^4}} \,d x \] Input:
int((d + e*x^2)^(3/2)/(a - c*x^4)^(1/2),x)
Output:
int((d + e*x^2)^(3/2)/(a - c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) d \] Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*e + int((sqrt (d + e*x**2)*sqrt(a - c*x**4))/(a - c*x**4),x)*d