Integrand size = 24, antiderivative size = 657 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\frac {c x \left (d-e x^2\right )}{6 a \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}}+\frac {e^2 \left (c d^2+3 a e^2\right ) x}{3 a d \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {e \left (7 c^2 d^4-27 a c d^2 e^2-12 a^2 e^4\right ) \sqrt {d+e x^2}}{12 a d \left (c d^2-a e^2\right )^3 x \sqrt {a-c x^4}}+\frac {c \left (5 c^2 d^4-15 a c d^2 e^2-22 a^2 e^4\right ) x \sqrt {d+e x^2}}{12 a^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt {c} e \left (7 c^2 d^4-27 a c d^2 e^2-12 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{12 a^2 d \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (5 c^2 d^4-9 a c d^2 e^2+12 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{12 a^{5/2} d \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/6*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(3/2)+1/3*e ^2*(3*a*e^2+c*d^2)*x/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2) -1/12*e*(-12*a^2*e^4-27*a*c*d^2*e^2+7*c^2*d^4)*(e*x^2+d)^(1/2)/a/d/(-a*e^2 +c*d^2)^3/x/(-c*x^4+a)^(1/2)+1/12*c*(-22*a^2*e^4-15*a*c*d^2*e^2+5*c^2*d^4) *x*(e*x^2+d)^(1/2)/a^2/(-a*e^2+c*d^2)^3/(-c*x^4+a)^(1/2)-1/12*c^(1/2)*e*(- 12*a^2*e^4-27*a*c*d^2*e^2+7*c^2*d^4)*(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^2/d/(c^(1/2)*d-a ^(1/2)*e)^3/(c^(1/2)*d+a^(1/2)*e)^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/12* c^(1/2)*(12*a^2*e^4-9*a*c*d^2*e^2+5*c^2*d^4)*(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^(5/2)/d/ (-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx \] Input:
Integrate[1/((d + e*x^2)^(3/2)*(a - c*x^4)^(5/2)),x]
Output:
Integrate[1/((d + e*x^2)^(3/2)*(a - c*x^4)^(5/2)), 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.
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{5/2} \left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{5/2} \left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[1/((d + e*x^2)^(3/2)*(a - c*x^4)^(5/2)),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]
\[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (-c \,x^{4}+a \right )^{\frac {5}{2}}}d x\]
Input:
int(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x)
Output:
int(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c^3*e^2*x^16 + 2*c^3*d*e*x^14 - 6*a*c^2*d*e*x^10 + (c^3*d^2 - 3*a*c^2*e^2)*x^12 + 6*a^2*c*d*e*x^6 - 3*(a *c^2*d^2 - a^2*c*e^2)*x^8 - 2*a^3*d*e*x^2 - a^3*d^2 + (3*a^2*c*d^2 - a^3*e ^2)*x^4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (a - c x^{4}\right )^{\frac {5}{2}} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x**2+d)**(3/2)/(-c*x**4+a)**(5/2),x)
Output:
Integral(1/((a - c*x**4)**(5/2)*(d + e*x**2)**(3/2)), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((-c*x^4 + a)^(5/2)*(e*x^2 + d)^(3/2)), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate(1/((-c*x^4 + a)^(5/2)*(e*x^2 + d)^(3/2)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{5/2}\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int(1/((a - c*x^4)^(5/2)*(d + e*x^2)^(3/2)),x)
Output:
int(1/((a - c*x^4)^(5/2)*(d + e*x^2)^(3/2)), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c^{3} e^{2} x^{16}-2 c^{3} d e \,x^{14}+3 a \,c^{2} e^{2} x^{12}-c^{3} d^{2} x^{12}+6 a \,c^{2} d e \,x^{10}-3 a^{2} c \,e^{2} x^{8}+3 a \,c^{2} d^{2} x^{8}-6 a^{2} c d e \,x^{6}+a^{3} e^{2} x^{4}-3 a^{2} c \,d^{2} x^{4}+2 a^{3} d e \,x^{2}+a^{3} d^{2}}d x \] Input:
int(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^(5/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**3*d**2 + 2*a**3*d*e*x**2 + a** 3*e**2*x**4 - 3*a**2*c*d**2*x**4 - 6*a**2*c*d*e*x**6 - 3*a**2*c*e**2*x**8 + 3*a*c**2*d**2*x**8 + 6*a*c**2*d*e*x**10 + 3*a*c**2*e**2*x**12 - c**3*d** 2*x**12 - 2*c**3*d*e*x**14 - c**3*e**2*x**16),x)