Integrand size = 24, antiderivative size = 580 \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {\left (a-\frac {c d^2}{e^2}\right ) x \sqrt {a-c x^4}}{7 d \left (d+e x^2\right )^{7/2}}+\frac {2 \left (\frac {3 a}{d^2}+\frac {5 c}{e^2}\right ) x \sqrt {a-c x^4}}{35 \left (d+e x^2\right )^{5/2}}-\frac {\left (5 c^2 d^4-9 a c d^2 e^2+8 a^2 e^4\right ) x \sqrt {a-c x^4}}{35 d^3 e^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2}}+\frac {16 a^2 e \left (2 c d^2-a e^2\right ) \sqrt {a-c x^4}}{35 d^3 \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}+\frac {16 a^2 \sqrt {c} e \left (2 c d^2-a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{35 d^4 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {4 a^{3/2} \sqrt {c} \left (5 c d^2-4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{35 d^4 \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/7*(a-c*d^2/e^2)*x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(7/2)+2/35*(3*a/d^2+5*c/e ^2)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(5/2)-1/35*(8*a^2*e^4-9*a*c*d^2*e^2+5*c^2 *d^4)*x*(-c*x^4+a)^(1/2)/d^3/e^2/(-a*e^2+c*d^2)/(e*x^2+d)^(3/2)+16/35*a^2* e*(-a*e^2+2*c*d^2)*(-c*x^4+a)^(1/2)/d^3/(-a*e^2+c*d^2)^2/x/(e*x^2+d)^(1/2) +16/35*a^2*c^(1/2)*e*(-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)*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^4/(c^(1/2)*d-a^ (1/2)*e)/(-a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+4/35*a^(3/2)*c^(1 /2)*(-4*a*e^2+5*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))/d^4/(-a*e^2+c*d^2)/(e*x^2+d)^(1/ 2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx \] Input:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(9/2),x]
Output:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(9/2), 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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}}dx\) |
Input:
Int[(a - c*x^4)^(3/2)/(d + e*x^2)^(9/2),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]
\[\int \frac {\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x\]
Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
Output:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="fricas")
Output:
integral((-c*x^4 + a)^(3/2)*sqrt(e*x^2 + d)/(e^5*x^10 + 5*d*e^4*x^8 + 10*d ^2*e^3*x^6 + 10*d^3*e^2*x^4 + 5*d^4*e*x^2 + d^5), x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((-c*x**4+a)**(3/2)/(e*x**2+d)**(9/2),x)
Output:
Timed out
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(9/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(9/2), x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{9/2}} \,d x \] Input:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(9/2),x)
Output:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(9/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*x + 2*sqrt(d + e*x**2)*sqrt(a - c *x**4)*a*c*e*x**3 - 3*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*d*x**5 - 96*i nt((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(4*a**2*d**5*e**2 + 20*a**2*d* *4*e**3*x**2 + 40*a**2*d**3*e**4*x**4 + 40*a**2*d**2*e**5*x**6 + 20*a**2*d *e**6*x**8 + 4*a**2*e**7*x**10 + 21*a*c*d**7 + 105*a*c*d**6*e*x**2 + 206*a *c*d**5*e**2*x**4 + 190*a*c*d**4*e**3*x**6 + 65*a*c*d**3*e**4*x**8 - 19*a* c*d**2*e**5*x**10 - 20*a*c*d*e**6*x**12 - 4*a*c*e**7*x**14 - 21*c**2*d**7* x**4 - 105*c**2*d**6*e*x**6 - 210*c**2*d**5*e**2*x**8 - 210*c**2*d**4*e**3 *x**10 - 105*c**2*d**3*e**4*x**12 - 21*c**2*d**2*e**5*x**14),x)*a**2*c**2* d**6*e**2 - 384*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(4*a**2*d**5* e**2 + 20*a**2*d**4*e**3*x**2 + 40*a**2*d**3*e**4*x**4 + 40*a**2*d**2*e**5 *x**6 + 20*a**2*d*e**6*x**8 + 4*a**2*e**7*x**10 + 21*a*c*d**7 + 105*a*c*d* *6*e*x**2 + 206*a*c*d**5*e**2*x**4 + 190*a*c*d**4*e**3*x**6 + 65*a*c*d**3* e**4*x**8 - 19*a*c*d**2*e**5*x**10 - 20*a*c*d*e**6*x**12 - 4*a*c*e**7*x**1 4 - 21*c**2*d**7*x**4 - 105*c**2*d**6*e*x**6 - 210*c**2*d**5*e**2*x**8 - 2 10*c**2*d**4*e**3*x**10 - 105*c**2*d**3*e**4*x**12 - 21*c**2*d**2*e**5*x** 14),x)*a**2*c**2*d**5*e**3*x**2 - 576*int((sqrt(d + e*x**2)*sqrt(a - c*x** 4)*x**4)/(4*a**2*d**5*e**2 + 20*a**2*d**4*e**3*x**2 + 40*a**2*d**3*e**4*x* *4 + 40*a**2*d**2*e**5*x**6 + 20*a**2*d*e**6*x**8 + 4*a**2*e**7*x**10 + 21 *a*c*d**7 + 105*a*c*d**6*e*x**2 + 206*a*c*d**5*e**2*x**4 + 190*a*c*d**4...