Integrand size = 24, antiderivative size = 639 \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {\left (a-\frac {c d^2}{e^2}\right ) x \sqrt {a-c x^4}}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 \left (\frac {a}{d^2}+\frac {2 c}{e^2}\right ) x \sqrt {a-c x^4}}{15 \left (d+e x^2\right )^{3/2}}+\frac {\left (15 c^2 d^4-11 a c d^2 e^2+8 a^2 e^4\right ) \sqrt {a-c x^4}}{15 d^2 e^3 \left (c d^2-a e^2\right ) x \sqrt {d+e x^2}}+\frac {\sqrt {c} \left (15 c^2 d^4-11 a c d^2 e^2+8 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 e^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \sqrt {c} \left (5 c d^2-8 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {c^2 \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/5*(a-c*d^2/e^2)*x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(5/2)+4/15*(a/d^2+2*c/e^2 )*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2)+1/15*(8*a^2*e^4-11*a*c*d^2*e^2+15*c^2 *d^4)*(-c*x^4+a)^(1/2)/d^2/e^3/(-a*e^2+c*d^2)/x/(e*x^2+d)^(1/2)+1/15*c^(1/ 2)*(8*a^2*e^4-11*a*c*d^2*e^2+15*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))/d^3/e^3/(c^(1/ 2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/15*a^(1/2)*c^(1/2)*(-8* 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^3/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2) +c^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)*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^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(7/2),x]
Output:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(7/2), 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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}}dx\) |
Input:
Int[(a - c*x^4)^(3/2)/(d + e*x^2)^(7/2),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]
\[\int \frac {\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]
Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="fricas")
Output:
integral((-c*x^4 + a)^(3/2)*sqrt(e*x^2 + d)/(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2 *e^2*x^4 + 4*d^3*e*x^2 + d^4), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (a - c x^{4}\right )^{\frac {3}{2}}}{\left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((-c*x**4+a)**(3/2)/(e*x**2+d)**(7/2),x)
Output:
Integral((a - c*x**4)**(3/2)/(d + e*x**2)**(7/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(7/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(7/2), x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(7/2),x)
Output:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(7/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
(10*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*x + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e*x**3 + 16*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(4* a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24*a**2*d**2*e**4*x**4 + 16*a**2 *d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d**5*e*x**2 + 86*a* c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 - 9*a*c*d**2*e**4*x**8 - 16*a*c*d *e**5*x**10 - 4*a*c*e**6*x**12 - 15*c**2*d**6*x**4 - 60*c**2*d**5*e*x**6 - 90*c**2*d**4*e**2*x**8 - 60*c**2*d**3*e**3*x**10 - 15*c**2*d**2*e**4*x**1 2),x)*a**2*c**2*d**3*e**4 + 48*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8 )/(4*a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24*a**2*d**2*e**4*x**4 + 16 *a**2*d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d**5*e*x**2 + 86*a*c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 - 9*a*c*d**2*e**4*x**8 - 16* a*c*d*e**5*x**10 - 4*a*c*e**6*x**12 - 15*c**2*d**6*x**4 - 60*c**2*d**5*e*x **6 - 90*c**2*d**4*e**2*x**8 - 60*c**2*d**3*e**3*x**10 - 15*c**2*d**2*e**4 *x**12),x)*a**2*c**2*d**2*e**5*x**2 + 48*int((sqrt(d + e*x**2)*sqrt(a - c* x**4)*x**8)/(4*a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24*a**2*d**2*e**4 *x**4 + 16*a**2*d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d**5 *e*x**2 + 86*a*c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 - 9*a*c*d**2*e**4* x**8 - 16*a*c*d*e**5*x**10 - 4*a*c*e**6*x**12 - 15*c**2*d**6*x**4 - 60*c** 2*d**5*e*x**6 - 90*c**2*d**4*e**2*x**8 - 60*c**2*d**3*e**3*x**10 - 15*c**2 *d**2*e**4*x**12),x)*a**2*c**2*d*e**6*x**4 + 16*int((sqrt(d + e*x**2)*s...