Integrand size = 24, antiderivative size = 719 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (d+e x^2\right )^{9/2}}{6 a \left (a-c x^4\right )^{3/2}}+\frac {x \left (5 d-4 e x^2\right ) \left (d+e x^2\right )^{7/2}}{12 a^2 \sqrt {a-c x^4}}+\frac {d e \left (11 c d^2-15 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{12 a^2 c^2 x}+\frac {e^2 \left (c d^2-2 a e^2\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 a^2 c^2}-\frac {7 d e^3 x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{12 a^2 c}-\frac {e^4 x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 a^2 c}+\frac {d e \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (11 c d^2-15 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{12 a^2 c \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {d \left (5 c^2 d^4-14 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{12 a^{5/2} c^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {e^5 \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{c^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/6*x*(e*x^2+d)^(9/2)/a/(-c*x^4+a)^(3/2)+1/12*x*(-4*e*x^2+5*d)*(e*x^2+d)^( 7/2)/a^2/(-c*x^4+a)^(1/2)+1/12*d*e*(-15*a*e^2+11*c*d^2)*(e*x^2+d)^(1/2)*(- c*x^4+a)^(1/2)/a^2/c^2/x+1/4*e^2*(-2*a*e^2+c*d^2)*x*(e*x^2+d)^(1/2)*(-c*x^ 4+a)^(1/2)/a^2/c^2-7/12*d*e^3*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/c-1 /3*e^4*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/c+1/12*d*e*(d+a^(1/2)*e/c^ (1/2))*(-15*a*e^2+11*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))/a^2/c/(e*x^2+d)^(1/2)/(-c*x ^4+a)^(1/2)+1/12*d*(21*a^2*e^4-14*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)/c^(3/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+e^5*(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 ))/c^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx \] Input:
Integrate[(d + e*x^2)^(9/2)/(a - c*x^4)^(5/2),x]
Output:
Integrate[(d + e*x^2)^(9/2)/(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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}}dx\) |
Input:
Int[(d + e*x^2)^(9/2)/(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]
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {9}{2}}}{\left (-c \,x^{4}+a \right )^{\frac {5}{2}}}d x\]
Input:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x)
Output:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x, algorithm="fricas")
Output:
integral(-(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)*sqrt (-c*x^4 + a)*sqrt(e*x^2 + d)/(c^3*x^12 - 3*a*c^2*x^8 + 3*a^2*c*x^4 - a^3), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(9/2)/(-c*x**4+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(9/2)/(-c*x^4 + a)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^(9/2)/(-c*x^4 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{9/2}}{{\left (a-c\,x^4\right )}^{5/2}} \,d x \] Input:
int((d + e*x^2)^(9/2)/(a - c*x^4)^(5/2),x)
Output:
int((d + e*x^2)^(9/2)/(a - c*x^4)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^(5/2),x)
Output:
( - 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**6*x + 135*sqrt(d + e*x** 2)*sqrt(a - c*x**4)*a*c*d**2*e**4*x + 50*sqrt(d + e*x**2)*sqrt(a - c*x**4) *a*c*d*e**5*x**3 + 80*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e**6*x**5 - 30 *sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*d**4*e**2*x - 10*sqrt(d + e*x**2)* sqrt(a - c*x**4)*c**2*d**3*e**3*x**3 - 55*sqrt(d + e*x**2)*sqrt(a - c*x**4 )*c**2*d**2*e**4*x**5 + 432*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/ (12*a**4*d*e**2 + 12*a**4*e**3*x**2 - 5*a**3*c*d**3 - 5*a**3*c*d**2*e*x**2 - 36*a**3*c*d*e**2*x**4 - 36*a**3*c*e**3*x**6 + 15*a**2*c**2*d**3*x**4 + 15*a**2*c**2*d**2*e*x**6 + 36*a**2*c**2*d*e**2*x**8 + 36*a**2*c**2*e**3*x* *10 - 15*a*c**3*d**3*x**8 - 15*a*c**3*d**2*e*x**10 - 12*a*c**3*d*e**2*x**1 2 - 12*a*c**3*e**3*x**14 + 5*c**4*d**3*x**12 + 5*c**4*d**2*e*x**14),x)*a** 4*c**2*e**9 - 360*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(12*a**4*d *e**2 + 12*a**4*e**3*x**2 - 5*a**3*c*d**3 - 5*a**3*c*d**2*e*x**2 - 36*a**3 *c*d*e**2*x**4 - 36*a**3*c*e**3*x**6 + 15*a**2*c**2*d**3*x**4 + 15*a**2*c* *2*d**2*e*x**6 + 36*a**2*c**2*d*e**2*x**8 + 36*a**2*c**2*e**3*x**10 - 15*a *c**3*d**3*x**8 - 15*a*c**3*d**2*e*x**10 - 12*a*c**3*d*e**2*x**12 - 12*a*c **3*e**3*x**14 + 5*c**4*d**3*x**12 + 5*c**4*d**2*e*x**14),x)*a**3*c**3*d** 2*e**7 - 864*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(12*a**4*d*e**2 + 12*a**4*e**3*x**2 - 5*a**3*c*d**3 - 5*a**3*c*d**2*e*x**2 - 36*a**3*c*d* e**2*x**4 - 36*a**3*c*e**3*x**6 + 15*a**2*c**2*d**3*x**4 + 15*a**2*c**2...