Integrand size = 24, antiderivative size = 627 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )^{7/2}}{2 a \sqrt {a-c x^4}}+\frac {e \left (3 c d^2+2 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c^2 x}+\frac {3 d e^2 x \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c}+\frac {e^3 x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c}+\frac {e \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 c d^2+2 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a c \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (c^2 d^4-6 a c d^2 e^2-2 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} c^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {7 d e^3 \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 c \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/2*x*(e*x^2+d)^(7/2)/a/(-c*x^4+a)^(1/2)+1/2*e*(2*a*e^2+3*c*d^2)*(e*x^2+d) ^(1/2)*(-c*x^4+a)^(1/2)/a/c^2/x+3/2*d*e^2*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/ 2)/a/c+1/2*e^3*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/c+1/2*e*(d+a^(1/2)*e /c^(1/2))*(2*a*e^2+3*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/c/(e*x^2+d)^(1/2)/(-c*x^4 +a)^(1/2)+1/2*(-2*a^2*e^4-6*a*c*d^2*e^2+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^(3/2 )/c^(3/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-7/2*d*e^3*(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/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx \] Input:
Integrate[(d + e*x^2)^(7/2)/(a - c*x^4)^(3/2),x]
Output:
Integrate[(d + e*x^2)^(7/2)/(a - c*x^4)^(3/2), 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}}dx\) |
Input:
Int[(d + e*x^2)^(7/2)/(a - c*x^4)^(3/2),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]
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {7}{2}}}{\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x)
Output:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {7}{2}}}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**(7/2)/(-c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x**2)**(7/2)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(7/2)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^(7/2)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)^(7/2)/(a - c*x^4)^(3/2),x)
Output:
int((d + e*x^2)^(7/2)/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^(3/2),x)
Output:
(16*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e**4*x + 6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d**2*e**2*x - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d*e**3*x* *3 - 7*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*c**2*d*e* *4 + 7*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*c**3*d*e**4 *x**4 - 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x **2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**3*e* *5 + 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*e** 5*x**4 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x **2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*c**2* d**4*e - 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x **2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*c**3*d* *4*e*x**4 - 16*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x* *2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**3*d*e **4 - 6*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2* a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*d**3*e** 2 + 16*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**2*d + a**2*e*x**2 - 2*a *c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*d*e**4...