Integrand size = 31, antiderivative size = 781 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {\left (\frac {A}{d}-\frac {B}{e}\right ) x \sqrt {a-c x^4}}{7 \left (d+e x^2\right )^{7/2}}+\frac {\left (3 B c d^3+4 A c d^2 e-a B d e^2-6 a A e^3\right ) x \sqrt {a-c x^4}}{35 d^2 e \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{5/2}}+\frac {2 \left (4 A e \left (c^2 d^4-6 a c d^2 e^2+3 a^2 e^4\right )+B \left (3 c^2 d^5+3 a c d^3 e^2+2 a^2 d e^4\right )\right ) x \sqrt {a-c x^4}}{105 d^3 e \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}+\frac {2 a \left (A e \left (77 c^2 d^4-69 a c d^2 e^2+24 a^2 e^4\right )-B \left (21 c^2 d^5+15 a c d^3 e^2-4 a^2 d e^4\right )\right ) \sqrt {a-c x^4}}{105 d^3 \left (c d^2-a e^2\right )^3 x \sqrt {d+e x^2}}-\frac {2 a \sqrt {c} \left (21 B c^2 d^5-77 A c^2 d^4 e+15 a B c d^3 e^2+69 a A c d^2 e^3-4 a^2 B d e^4-24 a^2 A e^5\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 d^4 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {a} \sqrt {c} \left (4 a B d e \left (3 c d^2-a e^2\right )-A \left (35 c^2 d^4-51 a c d^2 e^2+24 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 d^4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/7*(A/d-B/e)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2)+1/35*(-6*A*a*e^3+4*A*c*d^ 2*e-B*a*d*e^2+3*B*c*d^3)*x*(-c*x^4+a)^(1/2)/d^2/e/(-a*e^2+c*d^2)/(e*x^2+d) ^(5/2)+2/105*(4*A*e*(3*a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)+B*(2*a^2*d*e^4+3*a*c *d^3*e^2+3*c^2*d^5))*x*(-c*x^4+a)^(1/2)/d^3/e/(-a*e^2+c*d^2)^2/(e*x^2+d)^( 3/2)+2/105*a*(A*e*(24*a^2*e^4-69*a*c*d^2*e^2+77*c^2*d^4)-B*(-4*a^2*d*e^4+1 5*a*c*d^3*e^2+21*c^2*d^5))*(-c*x^4+a)^(1/2)/d^3/(-a*e^2+c*d^2)^3/x/(e*x^2+ d)^(1/2)-2/105*a*c^(1/2)*(-24*A*a^2*e^5+69*A*a*c*d^2*e^3-77*A*c^2*d^4*e-4* B*a^2*d*e^4+15*B*a*c*d^3*e^2+21*B*c^2*d^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)*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)^3/(c^(1/2)*d+a^(1/2)*e)^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-2 /105*a^(1/2)*c^(1/2)*(4*a*B*d*e*(-a*e^2+3*c*d^2)-A*(24*a^2*e^4-51*a*c*d^2* e^2+35*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))/d^4/(-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)/( -c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(9/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{9/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(9/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="fricas" )
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/(e*x**2+d)**(9/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="maxima" )
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{9/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(9/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(9/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2),x)
Output:
( - 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2*x - 7*sqrt(d + e*x**2)*sq rt(a - c*x**4)*b*c*d**2*x + 7*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d*e*x* *3 + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*e**2*x**5 - 48*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**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**3*c*d**4*e**5 - 192* 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**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**3*c*d* *3*e**6*x**2 - 288*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*...