Integrand size = 31, antiderivative size = 577 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {\left (\frac {A}{d}-\frac {B}{e}\right ) x \sqrt {a-c x^4}}{5 \left (d+e x^2\right )^{5/2}}+\frac {\left (3 B c d^3+2 A c d^2 e-a B d e^2-4 a A e^3\right ) x \sqrt {a-c x^4}}{15 d^2 e \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2}}-\frac {2 a \left (3 B c d^3-8 A c d^2 e+a B d e^2+4 a A e^3\right ) \sqrt {a-c x^4}}{15 d^2 \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}-\frac {2 a \sqrt {c} \left (3 B c d^3-8 A c d^2 e+a B d e^2+4 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {2 \sqrt {a} \sqrt {c} \left (5 A c d^2-a B d e-4 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/5*(A/d-B/e)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(5/2)+1/15*(-4*A*a*e^3+2*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) ^(3/2)-2/15*a*(4*A*a*e^3-8*A*c*d^2*e+B*a*d*e^2+3*B*c*d^3)*(-c*x^4+a)^(1/2) /d^2/(-a*e^2+c*d^2)^2/x/(e*x^2+d)^(1/2)-2/15*a*c^(1/2)*(4*A*a*e^3-8*A*c*d^ 2*e+B*a*d*e^2+3*B*c*d^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)*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/(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)+2/15*a^(1/2)*c^(1/2)*(-4*A*a*e ^2+5*A*c*d^2-B*a*d*e)*(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/(-a*e^2+c*d^2)/(e*x^2+d)^(1/2 )/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(7/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{7/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(7/2),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="fricas" )
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*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)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{\left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/(e*x**2+d)**(7/2),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(d + e*x**2)**(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="maxima" )
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(7/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
Output:
( - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*d**2*x + 5*sqrt(d + e*x**2)*sq rt(a - c*x**4)*a*b*d*e*x**3 + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2 *x**5 - 7*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e*x**5 - 3*sqrt(d + e*x* *2)*sqrt(a - c*x**4)*b*c*d**2*x**5 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x* *4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x** 6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c *d*e**3*x**10 - c*e**4*x**12),x)*a*b*c*d**3*e**3 + 12*int((sqrt(d + e*x**2 )*sqrt(a - c*x**4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2* e**2*x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)*a*b*c*d**2*e**4*x**2 + 12* int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d** 3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)*a*b*c* d*e**5*x**4 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(a*d**4 + 4* a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d** 4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3*x**10 - c*e**4* x**12),x)*a*b*c*e**6*x**6 - 14*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**1 0)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e* *4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3* x**10 - c*e**4*x**12),x)*a*c**2*d**4*e**2 - 42*int((sqrt(d + e*x**2)*sq...