Integrand size = 31, antiderivative size = 740 \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {(B d-A e) \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}{3 d e^3 \left (d+e x^2\right )^{3/2}}-\frac {\left (9 B c d^3-6 A c d^2 e-a B d e^2-2 a A e^3\right ) x \sqrt {a-c x^4}}{3 d^2 e^3 \sqrt {d+e x^2}}+\frac {\left (105 B c d^3-60 A c d^2 e-8 a B d e^2-16 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 d^2 e^4 x}-\frac {B c x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 e^3}+\frac {\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) \left (105 B c d^3-60 A c d^2 e-8 a B d e^2-16 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{24 d^2 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \sqrt {c} \left (35 B c d^3-20 A c d^2 e-8 a B d e^2-16 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{24 d^2 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {c \left (35 B c d^2-20 A c d e-12 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/3*(-A*e+B*d)*(-a*e^2+c*d^2)*x*(-c*x^4+a)^(1/2)/d/e^3/(e*x^2+d)^(3/2)-1/3 *(-2*A*a*e^3-6*A*c*d^2*e-B*a*d*e^2+9*B*c*d^3)*x*(-c*x^4+a)^(1/2)/d^2/e^3/( e*x^2+d)^(1/2)+1/24*(-16*A*a*e^3-60*A*c*d^2*e-8*B*a*d*e^2+105*B*c*d^3)*(e* x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d^2/e^4/x-1/4*B*c*x*(e*x^2+d)^(1/2)*(-c*x^4+ a)^(1/2)/e^3+1/24*c^(1/2)*(c^(1/2)*d+a^(1/2)*e)*(-16*A*a*e^3-60*A*c*d^2*e- 8*B*a*d*e^2+105*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^2/e^4/(e*x^2+d)^(1/2)/(-c*x^ 4+a)^(1/2)-1/24*a^(1/2)*c^(1/2)*(-16*A*a*e^3-20*A*c*d^2*e-8*B*a*d*e^2+35*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)*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^2/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/8*c* (-20*A*c*d*e-12*B*a*e^2+35*B*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)*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^4/(e*x^2+d)^(1 /2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(5/2),x]
Output:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{5/2}}dx\) |
Input:
Int[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(5/2),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="fricas" )
Output:
integral(-(B*c*x^6 + A*c*x^4 - B*a*x^2 - A*a)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (a - c x^{4}\right )^{\frac {3}{2}}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(3/2)/(e*x**2+d)**(5/2),x)
Output:
Integral((A + B*x**2)*(a - c*x**4)**(3/2)/(d + e*x**2)**(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(5/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(5/2),x)
Output:
( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*d*x + 6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e*x**3 - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d*x**5 + 12 *int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2 *x**8 - c*e**3*x**10),x)*a*b*c*d**2*e**2 + 24*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*a*b* c*d*e**3*x**2 + 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e *x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*a*b*c*e**4*x**4 + 20*int((sqrt( d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2* x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e **3*x**10),x)*a*c**2*d**3*e + 40*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x* *8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x** 4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*a*c**2*d**2*e**2* x**2 + 20*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2* e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3 *c*d*e**2*x**8 - c*e**3*x**10),x)*a*c**2*d*e**3*x**4 - 35*int((sqrt(d + e* x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3...