Integrand size = 31, antiderivative size = 933 \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {(B d-A e) \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}{7 d e^3 \left (d+e x^2\right )^{7/2}}-\frac {\left (17 B c d^3-10 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^3 \left (d+e x^2\right )^{5/2}}+\frac {\left (71 B c^2 d^5-15 A c^2 d^4 e-55 a B c d^3 e^2+27 a A c d^2 e^3-4 a^2 B d e^4-24 a^2 A e^5\right ) x \sqrt {a-c x^4}}{105 d^3 e^3 \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2}}+\frac {\left (48 a^2 A e^5 \left (2 c d^2-a e^2\right )+B \left (105 c^3 d^7-182 a c^2 d^5 e^2+37 a^2 c d^3 e^4-8 a^3 d e^6\right )\right ) \sqrt {a-c x^4}}{105 d^3 e^4 \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}+\frac {\sqrt {c} \left (48 a^2 A e^5 \left (2 c d^2-a e^2\right )+B \left (105 c^3 d^7-182 a c^2 d^5 e^2+37 a^2 c d^3 e^4-8 a^3 d e^6\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}} 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 e^4 \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 {\sqrt {a} \sqrt {c} \left (12 a A e^3 \left (5 c d^2-4 a e^2\right )-B \left (35 c^2 d^5-31 a c d^3 e^2+8 a^2 d 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 e^3 \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {B c^2 \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 )}{e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/7*(-A*e+B*d)*(-a*e^2+c*d^2)*x*(-c*x^4+a)^(1/2)/d/e^3/(e*x^2+d)^(7/2)-1/3 5*(-6*A*a*e^3-10*A*c*d^2*e-B*a*d*e^2+17*B*c*d^3)*x*(-c*x^4+a)^(1/2)/d^2/e^ 3/(e*x^2+d)^(5/2)+1/105*(-24*A*a^2*e^5+27*A*a*c*d^2*e^3-15*A*c^2*d^4*e-4*B *a^2*d*e^4-55*B*a*c*d^3*e^2+71*B*c^2*d^5)*x*(-c*x^4+a)^(1/2)/d^3/e^3/(-a*e ^2+c*d^2)/(e*x^2+d)^(3/2)+1/105*(48*a^2*A*e^5*(-a*e^2+2*c*d^2)+B*(-8*a^3*d *e^6+37*a^2*c*d^3*e^4-182*a*c^2*d^5*e^2+105*c^3*d^7))*(-c*x^4+a)^(1/2)/d^3 /e^4/(-a*e^2+c*d^2)^2/x/(e*x^2+d)^(1/2)+1/105*c^(1/2)*(48*a^2*A*e^5*(-a*e^ 2+2*c*d^2)+B*(-8*a^3*d*e^6+37*a^2*c*d^3*e^4-182*a*c^2*d^5*e^2+105*c^3*d^7) )*(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/e^4/(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)+1/105*a^(1/2)*c^(1/2)*(12*a*A*e^3*(-4*a*e^2+5*c*d ^2)-B*(8*a^2*d*e^4-31*a*c*d^3*e^2+35*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)*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/e^3/ (-a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+B*c^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)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(9/2),x]
Output:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{9/2}}dx\) |
Input:
Int[((A + B*x^2)*(a - c*x^4)^(3/2))/(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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),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^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. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(3/2)/(e*x**2+d)**(9/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{9/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(9/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(9/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(9/2),x)
Output:
( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*e*x + 15*sqrt(d + e*x**2)* sqrt(a - c*x**4)*a**2*c*d*x + 14*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c* e*x**3 + 14*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d*x**3 + 4*sqrt(d + e* x**2)*sqrt(a - c*x**4)*a*b*c*e*x**5 - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)* a*c**2*d*x**5 + 112*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(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 + 5*a*c*d**7 + 25*a*c *d**6*e*x**2 + 46*a*c*d**5*e**2*x**4 + 30*a*c*d**4*e**3*x**6 - 15*a*c*d**3 *e**4*x**8 - 35*a*c*d**2*e**5*x**10 - 20*a*c*d*e**6*x**12 - 4*a*c*e**7*x** 14 - 5*c**2*d**7*x**4 - 25*c**2*d**6*e*x**6 - 50*c**2*d**5*e**2*x**8 - 50* c**2*d**4*e**3*x**10 - 25*c**2*d**3*e**4*x**12 - 5*c**2*d**2*e**5*x**14),x )*a**2*b*c**2*d**4*e**4 + 448*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10 )/(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 + 5*a*c*d** 7 + 25*a*c*d**6*e*x**2 + 46*a*c*d**5*e**2*x**4 + 30*a*c*d**4*e**3*x**6 - 1 5*a*c*d**3*e**4*x**8 - 35*a*c*d**2*e**5*x**10 - 20*a*c*d*e**6*x**12 - 4*a* c*e**7*x**14 - 5*c**2*d**7*x**4 - 25*c**2*d**6*e*x**6 - 50*c**2*d**5*e**2* x**8 - 50*c**2*d**4*e**3*x**10 - 25*c**2*d**3*e**4*x**12 - 5*c**2*d**2*e** 5*x**14),x)*a**2*b*c**2*d**3*e**5*x**2 + 672*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(4*a**2*d**5*e**2 + 20*a**2*d**4*e**3*x**2 + 40*a**2*d...