Integrand size = 31, antiderivative size = 922 \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {(B d-A e) \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}{9 d e^3 \left (d+e x^2\right )^{9/2}}-\frac {\left (21 B c d^3-12 A c d^2 e-a B d e^2-8 a A e^3\right ) x \sqrt {a-c x^4}}{63 d^2 e^3 \left (d+e x^2\right )^{7/2}}+\frac {\left (35 B c^2 d^5-5 A c^2 d^4 e-29 a B c d^3 e^2+17 a A c d^2 e^3-2 a^2 B d e^4-16 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 )^{5/2}}-\frac {\left (2 A e \left (5 c^3 d^6-22 a c^2 d^4 e^2+65 a^2 c d^2 e^4-32 a^3 e^6\right )+B \left (35 c^3 d^7-82 a c^2 d^5 e^2+23 a^2 c d^3 e^4-8 a^3 d e^6\right )\right ) x \sqrt {a-c x^4}}{315 d^4 e^3 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}-\frac {4 a^2 \left (21 B c^2 d^5-93 A c^2 d^4 e+15 a B c d^3 e^2+93 a A c d^2 e^3-4 a^2 B d e^4-32 a^2 A e^5\right ) \sqrt {a-c x^4}}{315 d^4 \left (c d^2-a e^2\right )^3 x \sqrt {d+e x^2}}-\frac {4 a^2 \sqrt {c} \left (21 B c^2 d^5-93 A c^2 d^4 e+15 a B c d^3 e^2+93 a A c d^2 e^3-4 a^2 B d e^4-32 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 )}{315 d^5 \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 {4 a^{3/2} \sqrt {c} \left (4 a B d e \left (3 c d^2-a e^2\right )-A \left (45 c^2 d^4-69 a c d^2 e^2+32 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 )}{315 d^5 \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/9*(-A*e+B*d)*(-a*e^2+c*d^2)*x*(-c*x^4+a)^(1/2)/d/e^3/(e*x^2+d)^(9/2)-1/6 3*(-8*A*a*e^3-12*A*c*d^2*e-B*a*d*e^2+21*B*c*d^3)*x*(-c*x^4+a)^(1/2)/d^2/e^ 3/(e*x^2+d)^(7/2)+1/105*(-16*A*a^2*e^5+17*A*a*c*d^2*e^3-5*A*c^2*d^4*e-2*B* a^2*d*e^4-29*B*a*c*d^3*e^2+35*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)^(5/2)-1/315*(2*A*e*(-32*a^3*e^6+65*a^2*c*d^2*e^4-22*a*c ^2*d^4*e^2+5*c^3*d^6)+B*(-8*a^3*d*e^6+23*a^2*c*d^3*e^4-82*a*c^2*d^5*e^2+35 *c^3*d^7))*x*(-c*x^4+a)^(1/2)/d^4/e^3/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2)-4/3 15*a^2*(-32*A*a^2*e^5+93*A*a*c*d^2*e^3-93*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)*(-c*x^4+a)^(1/2)/d^4/(-a*e^2+c*d^2)^3/x/(e*x^2+d) ^(1/2)-4/315*a^2*c^(1/2)*(-32*A*a^2*e^5+93*A*a*c*d^2*e^3-93*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^5/(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)-4 /315*a^(3/2)*c^(1/2)*(4*a*B*d*e*(-a*e^2+3*c*d^2)-A*(32*a^2*e^4-69*a*c*d^2* e^2+45*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^5/(-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)/( -c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(11/2),x]
Output:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(11/2), 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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{11/2}}dx\) |
Input:
Int[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(11/2),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {11}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),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^6*x^12 + 6*d*e^5*x^10 + 15*d^2*e^4*x^8 + 20*d^3*e^3*x^6 + 15*d^4*e ^2*x^4 + 6*d^5*e*x^2 + d^6), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(3/2)/(e*x**2+d)**(11/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="maxima ")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{11/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(11/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(11/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
Output:
( - 54*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*d*e**2*x - 12*sqrt(d + e*x **2)*sqrt(a - c*x**4)*a**2*b*e**3*x**3 + 33*sqrt(d + e*x**2)*sqrt(a - c*x* *4)*a**2*c*d**2*e*x + 88*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c*d*e**2*x **3 - 42*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d**3*x + 119*sqrt(d + e*x **2)*sqrt(a - c*x**4)*a*b*c*d**2*e*x**3 + 90*sqrt(d + e*x**2)*sqrt(a - c*x **4)*a*b*c*d*e**2*x**5 - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*e**3*x **7 - 99*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**2*d**2*e*x**5 - 22*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**2*d*e**2*x**7 - 77*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*d**2*e*x**7 - 288*int((sqrt(d + e*x**2)*sqrt(a - c*x**4) *x**4)/(4*a**2*d**6*e**2 + 24*a**2*d**5*e**3*x**2 + 60*a**2*d**4*e**4*x**4 + 80*a**2*d**3*e**5*x**6 + 60*a**2*d**2*e**6*x**8 + 24*a**2*d*e**7*x**10 + 4*a**2*e**8*x**12 + 7*a*c*d**8 + 42*a*c*d**7*e*x**2 + 101*a*c*d**6*e**2* x**4 + 116*a*c*d**5*e**3*x**6 + 45*a*c*d**4*e**4*x**8 - 38*a*c*d**3*e**5*x **10 - 53*a*c*d**2*e**6*x**12 - 24*a*c*d*e**7*x**14 - 4*a*c*e**8*x**16 - 7 *c**2*d**8*x**4 - 42*c**2*d**7*e*x**6 - 105*c**2*d**6*e**2*x**8 - 140*c**2 *d**5*e**3*x**10 - 105*c**2*d**4*e**4*x**12 - 42*c**2*d**3*e**5*x**14 - 7* c**2*d**2*e**6*x**16),x)*a**4*b*d**5*e**6 - 1440*int((sqrt(d + e*x**2)*sqr t(a - c*x**4)*x**4)/(4*a**2*d**6*e**2 + 24*a**2*d**5*e**3*x**2 + 60*a**2*d **4*e**4*x**4 + 80*a**2*d**3*e**5*x**6 + 60*a**2*d**2*e**6*x**8 + 24*a**2* d*e**7*x**10 + 4*a**2*e**8*x**12 + 7*a*c*d**8 + 42*a*c*d**7*e*x**2 + 10...