Integrand size = 31, antiderivative size = 1002 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {\left (\frac {A}{d}-\frac {B}{e}\right ) x \sqrt {a-c x^4}}{9 \left (d+e x^2\right )^{9/2}}+\frac {\left (3 B c d^3+6 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 \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{7/2}}+\frac {2 \left (4 A e \left (c^2 d^4-4 a c d^2 e^2+2 a^2 e^4\right )+B \left (2 c^2 d^5+a c d^3 e^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 )^{5/2}}+\frac {2 \left (A e \left (8 c^3 d^6-103 a c^2 d^4 e^2+95 a^2 c d^2 e^4-32 a^3 e^6\right )+B \left (4 c^3 d^7+19 a c^2 d^5 e^2+13 a^2 c d^3 e^4-4 a^3 d e^6\right )\right ) x \sqrt {a-c x^4}}{315 d^4 e \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )^{3/2}}+\frac {2 a \left (2 A e \left (147 c^3 d^6-174 a c^2 d^4 e^2+123 a^2 c d^2 e^4-32 a^3 e^6\right )-B \left (63 c^3 d^7+90 a c^2 d^5 e^2-33 a^2 c d^3 e^4+8 a^3 d e^6\right )\right ) \sqrt {a-c x^4}}{315 d^4 \left (c d^2-a e^2\right )^4 x \sqrt {d+e x^2}}+\frac {2 a \sqrt {c} \left (2 A e \left (147 c^3 d^6-174 a c^2 d^4 e^2+123 a^2 c d^2 e^4-32 a^3 e^6\right )-B \left (63 c^3 d^7+90 a c^2 d^5 e^2-33 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 )}{315 d^5 \left (\sqrt {c} d-\sqrt {a} e\right )^4 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {a} \sqrt {c} \left (a B d e \left (51 c^2 d^4-27 a c d^2 e^2+8 a^2 e^4\right )-A \left (105 c^3 d^6-207 a c^2 d^4 e^2+198 a^2 c d^2 e^4-64 a^3 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}} \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 )^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/9*(A/d-B/e)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(9/2)+1/63*(-8*A*a*e^3+6*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) ^(7/2)+2/105*(4*A*e*(2*a^2*e^4-4*a*c*d^2*e^2+c^2*d^4)+B*(a^2*d*e^4+a*c*d^3 *e^2+2*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)^(5/2) +2/315*(A*e*(-32*a^3*e^6+95*a^2*c*d^2*e^4-103*a*c^2*d^4*e^2+8*c^3*d^6)+B*( -4*a^3*d*e^6+13*a^2*c*d^3*e^4+19*a*c^2*d^5*e^2+4*c^3*d^7))*x*(-c*x^4+a)^(1 /2)/d^4/e/(-a*e^2+c*d^2)^3/(e*x^2+d)^(3/2)+2/315*a*(2*A*e*(-32*a^3*e^6+123 *a^2*c*d^2*e^4-174*a*c^2*d^4*e^2+147*c^3*d^6)-B*(8*a^3*d*e^6-33*a^2*c*d^3* e^4+90*a*c^2*d^5*e^2+63*c^3*d^7))*(-c*x^4+a)^(1/2)/d^4/(-a*e^2+c*d^2)^4/x/ (e*x^2+d)^(1/2)+2/315*a*c^(1/2)*(2*A*e*(-32*a^3*e^6+123*a^2*c*d^2*e^4-174* a*c^2*d^4*e^2+147*c^3*d^6)-B*(8*a^3*d*e^6-33*a^2*c*d^3*e^4+90*a*c^2*d^5*e^ 2+63*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^5/(c^(1/2)*d-a^(1/2)*e)^4/(c^(1/2)*d+a ^(1/2)*e)^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-2/315*a^(1/2)*c^(1/2)*(a*B*d* e*(8*a^2*e^4-27*a*c*d^2*e^2+51*c^2*d^4)-A*(-64*a^3*e^6+198*a^2*c*d^2*e^4-2 07*a*c^2*d^4*e^2+105*c^3*d^6))*(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)^3/(e* x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(11/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{11/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(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]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {11}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x, algorithm="fricas ")
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + 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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/(e*x**2+d)**(11/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x, algorithm="maxima ")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{11/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(11/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(11/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(11/2),x)
Output:
( - 48*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*d*e**2*x + 4*sqrt(d + e*x**2) *sqrt(a - c*x**4)*a*b*e**3*x**3 - 63*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c *d**3*x + 63*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**3 + 36*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d*e**2*x**5 + 8*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*e**3*x**7 + 96*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 + 11 6*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**3*b*d**5*e**6 + 480*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**1...