Integrand size = 24, antiderivative size = 714 \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {\left (a-\frac {c d^2}{e^2}\right ) x \sqrt {a-c x^4}}{9 d \left (d+e x^2\right )^{9/2}}+\frac {4 \left (\frac {2 a}{d^2}+\frac {3 c}{e^2}\right ) x \sqrt {a-c x^4}}{63 \left (d+e x^2\right )^{7/2}}-\frac {\left (5 c^2 d^4-17 a c d^2 e^2+16 a^2 e^4\right ) x \sqrt {a-c x^4}}{105 d^3 e^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{5/2}}-\frac {2 \left (5 c^3 d^6-22 a c^2 d^4 e^2+65 a^2 c d^2 e^4-32 a^3 e^6\right ) x \sqrt {a-c x^4}}{315 d^4 e^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}+\frac {4 a^2 e \left (93 c^2 d^4-93 a c d^2 e^2+32 a^2 e^4\right ) \sqrt {a-c x^4}}{315 d^4 \left (c d^2-a e^2\right )^3 x \sqrt {d+e x^2}}+\frac {4 a^2 \sqrt {c} e \left (93 c^2 d^4-93 a c d^2 e^2+32 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 d^5 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {4 a^{3/2} \sqrt {c} \left (45 c^2 d^4-69 a c d^2 e^2+32 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 d^5 \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/9*(a-c*d^2/e^2)*x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(9/2)+4/63*(2*a/d^2+3*c/e ^2)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2)-1/105*(16*a^2*e^4-17*a*c*d^2*e^2+5* c^2*d^4)*x*(-c*x^4+a)^(1/2)/d^3/e^2/(-a*e^2+c*d^2)/(e*x^2+d)^(5/2)-2/315*( -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)*x*(-c*x^4+a)^(1/2 )/d^4/e^2/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2)+4/315*a^2*e*(32*a^2*e^4-93*a*c* d^2*e^2+93*c^2*d^4)*(-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)*e*(32*a^2*e^4-93*a*c*d^2*e^2+93*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)*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)*(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)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx \] Input:
Integrate[(a - c*x^4)^(3/2)/(d + e*x^2)^(11/2),x]
Output:
Integrate[(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.
\(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}}dx\) |
Input:
Int[(a - c*x^4)^(3/2)/(d + e*x^2)^(11/2),x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
\[\int \frac {\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {11}{2}}}d x\]
Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
Output:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="fricas")
Output:
integral((-c*x^4 + a)^(3/2)*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. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((-c*x**4+a)**(3/2)/(e*x**2+d)**(11/2),x)
Output:
Timed out
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(11/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)/(e*x^2 + d)^(11/2), x)
Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{11/2}} \,d x \] Input:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(11/2),x)
Output:
int((a - c*x^4)^(3/2)/(d + e*x^2)^(11/2), x)
\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {too large to display} \] Input:
int((-c*x^4+a)^(3/2)/(e*x^2+d)^(11/2),x)
Output:
(3*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*x + 8*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e*x**3 - 9*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*d*x**5 - 2* sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*e*x**7 - 96*int((sqrt(d + e*x**2)*s qrt(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**3*c*d**5*e**4 - 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 - 4 2*c**2*d**3*e**5*x**14 - 7*c**2*d**2*e**6*x**16),x)*a**3*c*d**4*e**5*x**2 - 960*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(4*a**2*d**6*e**2 + ...