Integrand size = 24, antiderivative size = 716 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=-\frac {\left (5 c d^2-32 a e^2\right ) \left (3 c d^2-4 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{1280 c e^3 x}+\frac {1}{640} d \left (236 a+\frac {5 c d^2}{e^2}\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (c d^2-32 a e^2\right ) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{160 e}-\frac {11}{80} c d x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {1}{10} c e x^7 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (5 c d^2-32 a e^2\right ) \left (3 c d^2-4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{1280 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (5 c^2 d^4+692 a c d^2 e^2+128 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{1280 \sqrt {c} e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {3 d \left (c d^2-12 a e^2\right ) \left (c d^2+4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{256 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/1280*(-32*a*e^2+5*c*d^2)*(-4*a*e^2+3*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^ (1/2)/c/e^3/x+1/640*d*(236*a+5*c*d^2/e^2)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/ 2)-1/160*(-32*a*e^2+c*d^2)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e-11/80*c* d*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/10*c*e*x^7*(e*x^2+d)^(1/2)*(-c*x^ 4+a)^(1/2)-1/1280*(d+a^(1/2)*e/c^(1/2))*(-32*a*e^2+5*c*d^2)*(-4*a*e^2+3*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)*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))/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/1280*a^(1/2 )*(128*a^2*e^4+692*a*c*d^2*e^2+5*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))/c^(1/2)/e^2/( e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-3/256*d*(-12*a*e^2+c*d^2)*(4*a*e^2+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^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx \] Input:
Integrate[(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2),x]
Output:
Integrate[(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2), 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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^{3/2}dx\) |
Input:
Int[(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2),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]
\[\int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (-c \,x^{4}+a \right )^{\frac {3}{2}}d x\]
Input:
int((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
Output:
int((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(-(c*e*x^6 + c*d*x^4 - a*e*x^2 - a*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d), x)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (a - c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x**2+d)**(3/2)*(-c*x**4+a)**(3/2),x)
Output:
Integral((a - c*x**4)**(3/2)*(d + e*x**2)**(3/2), x)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)*(e*x^2 + d)^(3/2), x)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int {\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:
int((a - c*x^4)^(3/2)*(d + e*x^2)^(3/2),x)
Output:
int((a - c*x^4)^(3/2)*(d + e*x^2)^(3/2), x)
\[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\frac {236 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a d \,e^{2} x +128 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a \,e^{3} x^{3}+5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c \,d^{3} x -4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c \,d^{2} e \,x^{3}-88 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c d \,e^{2} x^{5}-64 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c \,e^{3} x^{7}+128 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{4}-116 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{2} e^{2}+15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} d^{4}+424 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} d \,e^{3}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{3} e +404 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} d^{2} e^{2}-5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{4}}{640 e^{2}} \] Input:
int((e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
Output:
(236*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*d*e**2*x + 128*sqrt(d + e*x**2)*s qrt(a - c*x**4)*a*e**3*x**3 + 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d**3*x - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d**2*e*x**3 - 88*sqrt(d + e*x**2) *sqrt(a - c*x**4)*c*d*e**2*x**5 - 64*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*e **3*x**7 + 128*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x** 2 - c*d*x**4 - c*e*x**6),x)*a**2*e**4 - 116*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d**2*e**2 + 1 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c**2*d**4 + 424*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2 )/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*d*e**3 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a *c*d**3*e + 404*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*d**2*e**2 - 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d**4)/(640*e**2)