Integrand size = 31, antiderivative size = 868 \[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=-\frac {\left (10 A c d e \left (15 c d^2-68 a e^2\right )-B \left (105 c^2 d^4-332 a c d^2 e^2-384 a^2 e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3840 c e^4 x}-\frac {\left (35 B c d^3-50 A c d^2 e-108 a B d e^2-600 a A e^3\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{1920 e^3}+\frac {\left (7 B c d^2-10 A c d e+96 a B e^2\right ) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{480 e^2}-\frac {c (B d+10 A e) x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{80 e}-\frac {1}{10} B c x^7 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (10 A c d e \left (15 c d^2-68 a e^2\right )-B \left (105 c^2 d^4-332 a c d^2 e^2-384 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}} 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 )}{3840 \sqrt {c} e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (10 A c d e \left (5 c d^2+196 a e^2\right )-B \left (35 c^2 d^4-116 a c d^2 e^2-384 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 )}{3840 \sqrt {c} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (7 B c^2 d^5-10 A c^2 d^4 e-24 a B c d^3 e^2+48 a A c d^2 e^3+48 a^2 B d e^4+96 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}} \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^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/3840*(10*A*c*d*e*(-68*a*e^2+15*c*d^2)-B*(-384*a^2*e^4-332*a*c*d^2*e^2+1 05*c^2*d^4))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^4/x-1/1920*(-600*A*a*e^3 -50*A*c*d^2*e-108*B*a*d*e^2+35*B*c*d^3)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2) /e^3+1/480*(-10*A*c*d*e+96*B*a*e^2+7*B*c*d^2)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+ a)^(1/2)/e^2-1/80*c*(10*A*e+B*d)*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e-1/ 10*B*c*x^7*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/3840*(c^(1/2)*d+a^(1/2)*e)*( 10*A*c*d*e*(-68*a*e^2+15*c*d^2)-B*(-384*a^2*e^4-332*a*c*d^2*e^2+105*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))/c^(1/2)/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/3840 *a^(1/2)*(10*A*c*d*e*(196*a*e^2+5*c*d^2)-B*(-384*a^2*e^4-116*a*c*d^2*e^2+3 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^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2 )+1/256*(96*A*a^2*e^5+48*A*a*c*d^2*e^3-10*A*c^2*d^4*e+48*B*a^2*d*e^4-24*B* a*c*d^3*e^2+7*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)*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)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx \] Input:
Integrate[(A + B*x^2)*Sqrt[d + e*x^2]*(a - c*x^4)^(3/2),x]
Output:
Integrate[(A + B*x^2)*Sqrt[d + e*x^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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (A+B x^2\right ) \sqrt {d+e x^2} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (A+B x^2\right ) \sqrt {d+e x^2}dx\) |
Input:
Int[(A + B*x^2)*Sqrt[d + e*x^2]*(a - c*x^4)^(3/2),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]
\[\int \left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \left (-c \,x^{4}+a \right )^{\frac {3}{2}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),x)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),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), x)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (A + B x^{2}\right ) \left (a - c x^{4}\right )^{\frac {3}{2}} \sqrt {d + e x^{2}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(a - c*x**4)**(3/2)*sqrt(d + e*x**2), x)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)*sqrt(e*x^2 + d), x)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)*sqrt(e*x^2 + d), x)
Timed out. \[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}\,\sqrt {e\,x^2+d} \,d x \] Input:
int((A + B*x^2)*(a - c*x^4)^(3/2)*(d + e*x^2)^(1/2),x)
Output:
int((A + B*x^2)*(a - c*x^4)^(3/2)*(d + e*x^2)^(1/2), x)
\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(3/2),x)
Output:
(600*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**3*x + 108*sqrt(d + e*x**2)* sqrt(a - c*x**4)*a*b*d*e**2*x + 384*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b* e**3*x**3 + 50*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x - 40*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**3 - 240*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e**3*x**5 - 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**3*x + 28*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**3 - 24*sqrt(d + e*x**2 )*sqrt(a - c*x**4)*b*c*d*e**2*x**5 - 192*sqrt(d + e*x**2)*sqrt(a - c*x**4) *b*c*e**3*x**7 + 384*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*b*e**4 - 680*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*c*d* e**3 + 332*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*b*c*d**2*e**2 + 150*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**2*d**3*e - 105*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)*b*c**2*d**4 + 720*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**3*e**4 + 552*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*b*d*e**3 + 20*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*c*d**2*e**2 - 14*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)*...