Integrand size = 35, antiderivative size = 1243 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx =\text {Too large to display} \] Output:
1/384*(8*A*c*e*(32*a*c*e^2+3*b^2*e^2-22*b*c*d*e+15*c^2*d^2)-B*(105*c^3*d^3 +9*b^3*e^3-c^2*d*e*(-188*a*e+145*b*d)+15*b*c*e^2*(-4*a*e+b*d)))*(e*x^2+d)^ (1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e^4/x-1/192*(8*A*c*e*(-7*b*e+5*c*d)-B*(60* a*c*e^2+3*b^2*e^2-46*b*c*d*e+35*c^2*d^2))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a )^(1/2)/c/e^3-1/48*(-8*A*c*e-9*B*b*e+7*B*c*d)*x^3*(e*x^2+d)^(1/2)*(c*x^4+b *x^2+a)^(1/2)/e^2+1/8*B*c*x^5*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/e-1/76 8*(-4*a*c+b^2)^(1/2)*(8*A*c*e*(32*a*c*e^2+3*b^2*e^2-22*b*c*d*e+15*c^2*d^2) -B*(105*c^3*d^3+9*b^3*e^3-c^2*d*e*(-188*a*e+145*b*d)+15*b*c*e^2*(-4*a*e+b* d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*EllipticE(1 /2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^ (1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/c^2/e^4/(-a*(e+d/ x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/384*( -4*a*c+b^2)^(1/2)*(B*(35*c^3*d^3+9*b^3*e^3-c^2*d*e*(-68*a*e+53*b*d)+3*b*c* e^2*(-20*a*e+3*b*d))-8*A*c*e*(5*c^2*d^2+3*b^2*e^2-8*c*e*(2*a*e+b*d)))*(-a* (c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))* d-2*a*e))^(1/2)*x^3*EllipticF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2) *2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^( 1/2))*2^(1/2)/c^2/e^3/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/64*(-4*a*c+b ^2)^(1/2)*(8*A*c*e*(-b*e+c*d)*(5*c^2*d^2-b^2*e^2-4*c*e*(-3*a*e+b*d))-B*(35 *c^4*d^4+3*b^4*e^4+4*b^2*c*e^3*(-6*a*e+b*d)-12*c^3*d^2*e*(-6*a*e+5*b*d)...
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/Sqrt[d + e*x^2],x]
Output:
Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/Sqrt[d + e*x^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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 2260 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}}dx\) |
Input:
Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/Sqrt[d + e*x^2],x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\sqrt {e \,x^{2}+d}}d x\]
Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x)
Output:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="fr icas")
Output:
integral((B*c*x^6 + (B*b + A*c)*x^4 + (B*a + A*b)*x^2 + A*a)*sqrt(c*x^4 + b*x^2 + a)/sqrt(e*x^2 + d), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{\sqrt {d + e x^{2}}}\, dx \] Input:
integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d)**(1/2),x)
Output:
Integral((A + B*x**2)*(a + b*x**2 + c*x**4)**(3/2)/sqrt(d + e*x**2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="ma xima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/sqrt(e*x^2 + d), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="gi ac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/sqrt(e*x^2 + d), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{\sqrt {e\,x^2+d}} \,d x \] Input:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(1/2),x)
Output:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(1/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(1/2),x)
Output:
(116*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*e**2*x - 40*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c**2*d*e*x + 32*sqrt(d + e*x**2)*sqrt (a + b*x**2 + c*x**4)*a*c**2*e**2*x**3 + 3*sqrt(d + e*x**2)*sqrt(a + b*x** 2 + c*x**4)*b**3*e**2*x - 46*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b* *2*c*d*e*x + 36*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c*e**2*x** 3 + 35*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d**2*x - 28*sqrt( d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d*e*x**3 + 24*sqrt(d + e*x**2 )*sqrt(a + b*x**2 + c*x**4)*b*c**2*e**2*x**5 + 256*int((sqrt(d + e*x**2)*s qrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d *x**4 + c*e*x**6),x)*a**2*c**2*e**3 + 84*int((sqrt(d + e*x**2)*sqrt(a + b* x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c* e*x**6),x)*a*b**2*c*e**3 - 364*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x **4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x) *a*b*c**2*d*e**2 + 120*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x** 4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*c**3* d**2*e - 9*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a* e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b**4*e**3 - 15*int( (sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x* *2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b**3*c*d*e**2 + 145*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*...