Integrand size = 35, antiderivative size = 1311 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Output:
-1/5*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/d/e^3/(e*x^2+d )^(5/2)+1/15*(B*d*(13*c*d^2-e*(-a*e+7*b*d))-2*A*e*(4*c*d^2-e*(2*a*e+b*d))) *x*(c*x^4+b*x^2+a)^(1/2)/d^2/e^3/(e*x^2+d)^(3/2)+1/15*(A*e*(15*c^2*d^4-c*d ^2*e*(-11*a*e+10*b*d)-e^2*(-8*a^2*e^2+3*a*b*d*e+2*b^2*d^2))-B*d*(45*c^2*d^ 4-c*d^2*e*(-31*a*e+50*b*d)+e^2*(-2*a^2*e^2-3*a*b*d*e+8*b^2*d^2)))*x*(c*x^4 +b*x^2+a)^(1/2)/d^3/e^3/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)-1/30*(2*A*e*(1 5*c^2*d^4-c*d^2*e*(-11*a*e+10*b*d)-e^2*(-8*a^2*e^2+3*a*b*d*e+2*b^2*d^2))-B *d*(105*c^2*d^4-c*d^2*e*(-77*a*e+115*b*d)+2*e^2*(-2*a^2*e^2-3*a*b*d*e+8*b^ 2*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/d^3/e^4/(a*e^2-b*d*e+c*d^2) /x+1/60*(-4*a*c+b^2)^(1/2)*(2*A*e*(15*c^2*d^4-c*d^2*e*(-11*a*e+10*b*d)-e^2 *(-8*a^2*e^2+3*a*b*d*e+2*b^2*d^2))-B*d*(105*c^2*d^4-c*d^2*e*(-77*a*e+115*b *d)+2*e^2*(-2*a^2*e^2-3*a*b*d*e+8*b^2*d^2)))*(-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)/d^3/e^4/(a*e^2-b*d*e+c*d^2)/(-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)+1/30*(-4*a*c+b^2)^(1/ 2)*(B*d*(35*c*d^2-4*e*(a*e+2*b*d))-2*A*e*(5*c*d^2+e*(8*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...
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(7/2),x]
Output:
Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(7/2), 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.
\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 2260 |
\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}}dx\) |
Input:
Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(7/2),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]
Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),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)/(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2*e^2*x^4 + 4*d^3* e*x^2 + d^4), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="ma xima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="gi ac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(7/2),x)
Output:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x \] Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(7/2),x)