Integrand size = 34, antiderivative size = 789 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}+\frac {2 (a d (2 d e-5 c f)+b c (d e+2 c f)) x \sqrt {a+b x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {\left (2 b^2 c^2 \left (d^2 e^2-9 c d e f-4 c^2 f^2\right )-a^2 d^2 \left (8 d^2 e^2-29 c d e f+45 c^2 f^2\right )+a b c d \left (3 d^2 e^2-5 c d e f+50 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{15 c^3 (b c-a d) (d e-c f)^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {-b e+a f} \left (2 b^2 c^2 e \left (d^2 e^2-10 c d e f-15 c^2 f^2\right )-a^2 d \left (8 d^3 e^3-33 c d^2 e^2 f+58 c^2 d e f^2+15 c^3 f^3\right )+a b c \left (3 d^3 e^3-7 c d^2 e^2 f+85 c^2 d e f^2+15 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 \sqrt {a} c^2 (b c-a d) e (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {2 \sqrt {-b e+a f} \left (b c \left (d^2 e^2-10 c d e f-15 c^2 f^2\right )+2 a d \left (d^2 e^2-4 c d e f+15 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 \sqrt {a} c^2 (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
-1/5*(-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(5/2)/(f*x^2+e)^( 1/2)+2/15*(a*d*(-5*c*f+2*d*e)+b*c*(2*c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-c*f +d*e)^2/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2)+1/15*(2*b^2*c^2*(-4*c^2*f^2-9*c*d* e*f+d^2*e^2)-a^2*d^2*(45*c^2*f^2-29*c*d*e*f+8*d^2*e^2)+a*b*c*d*(50*c^2*f^2 -5*c*d*e*f+3*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^3/(-a*d+b*c)/(-c*f+d*e)^3/(d*x^ 2+c)^(1/2)/(f*x^2+e)^(1/2)+1/15*(a*f-b*e)^(1/2)*(2*b^2*c^2*e*(-15*c^2*f^2- 10*c*d*e*f+d^2*e^2)-a^2*d*(15*c^3*f^3+58*c^2*d*e*f^2-33*c*d^2*e^2*f+8*d^3* e^3)+a*b*c*(15*c^3*f^3+85*c^2*d*e*f^2-7*c*d^2*e^2*f+3*d^3*e^3))*(b*x^2+a)^ (1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticE((a*f-b*e)^(1/2)*x/a^(1/2)/ (f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/c^2/(-a*d+b*c)/ e/(-c*f+d*e)^4/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-2/15*(a*f-b *e)^(1/2)*(b*c*(-15*c^2*f^2-10*c*d*e*f+d^2*e^2)+2*a*d*(15*c^2*f^2-4*c*d*e* f+d^2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f -b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a ^(1/2)/c^2/(-c*f+d*e)^4/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:
Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)),x]
Output:
Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}}dx\) |
Input:
Int[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)),x]
Output:
$Aborted
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2)^(r_.), x_Symbol] :> Unintegrable[(a + b*x^2)^p*(c + d*x^2)^q*(e + f* x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {7}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]
Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="fr icas")
Output:
integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^4*f^2*x^12 + 2*(d^4*e*f + 2*c*d^3*f^2)*x^10 + (d^4*e^2 + 8*c*d^3*e*f + 6*c^2*d^2*f^2)* x^8 + 4*(c*d^3*e^2 + 3*c^2*d^2*e*f + c^3*d*f^2)*x^6 + c^4*e^2 + (6*c^2*d^2 *e^2 + 8*c^3*d*e*f + c^4*f^2)*x^4 + 2*(2*c^3*d*e^2 + c^4*e*f)*x^2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(7/2)/(f*x**2+e)**(3/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)^(3/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{7/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)),x)
Output:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x \] Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(3/2),x)