Integrand size = 30, antiderivative size = 319 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {3 a b^2 x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}+\frac {b^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d f}+\frac {b \left (6 a^2 d f-b^2 (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {e+f x^2}}\right )}{2 d^{3/2} f^{3/2}}-\frac {3 a b^2 \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {a^3 \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \] Output:
3*a*b^2*x*(d*x^2+c)^(1/2)/d/(f*x^2+e)^(1/2)+1/2*b^3*(d*x^2+c)^(1/2)*(f*x^2 +e)^(1/2)/d/f+1/2*b*(6*a^2*d*f-b^2*(c*f+d*e))*arctanh(f^(1/2)*(d*x^2+c)^(1 /2)/d^(1/2)/(f*x^2+e)^(1/2))/d^(3/2)/f^(3/2)-3*a*b^2*e^(1/2)*(d*x^2+c)^(1/ 2)*EllipticE(f^(1/2)*x/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))/d/f^(1 /2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+a^3*e^(1/2)*(d*x^2+c)^ (1/2)*InverseJacobiAM(arctan(f^(1/2)*x/e^(1/2)),(1-d*e/c/f)^(1/2))/c/f^(1/ 2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 6.78 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b \sqrt {\frac {d}{c}} \left (b^2 \sqrt {d} \sqrt {f} \left (c+d x^2\right ) \left (e+f x^2\right )-\left (-6 a^2 d f+b^2 (d e+c f)\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x^2}}{\sqrt {f} \sqrt {c+d x^2}}\right )\right )-6 i a b^2 d^{3/2} e \sqrt {f} \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+2 i a d^{3/2} \sqrt {f} \left (3 b^2 e-a^2 f\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{2 d^{3/2} \sqrt {\frac {d}{c}} f^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:
Integrate[(a + b*x)^3/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
(b*Sqrt[d/c]*(b^2*Sqrt[d]*Sqrt[f]*(c + d*x^2)*(e + f*x^2) - (-6*a^2*d*f + b^2*(d*e + c*f))*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x^2])/(Sqrt[f]*Sqrt[c + d*x^2])]) - (6*I)*a*b^2*d^(3/2)*e*Sqrt[f]*Sqrt[ 1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f) /(d*e)] + (2*I)*a*d^(3/2)*Sqrt[f]*(3*b^2*e - a^2*f)*Sqrt[1 + (d*x^2)/c]*Sq rt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(2*d^(3/ 2)*Sqrt[d/c]*f^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.99 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {3 a^2 b x}{\sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {3 a b^2 x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^3 x^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {3 a^2 b \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {e+f x^2}}\right )}{\sqrt {d} \sqrt {f}}-\frac {3 a b^2 \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {3 a b^2 x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {b^3 (c f+d e) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {e+f x^2}}\right )}{2 d^{3/2} f^{3/2}}+\frac {b^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d f}\) |
Input:
Int[(a + b*x)^3/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
(3*a*b^2*x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) + (b^3*Sqrt[c + d*x^2]*Sqr t[e + f*x^2])/(2*d*f) + (3*a^2*b*ArcTanh[(Sqrt[f]*Sqrt[c + d*x^2])/(Sqrt[d ]*Sqrt[e + f*x^2])])/(Sqrt[d]*Sqrt[f]) - (b^3*(d*e + c*f)*ArcTanh[(Sqrt[f] *Sqrt[c + d*x^2])/(Sqrt[d]*Sqrt[e + f*x^2])])/(2*d^(3/2)*f^(3/2)) - (3*a*b ^2*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e )/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2] ) + (a^3*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
Time = 8.47 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {b^{3} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{2 d f}+\frac {\left (\frac {b \left (6 a^{2} d f -b^{2} c f -d e \,b^{2}\right ) \ln \left (\frac {\frac {1}{2} c f +\frac {1}{2} d e +d f \,x^{2}}{\sqrt {d f}}+\sqrt {d f \,x^{4}+\left (c f +d e \right ) x^{2}+c e}\right )}{2 \sqrt {d f}}+\frac {2 a^{3} d f \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {6 a \,b^{2} d e \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{2 d f \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(368\) |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{2 d f}+\frac {a^{3} \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (3 a^{2} b -\frac {\left (c f +d e \right ) b^{3}}{2 d f}\right ) \ln \left (\frac {2 d f \,x^{2}+c f +d e}{\sqrt {d f}}+2 \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\right )}{2 \sqrt {d f}}-\frac {3 a \,b^{2} e \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(369\) |
| default | \(\frac {\left (2 \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, b^{3} d f \,x^{4}+4 \sqrt {d f}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{3} d f -12 \sqrt {d f}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d e +12 \sqrt {d f}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d e +2 \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, b^{3} c f \,x^{2}+2 \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, b^{3} d e \,x^{2}+6 \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \ln \left (\frac {2 d f \,x^{2}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \sqrt {d f}+c f +d e}{\sqrt {d f}}\right ) \sqrt {-\frac {d}{c}}\, a^{2} b d f -\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \ln \left (\frac {2 d f \,x^{2}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \sqrt {d f}+c f +d e}{\sqrt {d f}}\right ) \sqrt {-\frac {d}{c}}\, b^{3} c f -\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \ln \left (\frac {2 d f \,x^{2}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \sqrt {d f}+c f +d e}{\sqrt {d f}}\right ) \sqrt {-\frac {d}{c}}\, b^{3} d e +2 \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, b^{3} c e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{4 \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, f d \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(568\) |
Input:
int((b*x+a)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*b^3*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/f+1/2/d/f*(1/2*b*(6*a^2*d*f-b^2* c*f-b^2*d*e)*ln((1/2*c*f+1/2*d*e+d*f*x^2)/(d*f)^(1/2)+(d*f*x^4+(c*f+d*e)*x ^2+c*e)^(1/2))/(d*f)^(1/2)+2*a^3*d*f/(-d/c)^(1/2)*(1+1/c*x^2*d)^(1/2)*(1+f *x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2) ,(-1+(c*f+d*e)/e/d)^(1/2))-6*a*b^2*d*e/(-d/c)^(1/2)*(1+1/c*x^2*d)^(1/2)*(1 +f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*(EllipticF(x*(-d/c)^(1 /2),(-1+(c*f+d*e)/e/d)^(1/2))-EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^ (1/2))))*((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)
Time = 0.50 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {24 \, \sqrt {d f} a b^{2} d e^{2} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - 8 \, {\left (3 \, a b^{2} d e^{2} + a^{3} d f^{2}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) + {\left (b^{3} d e^{2} + {\left (b^{3} c - 6 \, a^{2} b d\right )} e f\right )} \sqrt {d f} x \log \left (-8 \, d^{2} f^{2} x^{4} - d^{2} e^{2} - 6 \, c d e f - c^{2} f^{2} - 8 \, {\left (d^{2} e f + c d f^{2}\right )} x^{2} - 4 \, {\left (2 \, d f x^{2} + d e + c f\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} \sqrt {d f}\right ) - 4 \, {\left (b^{3} d e f x + 6 \, a b^{2} d e f\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{8 \, d^{2} e f^{2} x} \] Input:
integrate((b*x+a)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="fricas")
Output:
-1/8*(24*sqrt(d*f)*a*b^2*d*e^2*x*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x ), c*f/(d*e)) - 8*(3*a*b^2*d*e^2 + a^3*d*f^2)*sqrt(d*f)*x*sqrt(-e/f)*ellip tic_f(arcsin(sqrt(-e/f)/x), c*f/(d*e)) + (b^3*d*e^2 + (b^3*c - 6*a^2*b*d)* e*f)*sqrt(d*f)*x*log(-8*d^2*f^2*x^4 - d^2*e^2 - 6*c*d*e*f - c^2*f^2 - 8*(d ^2*e*f + c*d*f^2)*x^2 - 4*(2*d*f*x^2 + d*e + c*f)*sqrt(d*x^2 + c)*sqrt(f*x ^2 + e)*sqrt(d*f)) - 4*(b^3*d*e*f*x + 6*a*b^2*d*e*f)*sqrt(d*x^2 + c)*sqrt( f*x^2 + e))/(d^2*e*f^2*x)
\[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a + b x\right )^{3}}{\sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:
integrate((b*x+a)**3/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
Output:
Integral((a + b*x)**3/(sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
\[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x+a)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x + a)^3/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
\[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x+a)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="giac")
Output:
integrate((b*x + a)^3/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (a+b\,x\right )}^3}{\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int((a + b*x)^3/((c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
Output:
int((a + b*x)^3/((c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {(a+b x)^3}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, b^{3} d f +\sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f -\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) b^{3} c f +\sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f -\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) b^{3} d e -6 \sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f +\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) a^{2} b d f +2 \sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f +\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) b^{3} c f +2 \sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f +\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) b^{3} d e +6 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, x^{2}}{d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}d x \right ) a \,b^{2} d^{2} f^{2}+2 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}d x \right ) a^{3} d^{2} f^{2}}{2 d^{2} f^{2}} \] Input:
int((b*x+a)^3/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
Output:
(sqrt(e + f*x**2)*sqrt(c + d*x**2)*b**3*d*f + sqrt(f)*sqrt(d)*log( - sqrt( d)*sqrt(c + d*x**2)*f - sqrt(f)*sqrt(e + f*x**2)*d)*b**3*c*f + sqrt(f)*sqr t(d)*log( - sqrt(d)*sqrt(c + d*x**2)*f - sqrt(f)*sqrt(e + f*x**2)*d)*b**3* d*e - 6*sqrt(f)*sqrt(d)*log( - sqrt(d)*sqrt(c + d*x**2)*f + sqrt(f)*sqrt(e + f*x**2)*d)*a**2*b*d*f + 2*sqrt(f)*sqrt(d)*log( - sqrt(d)*sqrt(c + d*x** 2)*f + sqrt(f)*sqrt(e + f*x**2)*d)*b**3*c*f + 2*sqrt(f)*sqrt(d)*log( - sqr t(d)*sqrt(c + d*x**2)*f + sqrt(f)*sqrt(e + f*x**2)*d)*b**3*d*e + 6*int((sq rt(e + f*x**2)*sqrt(c + d*x**2)*x**2)/(c*e + c*f*x**2 + d*e*x**2 + d*f*x** 4),x)*a*b**2*d**2*f**2 + 2*int((sqrt(e + f*x**2)*sqrt(c + d*x**2))/(c*e + c*f*x**2 + d*e*x**2 + d*f*x**4),x)*a**3*d**2*f**2)/(2*d**2*f**2)