Integrand size = 27, antiderivative size = 382 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]
e*arctanh((c*x^2+a)^(1/2)/a^(1/2))*a^(1/2)/d^2-(c*x^2+a)^(1/2)/d/x-1/2*f*a rctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a *f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2))*(2*c*d^2+a*(e^2-2*d*f+e*(- 4*d*f+e^2)^(1/2)))/d^2*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e* (-4*d*f+e^2)^(1/2)))^(1/2)+1/2*f*arctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e^2)^(1 /2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))) ^(1/2))*(2*c*d^2+a*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))/d^2*2^(1/2)/(-4*d*f+e ^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {d \sqrt {a+c x^2}-\sqrt {a} e x \log (x)+\sqrt {a} e x \log \left (-\sqrt {a}+\sqrt {a+c x^2}\right )+x \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c^2 d^2 \log (x)-a c e^2 \log (x)+a c d f \log (x)+c^2 d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+a c e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-a c d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-2 a^{3/2} e f \log (x) \text {$\#$1}+2 a^{3/2} e f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c d^2 \log (x) \text {$\#$1}^2+a e^2 \log (x) \text {$\#$1}^2-a d f \log (x) \text {$\#$1}^2-c d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{d^2 x} \]
-((d*Sqrt[a + c*x^2] - Sqrt[a]*e*x*Log[x] + Sqrt[a]*e*x*Log[-Sqrt[a] + Sqr t[a + c*x^2]] + x*RootSum[c^2*d + 2*Sqrt[a]*c*e*#1 - 2*c*d*#1^2 + 4*a*f*#1 ^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(c^2*d^2*Log[x]) - a*c*e^2*Log[x] + a *c*d*f*Log[x] + c^2*d^2*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] + a*c*e^2*L og[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] - a*c*d*f*Log[-Sqrt[a] + Sqrt[a + c* x^2] - x*#1] - 2*a^(3/2)*e*f*Log[x]*#1 + 2*a^(3/2)*e*f*Log[-Sqrt[a] + Sqrt [a + c*x^2] - x*#1]*#1 + c*d^2*Log[x]*#1^2 + a*e^2*Log[x]*#1^2 - a*d*f*Log [x]*#1^2 - c*d^2*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1]*#1^2 - a*e^2*Log[- Sqrt[a] + Sqrt[a + c*x^2] - x*#1]*#1^2 + a*d*f*Log[-Sqrt[a] + Sqrt[a + c*x ^2] - x*#1]*#1^2)/(-(Sqrt[a]*c*e) + 2*c*d*#1 - 4*a*f*#1 + 3*Sqrt[a]*e*#1^2 - 2*d*#1^3) & ])/(d^2*x))
Time = 1.45 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7279, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {a+c x^2} \left (-d f+e^2+e f x\right )}{d^2 \left (d+e x+f x^2\right )}-\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\sqrt {a+c x^2}}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f \left (a \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {f \left (a \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\sqrt {a+c x^2}}{d x}\) |
-(Sqrt[a + c*x^2]/(d*x)) - (f*(2*c*d^2 + a*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d *f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*S qrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*c*d^2 + a*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e ^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f ^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2
3.1.56.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(337)=674\).
Time = 0.76 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.03
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}}{d x}-\frac {\frac {4 f \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{\left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )}-\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}+a e f +c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (-e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}+\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}-a e f -c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}}{d}\) | \(776\) |
| default | \(\text {Expression too large to display}\) | \(1415\) |
-(c*x^2+a)^(1/2)/d/x-1/d*(4*f*a^(1/2)*e/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f +e^2)^(1/2))*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-(f*a*(-4*d*f+e^2)^(1/2) -c*d*(-4*d*f+e^2)^(1/2)+a*e*f+c*d*e)/(-4*d*f+e^2)^(1/2)/(-e+(-4*d*f+e^2)^( 1/2))*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)* ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^ (1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/ 2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2 )))^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*( -(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(- 4*d*f+e^2)^(1/2))))+(f*a*(-4*d*f+e^2)^(1/2)-c*d*(-4*d*f+e^2)^(1/2)-a*e*f-c *d*e)/(-4*d*f+e^2)^(1/2)/(e+(-4*d*f+e^2)^(1/2))*2^(1/2)/(((-4*d*f+e^2)^(1/ 2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f ^2-2*c*d*f+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1 /2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^( 1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f* (x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f +c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)))
Leaf count of result is larger than twice the leaf count of optimal. 2656 vs. \(2 (335) = 670\).
Time = 176.36 (sec) , antiderivative size = 5324, normalized size of antiderivative = 13.94 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {a + c x^{2}}}{x^{2} \left (d + e x + f x^{2}\right )}\, dx \]
\[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (f x^{2} + e x + d\right )} x^{2}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+a}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \]