Integrand size = 27, antiderivative size = 604 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}+\frac {\sqrt {c} (2 c d-3 a f) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d f}-\frac {\left (4 a c d^2 f^2+c^2 d^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+a^2 f^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (4 a c d^2 f^2+a^2 f^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+c^2 d^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]
-(c*x^2+a)^(3/2)/d/x+a^(3/2)*e*arctanh((c*x^2+a)^(1/2)/a^(1/2))/d^2+3/2*a* arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))*c^(1/2)/d+1/2*(-3*a*f+2*c*d)*arctanh(x* c^(1/2)/(c*x^2+a)^(1/2))*c^(1/2)/d/f-a*e*(c*x^2+a)^(1/2)/d^2+3/2*c*x*(c*x^ 2+a)^(1/2)/d+1/2*(-c*d*x+2*a*e)*(c*x^2+a)^(1/2)/d^2-1/2*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))*(4*a*c*d^2*f^2+c^2*d^2*(e^2-2*d*f-e*(-4*d*f +e^2)^(1/2))+a^2*f^2*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))/d^2/f*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*arcta nh(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))*(4*a*c*d^2*f^2+a^2*f^2*(e^2-2* d*f-e*(-4*d*f+e^2)^(1/2))+c^2*d^2*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))/d^2/f* 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 = 0.60 (sec) , antiderivative size = 497, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {2 a^{3/2} e f x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )+d \left (a f \sqrt {a+c x^2}+c^{3/2} d x \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )\right )+x \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^3 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^2 \sqrt {c} e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{d^2 f x} \]
-((2*a^(3/2)*e*f*x*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]] + d*(a*f *Sqrt[a + c*x^2] + c^(3/2)*d*x*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]]) + x*Ro otSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^ 3 + f*#1^4 & , (-(a*c^2*d^2*e*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]) + a^3*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] - 2*c^(5/2)*d^3*Log[-(S qrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 4*a*c^(3/2)*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 2*a^2*Sqrt[c]*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 2*a^2*Sqrt[c]*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2 ] - #1]*#1 + c^2*d^2*e*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - a^2 *e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d *#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(d^2*f*x))
Time = 2.63 (sec) , antiderivative size = 604, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (a+c x^2\right )^{3/2} \left (-d f+e^2+e f x\right )}{d^2 \left (d+e x+f x^2\right )}-\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {\left (a+c x^2\right )^{3/2}}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\left (a^2 f^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (a^2 f^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (2 c d-3 a f)}{2 d f}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}-\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {\sqrt {a+c x^2} (2 a e-c d x)}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 c x \sqrt {a+c x^2}}{2 d}\) |
-((a*e*Sqrt[a + c*x^2])/d^2) + (3*c*x*Sqrt[a + c*x^2])/(2*d) + ((2*a*e - c *d*x)*Sqrt[a + c*x^2])/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*Sqrt[c]*Ar cTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d) + (Sqrt[c]*(2*c*d - 3*a*f)*ArcTa nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + a^2*f^2*(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*f*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])]) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(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*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2* d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]] )/d^2
3.1.62.3.1 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]
Time = 0.78 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(-\frac {a \sqrt {c \,x^{2}+a}}{d x}+\frac {\frac {c^{\frac {3}{2}} d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{f}-\frac {4 f \,a^{\frac {3}{2}} 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 (2 a^{2} f^{3} \sqrt {-4 d f +e^{2}}-4 a c d \,f^{2} \sqrt {-4 d f +e^{2}}+2 c^{2} d^{2} f \sqrt {-4 d f +e^{2}}-2 \sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}-2 a^{2} e \,f^{3}-4 a c d e \,f^{2}+6 c^{2} d^{2} e f -2 c^{2} d \,e^{3}\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 )}{2 f^{2} \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 (-2 a^{2} f^{3} \sqrt {-4 d f +e^{2}}+4 a c d \,f^{2} \sqrt {-4 d f +e^{2}}-2 c^{2} d^{2} f \sqrt {-4 d f +e^{2}}+2 \sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}-2 a^{2} e \,f^{3}-4 a c d e \,f^{2}+6 c^{2} d^{2} e f -2 c^{2} d \,e^{3}\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 )}{2 f^{2} \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}\) | \(957\) |
| default | \(\text {Expression too large to display}\) | \(2494\) |
-a/d*(c*x^2+a)^(1/2)/x+1/d*(c^(3/2)*d/f*ln(x*c^(1/2)+(c*x^2+a)^(1/2))-4*f* a^(3/2)*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)-1/2*(2*a^2*f^3*(-4*d*f+e^2)^(1/2)-4*a*c*d*f^2*(-4*d*f +e^2)^(1/2)+2*c^2*d^2*f*(-4*d*f+e^2)^(1/2)-2*(-4*d*f+e^2)^(1/2)*c^2*d*e^2- 2*a^2*e*f^3-4*a*c*d*e*f^2+6*c^2*d^2*e*f-2*c^2*d*e^3)/f^2/(-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))-1/2*(-2*a^2*f^3*(-4*d*f+e^2)^(1/2)+4*a*c*d* f^2*(-4*d*f+e^2)^(1/2)-2*c^2*d^2*f*(-4*d*f+e^2)^(1/2)+2*(-4*d*f+e^2)^(1/2) *c^2*d*e^2-2*a^2*e*f^3-4*a*c*d*e*f^2+6*c^2*d^2*e*f-2*c^2*d*e^3)/f^2/(-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+...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )} x^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \]