Integrand size = 28, antiderivative size = 266 \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}+\frac {\sqrt {d} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \]
-arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/f/c^(1/2)+1/2*arctanh( 1/2*(b*d^(1/2)-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/ (c*d+a*f-b*d^(1/2)*f^(1/2))^(1/2))*d^(1/2)/f/(c*d+a*f-b*d^(1/2)*f^(1/2))^( 1/2)+1/2*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c* x^2+b*x+a)^(1/2)/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))*d^(1/2)/f/(c*d+a*f+b*d ^(1/2)*f^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\frac {\frac {2 \log \left (f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {c}}-d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 f} \]
((2*Log[f*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] - d*Root Sum[b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*Sqrt[c]*Log[-(Sqrt [c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/(2*f)
Time = 0.50 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2144, 25, 27, 1092, 219, 1316, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {x^2}{\left (d-f x^2\right ) \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle -\frac {\int -\frac {d}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}\) |
| \(\Big \downarrow \) 1316 |
| \(\displaystyle \frac {d \left (\frac {1}{2} \int \frac {1}{\sqrt {d} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \int \frac {1}{\sqrt {d} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {d \left (-\frac {\int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}-\frac {\int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {d} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {\text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {d} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} f}\) |
-(ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(Sqrt[c]*f)) + (d *(ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[ c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqr t[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/f
3.1.96.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[1/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sy mbol] :> Simp[1/2 Int[1/((a - Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x] , x] + Simp[1/2 Int[1/((a + Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.80 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f \sqrt {c}}-\frac {d \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 \sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}+\frac {d \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 \sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}\) | \(399\) |
-1/f*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-1/2*d/(d*f)^(1/2) /f/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1 /f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d ))^(1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/ f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f))+1/2*d/(d*f)^(1/ 2)/f/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2* c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2) *((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f) ^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f))
Timed out. \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=- \int \frac {x^{2}}{- d \sqrt {a + b x + c x^{2}} + f x^{2} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se e `assume?
Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {x^2}{\left (d-f\,x^2\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]