Integrand size = 28, antiderivative size = 287 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\sqrt {a+b x+c x^2}}{c f}+\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} f}-\frac {d \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {d \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \] Output:
-(c*x^2+b*x+a)^(1/2)/c/f+1/2*b*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a) ^(1/2))/c^(3/2)/f-1/2*d*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+(2*c*d^(1/2)-b* f^(1/2))*x)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/f^(3/2) /(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)+1/2*d*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2 )+(2*c*d^(1/2)+b*f^(1/2))*x)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+ a)^(1/2))/f^(3/2)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\frac {2 \sqrt {a+x (b+c x)}}{c}+\frac {b \log \left (c f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{3/2}}+d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {a \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-\log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \sqrt {c} d+2 c d \text {$\#$1}+a f \text {$\#$1}-f \text {$\#$1}^3}\&\right ]}{2 f} \] Input:
Integrate[x^3/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]
Output:
-1/2*((2*Sqrt[a + x*(b + c*x)])/c + (b*Log[c*f*(b + 2*c*x - 2*Sqrt[c]*Sqrt [a + x*(b + c*x)])])/c^(3/2) + d*RootSum[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 & , (a*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(-(b *Sqrt[c]*d) + 2*c*d*#1 + a*f*#1 - f*#1^3) & ])/f
Time = 0.81 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 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.
\(\displaystyle \int \frac {x^3}{\left (d-f x^2\right ) \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {d x}{f \left (d-f x^2\right ) \sqrt {a+b x+c x^2}}-\frac {x}{f \sqrt {a+b x+c x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} f}-\frac {d \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {d \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{c f}\) |
Input:
Int[x^3/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]
Output:
-(Sqrt[a + b*x + c*x^2]/(c*f)) + (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)*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*f^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (d*ArcT anh[(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*f^(3/2)*Sqrt[c*d + b* Sqrt[d]*Sqrt[f] + a*f])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.51 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}}{f}+\frac {d \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+\frac {d \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\) | \(409\) |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}}{c f}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}} f}+\frac {d \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+\frac {d \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\) | \(410\) |
Input:
int(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-1/f*(1/c*(c*x^2+b*x+a)^(1/2)-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+ b*x+a)^(1/2)))+1/2*d/f^2/((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*ln((2*(b*(d*f)^ (1/2)+a*f+c*d)/f+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/ 2)+a*f+c*d)/f)^(1/2)*(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d* f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f))+1/2*d/f^2 /(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+a*f+c*d)+1/f *(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+a*f+c*d)) ^(1/2)*(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f) +1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2))/(x+(d*f)^(1/2)/f))
Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=- \int \frac {x^{3}}{- d \sqrt {a + b x + c x^{2}} + f x^{2} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(x**3/(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)
Output:
-Integral(x**3/(-d*sqrt(a + b*x + c*x**2) + f*x**2*sqrt(a + b*x + c*x**2)) , x)
Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="maxima")
Output:
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. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {x^3}{\left (d-f\,x^2\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(x^3/((d - f*x^2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(x^3/((d - f*x^2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {x^{3}}{\sqrt {c \,x^{2}+b x +a}\, \left (-f \,x^{2}+d \right )}d x \] Input:
int(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)
Output:
int(x^3/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)