Integrand size = 26, antiderivative size = 282 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f}-\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}} \]
-1/2*b*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/f/c^(1/2)-(c*x^2 +b*x+a)^(1/2)/f-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))*(c*d+a*f-b* d^(1/2)*f^(1/2))^(1/2)/f^(3/2)+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))*(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2)/f^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.24 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {2 \sqrt {a+x (b+c x)}-\frac {b \log \left (f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {c}}+\text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 f} \]
-1/2*(2*Sqrt[a + x*(b + c*x)] - (b*Log[f*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x *(b + c*x)])])/Sqrt[c] + 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 & , (b^2*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c* x^2] - #1] - a*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a^2*f* Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*b*Sqrt[c]*d*Log[-(Sqrt[ c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b *x + c*x^2] - #1]*#1^2 + a*f*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 = 0.57 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1354, 27, 2144, 27, 1092, 219, 1366, 25, 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.
| \(\displaystyle \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 1354 |
| \(\displaystyle \frac {\int \frac {b f x^2+2 (c d+a f) x+b d}{2 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b f x^2+2 (c d+a f) x+b d}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 f (b d+(c d+a f) x)}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-b \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {b d+(c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-b \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 \int \frac {b d+(c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-2 b \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \int \frac {b d+(c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 1366 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a f+b \sqrt {d} \sqrt {f}+c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right ) \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a f+b \sqrt {d} \sqrt {f}+c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right ) \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right ) \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right ) \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}-\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right ) \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {f}}-\frac {\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {f}}\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {\sqrt {a+b x+c x^2}}{f}\) |
-(Sqrt[a + b*x + c*x^2]/f) + (-((b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c]) + 2*(-1/2*(Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*A rcTanh[(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])])/Sqrt[f] + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*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[f])))/(2*f)
3.1.79.3.1 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[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f _.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1)) Int[(a + b*x + c*x^ 2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*( c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4* a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q ))) Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 2*q))) Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d , e, f, g, h}, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(459\) vs. \(2(214)=428\).
Time = 0.75 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.63
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}}{f}-\frac {\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {\left (\sqrt {d f}\, a f +\sqrt {d f}\, c d +b d f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (\sqrt {d f}\, a f +\sqrt {d f}\, c d -b d f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 f}\) | \(460\) |
| default | \(-\frac {\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}}+\frac {\left (2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+b f}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+f a +c d \right ) \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 )}{f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 f}-\frac {\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}}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+b f}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+f a +c d \right ) \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 )}{f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 f}\) | \(768\) |
-(c*x^2+b*x+a)^(1/2)/f-1/2/f*(b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2) )/c^(1/2)-((d*f)^(1/2)*a*f+(d*f)^(1/2)*c*d+b*d*f)/(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))-((d*f)^(1/2)*a*f+(d*f)^(1/2)*c*d-b*d*f)/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (214) = 428\).
Time = 186.76 (sec) , antiderivative size = 1192, normalized size of antiderivative = 4.23 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Too large to display} \]
[1/4*(c*f*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) + 2*b*c*d*x + b^2*d + (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt((f ^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sqr t(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) - 2*b*c*d*x - b^2 *d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(-(f^3*sqrt(b^2*d/f ^5) - c*d - a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqr t(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3) + 2*b*c*d*x + b^2*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) + c*f*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f )/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt(-(f^3*sqrt(b ^2*d/f^5) - c*d - a*f)/f^3) - 2*b*c*d*x - b^2*d + (b*f^3*x + 2*a*f^3)*sqrt (b^2*d/f^5))/x) + b*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*sqrt(c*x^2 + b*x + a)*c)/(c*f) , 1/4*(c*f*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) + 2*b*c*d*x + b^2*d + (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(( f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sq rt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) - 2*b*c*d*x - b^ 2*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(-(f^3*sqrt(b^2*d/ f^5) - c*d - a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)...
\[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=- \int \frac {x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
Exception generated. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {x\,\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \]