Integrand size = 28, antiderivative size = 333 \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {b \sqrt {a+b x+c x^2}}{4 c f}-\frac {x \sqrt {a+b x+c x^2}}{2 f}+\frac {\left (b^2-\frac {4 c (2 c d+a f)}{f}\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2} f}+\frac {\sqrt {d} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^2}+\frac {\sqrt {d} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^2} \] Output:
-1/4*b*(c*x^2+b*x+a)^(1/2)/c/f-1/2*x*(c*x^2+b*x+a)^(1/2)/f+1/8*(b^2-4*c*(a *f+2*c*d)/f)*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^(1/2)*(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)*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^2+1/2*d^(1/2)*(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)*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^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.89 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} f (b+2 c x) \sqrt {a+x (b+c x)}+\left (-8 c^2 d+b^2 f-4 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-4 c^{3/2} d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{8 c^{3/2} f^2} \] Input:
Integrate[(x^2*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]
Output:
(-2*Sqrt[c]*f*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (-8*c^2*d + b^2*f - 4*a* c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 4*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 & , (b*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*c^(3/2)*d* Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a*Sqrt[c]*f*Log[-(Sq rt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + b*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) & ]) /(8*c^(3/2)*f^2)
Time = 0.83 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2140, 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.
\(\displaystyle \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx\) |
\(\Big \downarrow \) 2140 |
\(\displaystyle -\frac {\int -\frac {f \left (-f b^2+8 c^2 d+4 a c f\right ) x^2+8 b c d f x+\left (b^2+4 a c\right ) d f}{4 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {f \left (-f b^2+8 c^2 d+4 a c f\right ) x^2+8 b c d f x+\left (b^2+4 a c\right ) d f}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 2144 |
\(\displaystyle \frac {-\left (\left (4 a c f+b^2 (-f)+8 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx\right )-\frac {\int -\frac {8 c d f (c d+a f+b f x)}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8 c d \int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-\left (4 a c f+b^2 (-f)+8 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {8 c d \int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-2 \left (4 a c f+b^2 (-f)+8 c^2 d\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {8 c d \int \frac {c d+a f+b f x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 1366 |
\(\displaystyle \frac {8 c d \left (\frac {1}{2} \sqrt {f} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \sqrt {f} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {8 c d \left (\frac {1}{2} \sqrt {f} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \sqrt {f} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8 c d \left (\frac {1}{2} \left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {8 c d \left (\left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )-\left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {8 c d \left (\frac {\left (\frac {a f+c d}{\sqrt {d}}+b \sqrt {f}\right ) \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\left (b \sqrt {f}-\frac {a f+c d}{\sqrt {d}}\right ) \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 a c f+b^2 (-f)+8 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c f}\) |
Input:
Int[(x^2*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]
Output:
-1/4*((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(c*f) + (-(((8*c^2*d - b^2*f + 4* a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c]) + 8*c*d*(-1/2*((b*Sqrt[f] - (c*d + a*f)/Sqrt[d])*ArcTanh[(b*Sqrt[d] - 2*a*Sq rt[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[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ((b*Sqrt[ f] + (c*d + a*f)/Sqrt[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[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/(8*c*f^2)
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_) + (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_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x) *(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c *x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*( 2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2 *p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*((-b)*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A* f)*(2*p + 2*q + 3))))*x + (p*((-b)*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
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 = 2.54 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.45
method | result | size |
risch | \(-\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c f}-\frac {\frac {\left (4 a c f -b^{2} f +8 c^{2} d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f \sqrt {c}}-\frac {4 d c \left (b \sqrt {d f}-a f -c d \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {4 d c \left (b \sqrt {d f}+a f +c d \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}}{8 c f}\) | \(483\) |
default | \(-\frac {\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}}{f}-\frac {d \left (\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}}+\frac {\left (2 c \sqrt {d f}+f b \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+f b}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+a f +c d \right ) \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 )}{f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}\right )}{2 \sqrt {d f}\, f}+\frac {d \left (\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}}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+f b}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\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}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+a f +c d \right ) \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 )}{f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\right )}{2 \sqrt {d f}\, f}\) | \(849\) |
Input:
int(x^2*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)/f-1/8/c/f*(1/f*(4*a*c*f-b^2*f+8*c^2*d )*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-4*d*c*(b*(d*f)^(1/2) -a*f-c*d)/(d*f)^(1/2)/f/(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))-4*d*c*(b*(d*f)^(1/2)+a*f+c*d)/(d*f)^(1/2)/f/((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)))
Timed out. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Timed out} \] Input:
integrate(x^2*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=- \int \frac {x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \] Input:
integrate(x**2*(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)
Output:
-Integral(x**2*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x)
Exception generated. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(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. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(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. \[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {x^2\,\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \] Input:
int((x^2*(a + b*x + c*x^2)^(1/2))/(d - f*x^2),x)
Output:
int((x^2*(a + b*x + c*x^2)^(1/2))/(d - f*x^2), x)
\[ \int \frac {x^2 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{-f \,x^{2}+d}d x \] Input:
int(x^2*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)
Output:
int(x^2*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)