Integrand size = 30, antiderivative size = 331 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(b B+2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^{3/2}}+\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^{3/2}} \] Output:
-B*(c*x^2+b*x+a)^(1/2)/f-1/2*(2*A*c+B*b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c* x^2+b*x+a)^(1/2))/c^(1/2)/f-1/2*(B*d^(1/2)-A*f^(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))/d^(1/2)/f^(3/2)+1 /2*(B*d^(1/2)+A*f^(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))/d^(1/2)/f^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.21 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=-\frac {2 B \sqrt {a+x (b+c x)}-\frac {2 (b B+2 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+\text {RootSum}\left [c^2 d-b^2 f+4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2-4 a f \text {$\#$1}^2+d \text {$\#$1}^4\&,\frac {-A c^2 d \log (x)+A b^2 f \log (x)+a b B f \log (x)-a A c f \log (x)+A c^2 d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-A b^2 f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-a b B f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a A c f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 \sqrt {a} B c d \log (x) \text {$\#$1}-2 \sqrt {a} A b f \log (x) \text {$\#$1}-2 a^{3/2} B f \log (x) \text {$\#$1}+2 \sqrt {a} B c d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 \sqrt {a} A b f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a^{3/2} B f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+b B d \log (x) \text {$\#$1}^2+A c d \log (x) \text {$\#$1}^2+a A f \log (x) \text {$\#$1}^2-b B d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-A c d \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a A f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b f+c d \text {$\#$1}+2 a f \text {$\#$1}-d \text {$\#$1}^3}\&\right ]}{2 f} \] Input:
Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]
Output:
-1/2*(2*B*Sqrt[a + x*(b + c*x)] - (2*(b*B + 2*A*c)*ArcTanh[(Sqrt[c]*x)/(Sq rt[a] - Sqrt[a + x*(b + c*x)])])/Sqrt[c] + RootSum[c^2*d - b^2*f + 4*Sqrt[ a]*b*f*#1 - 2*c*d*#1^2 - 4*a*f*#1^2 + d*#1^4 & , (-(A*c^2*d*Log[x]) + A*b^ 2*f*Log[x] + a*b*B*f*Log[x] - a*A*c*f*Log[x] + A*c^2*d*Log[-Sqrt[a] + Sqrt [a + b*x + c*x^2] - x*#1] - A*b^2*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - a*b*B*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + a*A*c*f*Lo g[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*Sqrt[a]*B*c*d*Log[x]*#1 - 2 *Sqrt[a]*A*b*f*Log[x]*#1 - 2*a^(3/2)*B*f*Log[x]*#1 + 2*Sqrt[a]*B*c*d*Log[- Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + 2*Sqrt[a]*A*b*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + 2*a^(3/2)*B*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + b*B*d*Log[x]*#1^2 + A*c*d*Log[x]*#1^2 + a*A*f*L og[x]*#1^2 - b*B*d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - A*c *d*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - a*A*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2)/(-(Sqrt[a]*b*f) + c*d*#1 + 2*a*f*#1 - d*#1^3) & ])/f
Time = 0.80 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, 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.
\(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx\) |
\(\Big \downarrow \) 1354 |
\(\displaystyle \frac {\int \frac {(b B+2 A c) f x^2+2 (B c d+A b f+a B f) x+b B d+2 a A f}{2 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(b B+2 A c) f x^2+2 (B c d+A b f+a B f) x+b B d+2 a A f}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 2144 |
\(\displaystyle \frac {-\frac {\int -\frac {2 f (b B d+A c d+a A f+(B c d+A b f+a B f) x)}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\left ((2 A c+b B) \int \frac {1}{\sqrt {c x^2+b x+a}}dx\right )}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \int \frac {b B d+A c d+a A f+(B c d+A b f+a B f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-(2 A c+b B) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 \int \frac {b B d+A c d+a A f+(B c d+A b f+a B f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-2 (2 A c+b B) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \int \frac {b B d+A c d+a A f+(B c d+A b f+a B f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 1366 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \left (-\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \left (-\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (\frac {\left (\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}-\frac {\left (-\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}\right )-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {2 \left (\frac {\left (-\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}-\frac {\left (\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}\right )-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \left (\frac {\left (\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {f} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\left (-\frac {\sqrt {f} (a A f+A c d+b B d)}{\sqrt {d}}+a B f+A b f+B c d\right ) \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {f} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )-\frac {(2 A c+b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{2 f}-\frac {B \sqrt {a+b x+c x^2}}{f}\) |
Input:
Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]
Output:
-((B*Sqrt[a + b*x + c*x^2])/f) + (-(((b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2* Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c]) + 2*(-1/2*((B*c*d + A*b*f + a*B* f - (Sqrt[f]*(b*B*d + A*c*d + a*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[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((B*c*d + A*b*f + a*B*f + (Sqrt[f]*(b*B*d + A*c*d + a*A*f))/Sqrt[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*Sqrt[f]*Sqrt[c*d + b* Sqrt[d]*Sqrt[f] + a*f])))/(2*f)
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. \(540\) vs. \(2(251)=502\).
Time = 2.71 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.63
method | result | size |
risch | \(-\frac {B \sqrt {c \,x^{2}+b x +a}}{f}-\frac {\frac {B b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {\left (A \sqrt {d f}\, b f -a A \,f^{2}-A c d f +B \sqrt {d f}\, a f +B \sqrt {d f}\, c d -B b d f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {\left (A \sqrt {d f}\, b f +a A \,f^{2}+A c d f +B \sqrt {d f}\, a f +B \sqrt {d f}\, c d +B b d f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+2 A \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 f}\) | \(541\) |
default | \(-\frac {\left (A f +B \sqrt {d f}\right ) \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 {\left (A f -B \sqrt {d f}\right ) \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}\) | \(801\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-B*(c*x^2+b*x+a)^(1/2)/f-1/2/f*(B*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^( 1/2))/c^(1/2)-(A*(d*f)^(1/2)*b*f-a*A*f^2-A*c*d*f+B*(d*f)^(1/2)*a*f+B*(d*f) ^(1/2)*c*d-B*b*d*f)/(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))-(A*(d*f)^(1/2)*b*f+a*A*f^2+A*c*d*f+B*(d*f)^(1/2)*a*f+B*(d* f)^(1/2)*c*d+B*b*d*f)/(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) )+2*A*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=- \int \frac {A \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {B x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)
Output:
-Integral(A*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(B*x*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x)
Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(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. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((B*x+A)*(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. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/(d - f*x^2),x)
Output:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/(d - f*x^2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d-f x^2} \, dx=\int \frac {\left (B x +A \right ) \sqrt {c \,x^{2}+b x +a}}{-f \,x^{2}+d}d x \] Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)
Output:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x)