Integrand size = 36, antiderivative size = 188 \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^2 d^{3/2} f^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^2 \sqrt {b c-a d} \sqrt {b e-a f}} \]
-(2*a*C*d*f+b*(-2*B*d*f+C*c*f+C*d*e))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2 )/(f*x+e)^(1/2))/b^2/d^(3/2)/f^(3/2)-2*(A*b^2-a*(B*b-C*a))*arctanh((-a*f+b *e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^2/(-a*d+b*c)^(1/ 2)/(-a*f+b*e)^(1/2)+C*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d/f
Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\frac {b C \sqrt {c+d x} \sqrt {e+f x}}{d f}+\frac {2 \left (A b^2+a (-b B+a C)\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{\sqrt {b c-a d} \sqrt {-b e+a f}}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{d^{3/2} f^{3/2}}}{b^2} \]
((b*C*Sqrt[c + d*x]*Sqrt[e + f*x])/(d*f) + (2*(A*b^2 + a*(-(b*B) + a*C))*A rcTan[(Sqrt[b*c - a*d]*Sqrt[e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])]) /(Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]) - ((2*a*C*d*f + b*(C*d*e + c*C*f - 2 *B*d*f))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(d^(3/2 )*f^(3/2)))/b^2
Time = 0.49 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2118, 27, 175, 66, 104, 221}
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.
\(\displaystyle \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
\(\Big \downarrow \) 2118 |
\(\displaystyle \frac {\int \frac {b (2 A b d f-a C (d e+c f)-(2 a C d f+b (C d e+c C f-2 B d f)) x)}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b^2 d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 A b d f-a C (d e+c f)-(2 a C d f+b (C d e+c C f-2 B d f)) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 b d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {\frac {2 d f \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {(2 a C d f+b (-2 B d f+c C f+C d e)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}}{2 b d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {\frac {2 d f \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {2 (2 a C d f+b (-2 B d f+c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}}{2 b d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {\frac {4 d f \left (A b^2-a (b B-a C)\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {2 (2 a C d f+b (-2 B d f+c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}}{2 b d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {4 d f \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right )}{b \sqrt {b c-a d} \sqrt {b e-a f}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) (2 a C d f+b (-2 B d f+c C f+C d e))}{b \sqrt {d} \sqrt {f}}}{2 b d f}+\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b d f}\) |
(C*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*d*f) + ((-2*(2*a*C*d*f + b*(C*d*e + c*C *f - 2*B*d*f))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/( b*Sqrt[d]*Sqrt[f]) - (4*(A*b^2 - a*(b*B - a*C))*d*f*ArcTanh[(Sqrt[b*e - a* f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b*Sqrt[b*c - a*d]*Sqr t[b*e - a*f]))/(2*b*d*f)
3.1.57.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(745\) vs. \(2(160)=320\).
Time = 1.69 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.97
method | result | size |
default | \(-\frac {\left (2 A \sqrt {d f}\, \ln \left (\frac {-2 a d f x +b c f x +b d e x +2 \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, b -a c f -a d e +2 b c e}{b x +a}\right ) b^{2} d f -2 B \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, b^{2} d f -2 B \sqrt {d f}\, \ln \left (\frac {-2 a d f x +b c f x +b d e x +2 \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, b -a c f -a d e +2 b c e}{b x +a}\right ) a b d f +2 C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, a b d f +C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, b^{2} c f +C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, b^{2} d e +2 C \sqrt {d f}\, \ln \left (\frac {-2 a d f x +b c f x +b d e x +2 \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, b -a c f -a d e +2 b c e}{b x +a}\right ) a^{2} d f -2 C \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}\, \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, b^{2}\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, d \sqrt {d f}\, b^{3} \sqrt {\frac {a^{2} d f -a c f b -a b d e +b^{2} c e}{b^{2}}}\, f}\) | \(746\) |
-1/2*(2*A*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a *b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/ (b*x+a))*b^2*d*f-2*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2) +c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^2*d *f-2*B*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b* d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b* x+a))*a*b*d*f+2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c* f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b*d*f+ C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/ 2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^2*c*f+C*ln(1/2*(2*d*f* x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a* b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^2*d*e+2*C*(d*f)^(1/2)*ln((-2*a*d*f*x+b *c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f *x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*d*f-2*C*((d*x+c)*(f*x+e)) ^(1/2)*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^2)*(d*x +c)^(1/2)*(f*x+e)^(1/2)/((d*x+c)*(f*x+e))^(1/2)/d/(d*f)^(1/2)/b^3/((a^2*d* f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/f
Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {A + B x + C x^{2}}{\left (a + b x\right ) \sqrt {c + d x} \sqrt {e + f x}}\, dx \]
Exception generated. \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((-(2*a*d*f)/b^2)>0)', see `assu me?` for m
Exception generated. \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Hanged} \]