Integrand size = 37, antiderivative size = 122 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \arcsin (d x)}{d f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}} \]
-(-B*f+C*e)*arcsin(d*x)/d/f^2+(A*f^2-B*e*f+C*e^2)*arctan((d^2*e*x+f)/(d^2* e^2-f^2)^(1/2)/(-d^2*x^2+1)^(1/2))/f^2/(d^2*e^2-f^2)^(1/2)-C*(-d^2*x^2+1)^ (1/2)/d^2/f
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\frac {-\frac {C f \sqrt {1-d^2 x^2}}{d^2}+\frac {2 (-C e+B f) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d}-\frac {2 \sqrt {d^2 e^2-f^2} \left (C e^2+f (-B e+A f)\right ) \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{(d e-f) (d e+f)}}{f^2} \]
(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + (2*(-(C*e) + B*f)*ArcTan[(d*x)/(-1 + Sqr t[1 - d^2*x^2])])/d - (2*Sqrt[d^2*e^2 - f^2]*(C*e^2 + f*(-(B*e) + A*f))*Ar cTan[(Sqrt[d^2*e^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/((d*e - f)* (d*e + f)))/f^2
Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2112, 2185, 25, 27, 719, 223, 488, 217}
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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {d x+1} (e+f x)} \, dx\) |
| \(\Big \downarrow \) 2112 |
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {1-d^2 x^2} (e+f x)}dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {\int -\frac {d^2 f (A f-(C e-B f) x)}{(e+f x) \sqrt {1-d^2 x^2}}dx}{d^2 f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d^2 f (A f-(C e-B f) x)}{(e+f x) \sqrt {1-d^2 x^2}}dx}{d^2 f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A f-(C e-B f) x}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {\left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {(C e-B f) \int \frac {1}{\sqrt {1-d^2 x^2}}dx}{f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {\left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {-\frac {\left (A f^2-B e f+C e^2\right ) \int \frac {1}{-d^2 e^2+f^2-\frac {\left (e x d^2+f\right )^2}{1-d^2 x^2}}d\frac {e x d^2+f}{\sqrt {1-d^2 x^2}}}{f}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (A f^2-B e f+C e^2\right ) \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f \sqrt {d^2 e^2-f^2}}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\) |
-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) + (-(((C*e - B*f)*ArcSin[d*x])/(d*f)) + ( (C*e^2 - B*e*f + A*f^2)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f*Sqrt[d^2*e^2 - f^2]))/f
3.1.12.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; F reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] & & EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(114)=228\).
Time = 1.66 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.37
| method | result | size |
| risch | \(\frac {C \sqrt {d x +1}\, \left (d x -1\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{f \,d^{2} \sqrt {-\left (d x +1\right ) \left (d x -1\right )}\, \sqrt {-d x +1}}+\frac {\left (\frac {\left (B f -C e \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{f \sqrt {d^{2}}}-\frac {\left (A \,f^{2}-B e f +C \,e^{2}\right ) \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{f \sqrt {-d x +1}\, \sqrt {d x +1}}\) | \(289\) |
| default | \(\frac {\left (-A \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} f^{2}+B \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e f -C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e^{2}+B \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d \,f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \,\operatorname {csgn}\left (d \right ) f^{2} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d e f \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\right ) \sqrt {-d x +1}\, \sqrt {d x +1}\, \operatorname {csgn}\left (d \right )}{\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f^{3} \sqrt {-d^{2} x^{2}+1}\, d^{2}}\) | \(373\) |
C/f/d^2*(d*x+1)^(1/2)*(d*x-1)/(-(d*x+1)*(d*x-1))^(1/2)*((-d*x+1)*(d*x+1))^ (1/2)/(-d*x+1)^(1/2)+1/f*((B*f-C*e)/f/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d ^2*x^2+1)^(1/2))-(A*f^2-B*e*f+C*e^2)/f^2/(-(d^2*e^2-f^2)/f^2)^(1/2)*ln((-2 *(d^2*e^2-f^2)/f^2+2/f*d^2*e*(x+e/f)+2*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*(x +e/f)^2+2/f*d^2*e*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2))/(x+e/f)))*((-d*x+1)*(d *x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (114) = 228\).
Time = 4.22 (sec) , antiderivative size = 493, normalized size of antiderivative = 4.04 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\left [-\frac {{\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {-d^{2} e^{2} + f^{2}} \log \left (\frac {d^{2} e f x + f^{2} - \sqrt {-d^{2} e^{2} + f^{2}} {\left (d^{2} e x + f\right )} - {\left (\sqrt {-d^{2} e^{2} + f^{2}} \sqrt {-d x + 1} f + {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d x + 1}\right )} \sqrt {d x + 1}}{f x + e}\right ) + {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}, \frac {2 \, {\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {d^{2} e^{2} - f^{2}} \arctan \left (-\frac {\sqrt {d^{2} e^{2} - f^{2}} \sqrt {d x + 1} \sqrt {-d x + 1} e - \sqrt {d^{2} e^{2} - f^{2}} {\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) - {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}\right ] \]
[-((C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*sqrt(-d^2*e^2 + f^2)*log((d^2*e*f*x + f^2 - sqrt(-d^2*e^2 + f^2)*(d^2*e*x + f) - (sqrt(-d^2*e^2 + f^2)*sqrt(- d*x + 1)*f + (d^2*e^2 - f^2)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) + ( C*d^2*e^2*f - C*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 2*(C*d^3*e^3 - B*d^3*e ^2*f - C*d*e*f^2 + B*d*f^3)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x )))/(d^4*e^2*f^2 - d^2*f^4), (2*(C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*sqrt(d ^2*e^2 - f^2)*arctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*e - sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) - (C*d^2*e^2*f - C*f ^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(C*d^3*e^3 - B*d^3*e^2*f - C*d*e*f^2 + B*d*f^3)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^4*e^2*f^2 - d^2*f^4)]
\[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\int \frac {A + B x + C x^{2}}{\left (e + f x\right ) \sqrt {- d x + 1} \sqrt {d x + 1}}\, dx \]
Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 0.01 (sec) , antiderivative size = 5803, normalized size of antiderivative = 47.57 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Too large to display} \]
(4*C*e*atan((37748736*C^5*d^4*e^10*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2 ) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e ^8*f^2)) + (67108864*C^5*e^6*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8 *f^2)) - (100663296*C^5*d^2*e^8*f^2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/ 2) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2* e^8*f^2))))/(d*f^2) - (4*B*atan((67108864*B^5*e*f^4*((1 - d*x)^(1/2) - 1)) /(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 10066 3296*B^5*d^2*e^3*f^2)) + (37748736*B^5*d^4*e^5*((1 - d*x)^(1/2) - 1))/(((d *x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296* B^5*d^2*e^3*f^2)) - (100663296*B^5*d^2*e^3*f^2*((1 - d*x)^(1/2) - 1))/(((d *x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296* B^5*d^2*e^3*f^2))))/(d*f) - (8*C*((1 - d*x)^(1/2) - 1)^2)/(f*((d*x + 1)^(1 /2) - 1)^2*(d^2 + (2*d^2*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (d^2*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4)) - (A*atan((f^2*1 i - d^2*e^2*1i - (f^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2 + (d^2*e^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2)/(f*(f + d* e)^(1/2)*(f - d*e)^(1/2) - (f*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))/((d*x + 1)^(1/2) - 1)^2 + (2*d*e*((1 - d*x)^(1/2) - 1)*(f + d *e)^(1/2)*(f - d*e)^(1/2))/((d*x + 1)^(1/2) - 1)))*2i)/((f + d*e)^(1/2)...