Integrand size = 30, antiderivative size = 82 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{\sqrt {c} \sqrt {b e-a f} \sqrt {b e+a f}} \] Output:
2*arctan(c^(1/2)*(a*f+b*e)^(1/2)*(b*x+a)^(1/2)/(-a*f+b*e)^(1/2)/(-b*c*x+a* c)^(1/2))/c^(1/2)/(-a*f+b*e)^(1/2)/(a*f+b*e)^(1/2)
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\frac {2 \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{\sqrt {b e-a f} \sqrt {b e+a f} \sqrt {c (a-b x)}} \] Input:
Integrate[1/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]
Output:
(2*Sqrt[a - b*x]*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*S qrt[a - b*x])])/(Sqrt[b*e - a*f]*Sqrt[b*e + a*f]*Sqrt[c*(a - b*x)])
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {104, 218}
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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x} (e+f x) \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 2 \int \frac {1}{b e-a f+\frac {c (b e+a f) (a+b x)}{a c-b c x}}d\frac {\sqrt {a+b x}}{\sqrt {a c-b c x}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a+b x} \sqrt {a f+b e}}{\sqrt {a c-b c x} \sqrt {b e-a f}}\right )}{\sqrt {c} \sqrt {b e-a f} \sqrt {a f+b e}}\) |
Input:
Int[1/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]
Output:
(2*ArcTan[(Sqrt[c]*Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[a* c - b*c*x])])/(Sqrt[c]*Sqrt[b*e - a*f]*Sqrt[b*e + a*f])
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.55
method | result | size |
default | \(-\frac {\ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) \sqrt {c \left (-b x +a \right )}\, \sqrt {b x +a}}{\sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}\) | \(127\) |
Input:
int(1/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x,method=_RETURNVERBOSE)
Output:
-ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2) )^(1/2)*f)/(f*x+e))*(c*(-b*x+a))^(1/2)*(b*x+a)^(1/2)/(c*(a^2*f^2-b^2*e^2)/ f^2)^(1/2)/(c*(-b^2*x^2+a^2))^(1/2)/f
Time = 0.12 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.62 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\left [-\frac {\sqrt {-b^{2} c e^{2} + a^{2} c f^{2}} \log \left (\frac {2 \, a^{2} b^{2} c e f x - a^{2} b^{2} c e^{2} + 2 \, a^{4} c f^{2} + {\left (2 \, b^{4} c e^{2} - a^{2} b^{2} c f^{2}\right )} x^{2} - 2 \, \sqrt {-b^{2} c e^{2} + a^{2} c f^{2}} {\left (b^{2} e x + a^{2} f\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{f^{2} x^{2} + 2 \, e f x + e^{2}}\right )}{2 \, {\left (b^{2} c e^{2} - a^{2} c f^{2}\right )}}, \frac {\arctan \left (\frac {\sqrt {b^{2} c e^{2} - a^{2} c f^{2}} {\left (b^{2} e x + a^{2} f\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{a^{2} b^{2} c e^{2} - a^{4} c f^{2} - {\left (b^{4} c e^{2} - a^{2} b^{2} c f^{2}\right )} x^{2}}\right )}{\sqrt {b^{2} c e^{2} - a^{2} c f^{2}}}\right ] \] Input:
integrate(1/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x, algorithm="fricas" )
Output:
[-1/2*sqrt(-b^2*c*e^2 + a^2*c*f^2)*log((2*a^2*b^2*c*e*f*x - a^2*b^2*c*e^2 + 2*a^4*c*f^2 + (2*b^4*c*e^2 - a^2*b^2*c*f^2)*x^2 - 2*sqrt(-b^2*c*e^2 + a^ 2*c*f^2)*(b^2*e*x + a^2*f)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(f^2*x^2 + 2* e*f*x + e^2))/(b^2*c*e^2 - a^2*c*f^2), arctan(sqrt(b^2*c*e^2 - a^2*c*f^2)* (b^2*e*x + a^2*f)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^2*b^2*c*e^2 - a^4*c* f^2 - (b^4*c*e^2 - a^2*b^2*c*f^2)*x^2))/sqrt(b^2*c*e^2 - a^2*c*f^2)]
\[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\int \frac {1}{\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x} \left (e + f x\right )}\, dx \] Input:
integrate(1/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2)/(f*x+e),x)
Output:
Integral(1/(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(e + f*x)), x)
Exception generated. \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),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((a*f-b*e)>0)', see `assume?` for more deta
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=-\frac {2 \, \sqrt {-c} \arctan \left (-\frac {2 \, b c e - {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} f}{2 \, \sqrt {-b^{2} e^{2} + a^{2} f^{2}} c}\right )}{\sqrt {-b^{2} e^{2} + a^{2} f^{2}} c} \] Input:
integrate(1/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x, algorithm="giac")
Output:
-2*sqrt(-c)*arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*f)/(sqrt(-b^2*e^2 + a^2*f^2)*c))/(sqrt(-b^2*e^2 + a^2*f^2 )*c)
Time = 2.94 (sec) , antiderivative size = 342, normalized size of antiderivative = 4.17 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=-\frac {a\,\mathrm {atan}\left (\frac {-{\left (a\,c\right )}^{3/2}\,\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}\,1{}\mathrm {i}+a\,c\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}\,2{}\mathrm {i}+a\,c\,\sqrt {a\,c}\,\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}\,1{}\mathrm {i}+b\,c\,x\,\sqrt {a\,c}\,\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}\,2{}\mathrm {i}-\sqrt {a}\,c\,\sqrt {a\,c}\,\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}\,\sqrt {a+b\,x}\,2{}\mathrm {i}}{2\,a^{5/2}\,b\,c^2\,e-2\,a^3\,c^2\,f\,\sqrt {a+b\,x}-2\,a^2\,b\,c^2\,e\,\sqrt {a+b\,x}+2\,a^{5/2}\,b\,c^2\,f\,x+2\,a^{5/2}\,c\,f\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a\,c}-2\,a^{3/2}\,b\,c\,e\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a\,c}+2\,a\,b\,c\,e\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a\,c}\,\sqrt {a+b\,x}}\right )\,2{}\mathrm {i}}{\sqrt {a^4\,c\,f^2-a^2\,b^2\,c\,e^2}} \] Input:
int(1/((e + f*x)*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
-(a*atan((a*c*(a*c - b*c*x)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*2i - ( a*c)^(3/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*1i + a*c*(a*c)^(1/2)*(a^4*c*f ^2 - a^2*b^2*c*e^2)^(1/2)*1i + b*c*x*(a*c)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^ 2)^(1/2)*2i - a^(1/2)*c*(a*c)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*(a + b*x)^(1/2)*2i)/(2*a^(5/2)*b*c^2*e - 2*a^3*c^2*f*(a + b*x)^(1/2) - 2*a^2*b *c^2*e*(a + b*x)^(1/2) + 2*a^(5/2)*b*c^2*f*x + 2*a^(5/2)*c*f*(a*c - b*c*x) ^(1/2)*(a*c)^(1/2) - 2*a^(3/2)*b*c*e*(a*c - b*c*x)^(1/2)*(a*c)^(1/2) + 2*a *b*c*e*(a*c - b*c*x)^(1/2)*(a*c)^(1/2)*(a + b*x)^(1/2)))*2i)/(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)
Time = 0.19 (sec) , antiderivative size = 979, normalized size of antiderivative = 11.94 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx =\text {Too large to display} \] Input:
int(1/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x)
Output:
(sqrt(c)*sqrt(a*f + b*e)*( - 2*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt (a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin( sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)* sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e))) - 2*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2 ) - 3*a*f + b*e)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)* sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a*f + 2*sqrt(2*sq rt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 )))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e) *sqrt(a*f + b*e)))*b*e - sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) + 3*a*f - b*e)*sqrt(a*f - b*e)*sqrt(2)*log(sqrt(a*f + b*e)*t an(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - sqrt(2*sqrt(f)*sqrt(a)*sqrt( a*f - b*e)*sqrt(2) + 3*a*f - b*e)) + sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a )*sqrt(a*f - b*e)*sqrt(2) + 3*a*f - b*e)*sqrt(a*f - b*e)*sqrt(2)*log(sqrt( a*f + b*e)*tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + sqrt(2*sqrt(f)*s qrt(a)*sqrt(a*f - b*e)*sqrt(2) + 3*a*f - b*e)) + sqrt(2*sqrt(f)*sqrt(a)*sq rt(a*f - b*e)*sqrt(2) + 3*a*f - b*e)*log(sqrt(a*f + b*e)*tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sq...