Integrand size = 14, antiderivative size = 77 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\frac {2 \left (2 a+b \sqrt {x}\right )}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}-\frac {4 b \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:
2*(2*a+b*x^(1/2))/(-4*a*c+b^2)/(a+b*x^(1/2)+c*x)-4*b*arctanh((b+2*c*x^(1/2 ))/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\frac {2 \left (2 a+b \sqrt {x}\right )}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}-\frac {4 b \arctan \left (\frac {b+2 c \sqrt {x}}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \] Input:
Integrate[(a + b*Sqrt[x] + c*x)^(-2),x]
Output:
(2*(2*a + b*Sqrt[x]))/((b^2 - 4*a*c)*(a + b*Sqrt[x] + c*x)) - (4*b*ArcTan[ (b + 2*c*Sqrt[x])/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1680, 1159, 1083, 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.
\(\displaystyle \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+c x+b \sqrt {x}\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle 2 \left (\frac {b \int \frac {1}{a+c x+b \sqrt {x}}d\sqrt {x}}{b^2-4 a c}+\frac {2 a+b \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 2 \left (\frac {2 a+b \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}-\frac {2 b \int \frac {1}{b^2-4 a c-x}d\left (b+2 c \sqrt {x}\right )}{b^2-4 a c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {2 a+b \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}-\frac {2 b \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\right )\) |
Input:
Int[(a + b*Sqrt[x] + c*x)^(-2),x]
Output:
2*((2*a + b*Sqrt[x])/((b^2 - 4*a*c)*(a + b*Sqrt[x] + c*x)) - (2*b*ArcTanh[ (b + 2*c*Sqrt[x])/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2))
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k* n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && Fr actionQ[n]
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-2 b \sqrt {x}-4 a}{\left (4 a c -b^{2}\right ) \left (a +b \sqrt {x}+c x \right )}-\frac {4 b \arctan \left (\frac {b +2 c \sqrt {x}}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(75\) |
default | \(b^{2} \left (\frac {\left (-2 a c +b^{2}\right ) x -2 a^{2}}{\left (4 a \,b^{2} c -b^{4}\right ) \left (c^{2} x^{2}+2 a c x -b^{2} x +a^{2}\right )}+\frac {2 \left (-2 a c +b^{2}\right ) \arctan \left (\frac {2 c^{2} x +2 a c -b^{2}}{\sqrt {4 a \,b^{2} c -b^{4}}}\right )}{\left (4 a \,b^{2} c -b^{4}\right )^{\frac {3}{2}}}\right )-\frac {b \sqrt {x}}{\left (a +b \sqrt {x}+c x \right ) \left (4 a c -b^{2}\right )}-\frac {b \sqrt {x}}{\left (-b \sqrt {x}+c x +a \right ) \left (4 a c -b^{2}\right )}+a^{2} \left (\frac {2 c^{2} x +2 a c -b^{2}}{\left (4 a \,b^{2} c -b^{4}\right ) \left (c^{2} x^{2}+2 a c x -b^{2} x +a^{2}\right )}+\frac {4 c^{2} \arctan \left (\frac {2 c^{2} x +2 a c -b^{2}}{\sqrt {4 a \,b^{2} c -b^{4}}}\right )}{\left (4 a \,b^{2} c -b^{4}\right )^{\frac {3}{2}}}\right )-\frac {2 b \arctan \left (\frac {b +2 c \sqrt {x}}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+c^{2} \left (\frac {\frac {\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{c^{2} b^{2} \left (4 a c -b^{2}\right )}+\frac {a^{2} \left (2 a c -b^{2}\right )}{b^{2} c^{2} \left (4 a c -b^{2}\right )}}{c^{2} x^{2}+2 a c x -b^{2} x +a^{2}}+\frac {4 a^{2} \arctan \left (\frac {2 c^{2} x +2 a c -b^{2}}{\sqrt {4 a \,b^{2} c -b^{4}}}\right )}{b^{2} \left (4 a c -b^{2}\right ) \sqrt {4 a \,b^{2} c -b^{4}}}\right )+2 a c \left (\frac {\left (-2 a c +b^{2}\right ) x -2 a^{2}}{\left (4 a \,b^{2} c -b^{4}\right ) \left (c^{2} x^{2}+2 a c x -b^{2} x +a^{2}\right )}+\frac {2 \left (-2 a c +b^{2}\right ) \arctan \left (\frac {2 c^{2} x +2 a c -b^{2}}{\sqrt {4 a \,b^{2} c -b^{4}}}\right )}{\left (4 a \,b^{2} c -b^{4}\right )^{\frac {3}{2}}}\right )+\frac {2 a}{\left (-b \sqrt {x}+c x +a \right ) \left (4 a c -b^{2}\right )}-\frac {2 b \arctan \left (\frac {2 c \sqrt {x}-b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}-\frac {2 a}{\left (a +b \sqrt {x}+c x \right ) \left (4 a c -b^{2}\right )}\) | \(716\) |
Input:
int(1/(a+b*x^(1/2)+c*x)^2,x,method=_RETURNVERBOSE)
Output:
2*(-b*x^(1/2)-2*a)/(4*a*c-b^2)/(a+b*x^(1/2)+c*x)-4*b/(4*a*c-b^2)^(3/2)*arc tan((b+2*c*x^(1/2))/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (67) = 134\).
Time = 0.09 (sec) , antiderivative size = 565, normalized size of antiderivative = 7.34 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\left [\frac {2 \, {\left (2 \, a^{2} b^{2} - 8 \, a^{3} c - {\left (b c^{2} x^{2} + a^{2} b - {\left (b^{3} - 2 \, a b c\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{3} x^{2} - b^{2} c x + a b^{2} - 2 \, a^{2} c - {\left (b c x - a b\right )} \sqrt {b^{2} - 4 \, a c} - {\left (b^{3} - 4 \, a b c - {\left (2 \, c^{2} x - b^{2} + 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {x}}{c^{2} x^{2} + a^{2} - {\left (b^{2} - 2 \, a c\right )} x}\right ) - {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x - {\left (a b^{3} - 4 \, a^{2} b c - {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt {x}\right )}}{a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} - {\left (b^{6} - 10 \, a b^{4} c + 32 \, a^{2} b^{2} c^{2} - 32 \, a^{3} c^{3}\right )} x}, \frac {2 \, {\left (2 \, a^{2} b^{2} - 8 \, a^{3} c - 2 \, {\left (b c^{2} x^{2} + a^{2} b - {\left (b^{3} - 2 \, a b c\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c \sqrt {x} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x - {\left (a b^{3} - 4 \, a^{2} b c - {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt {x}\right )}}{a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} - {\left (b^{6} - 10 \, a b^{4} c + 32 \, a^{2} b^{2} c^{2} - 32 \, a^{3} c^{3}\right )} x}\right ] \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^2,x, algorithm="fricas")
Output:
[2*(2*a^2*b^2 - 8*a^3*c - (b*c^2*x^2 + a^2*b - (b^3 - 2*a*b*c)*x)*sqrt(b^2 - 4*a*c)*log((2*c^3*x^2 - b^2*c*x + a*b^2 - 2*a^2*c - (b*c*x - a*b)*sqrt( b^2 - 4*a*c) - (b^3 - 4*a*b*c - (2*c^2*x - b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)) *sqrt(x))/(c^2*x^2 + a^2 - (b^2 - 2*a*c)*x)) - (b^4 - 6*a*b^2*c + 8*a^2*c^ 2)*x - (a*b^3 - 4*a^2*b*c - (b^3*c - 4*a*b*c^2)*x)*sqrt(x))/(a^2*b^4 - 8*a ^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 - (b^6 - 10*a*b^4*c + 32*a^2*b^2*c^2 - 32*a^3*c^3)*x), 2*(2*a^2*b^2 - 8*a^3*c - 2*( b*c^2*x^2 + a^2*b - (b^3 - 2*a*b*c)*x)*sqrt(-b^2 + 4*a*c)*arctan(-(2*sqrt( -b^2 + 4*a*c)*c*sqrt(x) + sqrt(-b^2 + 4*a*c)*b)/(b^2 - 4*a*c)) - (b^4 - 6* a*b^2*c + 8*a^2*c^2)*x - (a*b^3 - 4*a^2*b*c - (b^3*c - 4*a*b*c^2)*x)*sqrt( x))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2* c^4)*x^2 - (b^6 - 10*a*b^4*c + 32*a^2*b^2*c^2 - 32*a^3*c^3)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 1358 vs. \(2 (70) = 140\).
Time = 38.16 (sec) , antiderivative size = 1358, normalized size of antiderivative = 17.64 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2)+c*x)**2,x)
Output:
Piecewise((2*a*log(a/b + sqrt(x))/(a*b**2 + b**3*sqrt(x)) + 2*a/(a*b**2 + b**3*sqrt(x)) + 2*b*sqrt(x)*log(a/b + sqrt(x))/(a*b**2 + b**3*sqrt(x)), Eq (c, 0)), (-4*b/(3*b**3 + 18*b**2*c*sqrt(x) + 36*b*c**2*x + 24*c**3*x**(3/2 )) - 24*c*sqrt(x)/(3*b**3 + 18*b**2*c*sqrt(x) + 36*b*c**2*x + 24*c**3*x**( 3/2)), Eq(a, b**2/(4*c))), (-2*a*b*log(b/(2*c) + sqrt(x) - sqrt(-4*a*c + b **2)/(2*c))/(4*a**2*c*sqrt(-4*a*c + b**2) - a*b**2*sqrt(-4*a*c + b**2) + 4 *a*b*c*sqrt(x)*sqrt(-4*a*c + b**2) + 4*a*c**2*x*sqrt(-4*a*c + b**2) - b**3 *sqrt(x)*sqrt(-4*a*c + b**2) - b**2*c*x*sqrt(-4*a*c + b**2)) + 2*a*b*log(b /(2*c) + sqrt(x) + sqrt(-4*a*c + b**2)/(2*c))/(4*a**2*c*sqrt(-4*a*c + b**2 ) - a*b**2*sqrt(-4*a*c + b**2) + 4*a*b*c*sqrt(x)*sqrt(-4*a*c + b**2) + 4*a *c**2*x*sqrt(-4*a*c + b**2) - b**3*sqrt(x)*sqrt(-4*a*c + b**2) - b**2*c*x* sqrt(-4*a*c + b**2)) - 4*a*sqrt(-4*a*c + b**2)/(4*a**2*c*sqrt(-4*a*c + b** 2) - a*b**2*sqrt(-4*a*c + b**2) + 4*a*b*c*sqrt(x)*sqrt(-4*a*c + b**2) + 4* a*c**2*x*sqrt(-4*a*c + b**2) - b**3*sqrt(x)*sqrt(-4*a*c + b**2) - b**2*c*x *sqrt(-4*a*c + b**2)) - 2*b**2*sqrt(x)*log(b/(2*c) + sqrt(x) - sqrt(-4*a*c + b**2)/(2*c))/(4*a**2*c*sqrt(-4*a*c + b**2) - a*b**2*sqrt(-4*a*c + b**2) + 4*a*b*c*sqrt(x)*sqrt(-4*a*c + b**2) + 4*a*c**2*x*sqrt(-4*a*c + b**2) - b**3*sqrt(x)*sqrt(-4*a*c + b**2) - b**2*c*x*sqrt(-4*a*c + b**2)) + 2*b**2* sqrt(x)*log(b/(2*c) + sqrt(x) + sqrt(-4*a*c + b**2)/(2*c))/(4*a**2*c*sqrt( -4*a*c + b**2) - a*b**2*sqrt(-4*a*c + b**2) + 4*a*b*c*sqrt(x)*sqrt(-4*a...
Exception generated. \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^2,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(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\frac {4 \, b \arctan \left (\frac {2 \, c \sqrt {x} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, {\left (b \sqrt {x} + 2 \, a\right )}}{{\left (b^{2} - 4 \, a c\right )} {\left (c x + b \sqrt {x} + a\right )}} \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^2,x, algorithm="giac")
Output:
4*b*arctan((2*c*sqrt(x) + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)) + 2*(b*sqrt(x) + 2*a)/((b^2 - 4*a*c)*(c*x + b*sqrt(x) + a))
Time = 18.80 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=-\frac {\frac {4\,a}{4\,a\,c-b^2}+\frac {2\,b\,\sqrt {x}}{4\,a\,c-b^2}}{a+c\,x+b\,\sqrt {x}}-\frac {4\,b\,\mathrm {atan}\left (\frac {\left (\frac {2\,b^2}{{\left (4\,a\,c-b^2\right )}^{3/2}}+\frac {4\,b\,c\,\sqrt {x}}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,b}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:
int(1/(a + c*x + b*x^(1/2))^2,x)
Output:
- ((4*a)/(4*a*c - b^2) + (2*b*x^(1/2))/(4*a*c - b^2))/(a + c*x + b*x^(1/2) ) - (4*b*atan((((2*b^2)/(4*a*c - b^2)^(3/2) + (4*b*c*x^(1/2))/(4*a*c - b^2 )^(3/2))*(4*a*c - b^2))/(2*b)))/(4*a*c - b^2)^(3/2)
Time = 0.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^2} \, dx=\frac {-4 \sqrt {x}\, \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2}-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) a b -4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b c x -8 a^{2} c +2 a \,b^{2}+8 a \,c^{2} x -2 b^{2} c x}{16 \sqrt {x}\, a^{2} b \,c^{2}-8 \sqrt {x}\, a \,b^{3} c +\sqrt {x}\, b^{5}+16 a^{3} c^{2}-8 a^{2} b^{2} c +16 a^{2} c^{3} x +a \,b^{4}-8 a \,b^{2} c^{2} x +b^{4} c x} \] Input:
int(1/(a+b*x^(1/2)+c*x)^2,x)
Output:
(2*( - 2*sqrt(x)*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b* *2))*b**2 - 2*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b**2) )*a*b - 2*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b**2))*b* c*x - 4*a**2*c + a*b**2 + 4*a*c**2*x - b**2*c*x))/(16*sqrt(x)*a**2*b*c**2 - 8*sqrt(x)*a*b**3*c + sqrt(x)*b**5 + 16*a**3*c**2 - 8*a**2*b**2*c + 16*a* *2*c**3*x + a*b**4 - 8*a*b**2*c**2*x + b**4*c*x)