Integrand size = 14, antiderivative size = 114 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=\frac {2 a+b \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )^2}-\frac {3 b \left (b+2 c \sqrt {x}\right )}{\left (b^2-4 a c\right )^2 \left (a+b \sqrt {x}+c x\right )}+\frac {12 b c \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:
(2*a+b*x^(1/2))/(-4*a*c+b^2)/(a+b*x^(1/2)+c*x)^2-3*b*(b+2*c*x^(1/2))/(-4*a *c+b^2)^2/(a+b*x^(1/2)+c*x)+12*b*c*arctanh((b+2*c*x^(1/2))/(-4*a*c+b^2)^(1 /2))/(-4*a*c+b^2)^(5/2)
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=-\frac {8 a^2 c+a b \left (b+10 c \sqrt {x}\right )+b \sqrt {x} \left (2 b^2+9 b c \sqrt {x}+6 c^2 x\right )}{\left (b^2-4 a c\right )^2 \left (a+b \sqrt {x}+c x\right )^2}-\frac {12 b c \arctan \left (\frac {b+2 c \sqrt {x}}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}} \] Input:
Integrate[(a + b*Sqrt[x] + c*x)^(-3),x]
Output:
-((8*a^2*c + a*b*(b + 10*c*Sqrt[x]) + b*Sqrt[x]*(2*b^2 + 9*b*c*Sqrt[x] + 6 *c^2*x))/((b^2 - 4*a*c)^2*(a + b*Sqrt[x] + c*x)^2)) - (12*b*c*ArcTan[(b + 2*c*Sqrt[x])/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1680, 1159, 1086, 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.
\(\displaystyle \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+c x+b \sqrt {x}\right )^3}d\sqrt {x}\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle 2 \left (\frac {3 b \int \frac {1}{\left (a+c x+b \sqrt {x}\right )^2}d\sqrt {x}}{2 \left (b^2-4 a c\right )}+\frac {2 a+b \sqrt {x}}{2 \left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )^2}\right )\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle 2 \left (\frac {3 b \left (-\frac {2 c \int \frac {1}{a+c x+b \sqrt {x}}d\sqrt {x}}{b^2-4 a c}-\frac {b+2 c \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}\right )}{2 \left (b^2-4 a c\right )}+\frac {2 a+b \sqrt {x}}{2 \left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )^2}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 2 \left (\frac {3 b \left (\frac {4 c \int \frac {1}{b^2-4 a c-x}d\left (b+2 c \sqrt {x}\right )}{b^2-4 a c}-\frac {b+2 c \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}\right )}{2 \left (b^2-4 a c\right )}+\frac {2 a+b \sqrt {x}}{2 \left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {3 b \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c \sqrt {x}}{\left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )}\right )}{2 \left (b^2-4 a c\right )}+\frac {2 a+b \sqrt {x}}{2 \left (b^2-4 a c\right ) \left (a+b \sqrt {x}+c x\right )^2}\right )\) |
Input:
Int[(a + b*Sqrt[x] + c*x)^(-3),x]
Output:
2*((2*a + b*Sqrt[x])/(2*(b^2 - 4*a*c)*(a + b*Sqrt[x] + c*x)^2) + (3*b*(-(( b + 2*c*Sqrt[x])/((b^2 - 4*a*c)*(a + b*Sqrt[x] + c*x))) + (4*c*ArcTanh[(b + 2*c*Sqrt[x])/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c)) )
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
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 = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {-b \sqrt {x}-2 a}{\left (4 a c -b^{2}\right ) \left (a +b \sqrt {x}+c x \right )^{2}}-\frac {3 b \left (\frac {b +2 c \sqrt {x}}{\left (4 a c -b^{2}\right ) \left (a +b \sqrt {x}+c x \right )}+\frac {4 c \arctan \left (\frac {b +2 c \sqrt {x}}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 a c -b^{2}}\) | \(123\) |
default | \(\text {Expression too large to display}\) | \(4753\) |
Input:
int(1/(a+b*x^(1/2)+c*x)^3,x,method=_RETURNVERBOSE)
Output:
(-b*x^(1/2)-2*a)/(4*a*c-b^2)/(a+b*x^(1/2)+c*x)^2-3*b/(4*a*c-b^2)*((b+2*c*x ^(1/2))/(4*a*c-b^2)/(a+b*x^(1/2)+c*x)+4*c/(4*a*c-b^2)^(3/2)*arctan((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. 615 vs. \(2 (100) = 200\).
Time = 0.12 (sec) , antiderivative size = 1299, normalized size of antiderivative = 11.39 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^3,x, algorithm="fricas")
Output:
[-(a^3*b^4 + 4*a^4*b^2*c - 32*a^5*c^2 - 3*(b^4*c^3 - 4*a*b^2*c^4)*x^3 + (5 *b^6*c - 33*a*b^4*c^2 + 60*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 - 6*(b*c^5*x^4 + a^4*b*c - 2*(b^3*c^3 - 2*a*b*c^4)*x^3 + (b^5*c - 4*a*b^3*c^2 + 6*a^2*b*c^3 )*x^2 - 2*(a^2*b^3*c - 2*a^3*b*c^2)*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)) - (3*a*b^6 - 11*a^2*b^4*c - 20*a^3*b^2*c^2 + 64*a^4*c^3) *x - 2*(3*a^3*b^3*c - 12*a^4*b*c^2 - 3*(b^3*c^4 - 4*a*b*c^5)*x^3 + (5*b^5* c^2 - 31*a*b^3*c^3 + 44*a^2*b*c^4)*x^2 - (b^7 - 7*a*b^5*c + 17*a^2*b^3*c^2 - 20*a^3*b*c^3)*x)*sqrt(x))/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64 *a^7*c^3 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^4 - 2* (b^8*c^2 - 14*a*b^6*c^3 + 72*a^2*b^4*c^4 - 160*a^3*b^2*c^5 + 128*a^4*c^6)* x^3 + (b^10 - 16*a*b^8*c + 102*a^2*b^6*c^2 - 328*a^3*b^4*c^3 + 544*a^4*b^2 *c^4 - 384*a^5*c^5)*x^2 - 2*(a^2*b^8 - 14*a^3*b^6*c + 72*a^4*b^4*c^2 - 160 *a^5*b^2*c^3 + 128*a^6*c^4)*x), -(a^3*b^4 + 4*a^4*b^2*c - 32*a^5*c^2 - 3*( b^4*c^3 - 4*a*b^2*c^4)*x^3 + (5*b^6*c - 33*a*b^4*c^2 + 60*a^2*b^2*c^3 - 32 *a^3*c^4)*x^2 - 12*(b*c^5*x^4 + a^4*b*c - 2*(b^3*c^3 - 2*a*b*c^4)*x^3 + (b ^5*c - 4*a*b^3*c^2 + 6*a^2*b*c^3)*x^2 - 2*(a^2*b^3*c - 2*a^3*b*c^2)*x)*sqr t(-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)) - (3*a*b^6 - 11*a^2*b^4*c - 20*a^3*b^2*c^2 + 64*a^...
Timed out. \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*x**(1/2)+c*x)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^3,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 = 139, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=-\frac {12 \, b c \arctan \left (\frac {2 \, c \sqrt {x} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, b c^{2} x^{\frac {3}{2}} + 9 \, b^{2} c x + 2 \, b^{3} \sqrt {x} + 10 \, a b c \sqrt {x} + a b^{2} + 8 \, a^{2} c}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (c x + b \sqrt {x} + a\right )}^{2}} \] Input:
integrate(1/(a+b*x^(1/2)+c*x)^3,x, algorithm="giac")
Output:
-12*b*c*arctan((2*c*sqrt(x) + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 1 6*a^2*c^2)*sqrt(-b^2 + 4*a*c)) - (6*b*c^2*x^(3/2) + 9*b^2*c*x + 2*b^3*sqrt (x) + 10*a*b*c*sqrt(x) + a*b^2 + 8*a^2*c)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)* (c*x + b*sqrt(x) + a)^2)
Time = 0.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.21 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=-\frac {\frac {8\,c\,a^2+a\,b^2}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {2\,b\,\sqrt {x}\,\left (b^2+5\,a\,c\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {9\,b^2\,c\,x}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {6\,b\,c^2\,x^{3/2}}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^2+2\,a\,b\,\sqrt {x}+2\,b\,c\,x^{3/2}}-\frac {12\,b\,c\,\mathrm {atan}\left (\frac {\left (\frac {6\,b^2\,c}{{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {12\,b\,c^2\,\sqrt {x}}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,b\,c}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \] Input:
int(1/(a + c*x + b*x^(1/2))^3,x)
Output:
- ((a*b^2 + 8*a^2*c)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (2*b*x^(1/2)*(5*a*c + b^2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (9*b^2*c*x)/(b^4 + 16*a^2*c^2 - 8 *a*b^2*c) + (6*b*c^2*x^(3/2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(x*(2*a*c + b^2) + a^2 + c^2*x^2 + 2*a*b*x^(1/2) + 2*b*c*x^(3/2)) - (12*b*c*atan((((6* b^2*c)/(4*a*c - b^2)^(5/2) + (12*b*c^2*x^(1/2))/(4*a*c - b^2)^(5/2))*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(6*b*c)))/(4*a*c - b^2)^(5/2)
Time = 0.30 (sec) , antiderivative size = 569, normalized size of antiderivative = 4.99 \[ \int \frac {1}{\left (a+b \sqrt {x}+c x\right )^3} \, dx=\frac {-24 \sqrt {x}\, \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c -24 \sqrt {x}\, \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{2} x -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b c -24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} x -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c x -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{3} x^{2}-16 \sqrt {x}\, a^{2} b \,c^{2}-4 \sqrt {x}\, a \,b^{3} c +2 \sqrt {x}\, b^{5}-20 a^{3} c^{2}+a^{2} b^{2} c +24 a^{2} c^{3} x +a \,b^{4}-30 a \,b^{2} c^{2} x +12 a \,c^{4} x^{2}+6 b^{4} c x -3 b^{2} c^{3} x^{2}}{128 \sqrt {x}\, a^{4} b \,c^{3}-96 \sqrt {x}\, a^{3} b^{3} c^{2}+128 \sqrt {x}\, a^{3} b \,c^{4} x +24 \sqrt {x}\, a^{2} b^{5} c -96 \sqrt {x}\, a^{2} b^{3} c^{3} x -2 \sqrt {x}\, a \,b^{7}+24 \sqrt {x}\, a \,b^{5} c^{2} x -2 \sqrt {x}\, b^{7} c x +64 a^{5} c^{3}-48 a^{4} b^{2} c^{2}+128 a^{4} c^{4} x +12 a^{3} b^{4} c -32 a^{3} b^{2} c^{3} x +64 a^{3} c^{5} x^{2}-a^{2} b^{6}-24 a^{2} b^{4} c^{2} x -48 a^{2} b^{2} c^{4} x^{2}+10 a \,b^{6} c x +12 a \,b^{4} c^{3} x^{2}-b^{8} x -b^{6} c^{2} x^{2}} \] Input:
int(1/(a+b*x^(1/2)+c*x)^3,x)
Output:
( - 24*sqrt(x)*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b**2 ))*a*b**2*c - 24*sqrt(x)*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4* a*c - b**2))*b**2*c**2*x - 12*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sq rt(4*a*c - b**2))*a**2*b*c - 24*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/ sqrt(4*a*c - b**2))*a*b*c**2*x - 12*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b**2))*b**3*c*x - 12*sqrt(4*a*c - b**2)*atan((2*sqrt(x)*c + b)/sqrt(4*a*c - b**2))*b*c**3*x**2 - 16*sqrt(x)*a**2*b*c**2 - 4*sqrt(x) *a*b**3*c + 2*sqrt(x)*b**5 - 20*a**3*c**2 + a**2*b**2*c + 24*a**2*c**3*x + a*b**4 - 30*a*b**2*c**2*x + 12*a*c**4*x**2 + 6*b**4*c*x - 3*b**2*c**3*x** 2)/(128*sqrt(x)*a**4*b*c**3 - 96*sqrt(x)*a**3*b**3*c**2 + 128*sqrt(x)*a**3 *b*c**4*x + 24*sqrt(x)*a**2*b**5*c - 96*sqrt(x)*a**2*b**3*c**3*x - 2*sqrt( x)*a*b**7 + 24*sqrt(x)*a*b**5*c**2*x - 2*sqrt(x)*b**7*c*x + 64*a**5*c**3 - 48*a**4*b**2*c**2 + 128*a**4*c**4*x + 12*a**3*b**4*c - 32*a**3*b**2*c**3* x + 64*a**3*c**5*x**2 - a**2*b**6 - 24*a**2*b**4*c**2*x - 48*a**2*b**2*c** 4*x**2 + 10*a*b**6*c*x + 12*a*b**4*c**3*x**2 - b**8*x - b**6*c**2*x**2)