Integrand size = 16, antiderivative size = 112 \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=-\frac {b \left (b+2 c \sqrt {x}\right ) \sqrt {a+b \sqrt {x}+c x}}{4 c^2}+\frac {2 \left (a+b \sqrt {x}+c x\right )^{3/2}}{3 c}+\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{8 c^{5/2}} \] Output:
-1/4*b*(b+2*c*x^(1/2))*(a+b*x^(1/2)+c*x)^(1/2)/c^2+2/3*(a+b*x^(1/2)+c*x)^( 3/2)/c+1/8*b*(-4*a*c+b^2)*arctanh(1/2*(b+2*c*x^(1/2))/c^(1/2)/(a+b*x^(1/2) +c*x)^(1/2))/c^(5/2)
Time = 0.33 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\frac {\sqrt {a+b \sqrt {x}+c x} \left (-3 b^2+2 b c \sqrt {x}+8 c (a+c x)\right )}{12 c^2}-\frac {\left (b^3-4 a b c\right ) \log \left (c^2 \left (b+2 c \sqrt {x}-2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}\right )\right )}{8 c^{5/2}} \] Input:
Integrate[Sqrt[a + b*Sqrt[x] + c*x],x]
Output:
(Sqrt[a + b*Sqrt[x] + c*x]*(-3*b^2 + 2*b*c*Sqrt[x] + 8*c*(a + c*x)))/(12*c ^2) - ((b^3 - 4*a*b*c)*Log[c^2*(b + 2*c*Sqrt[x] - 2*Sqrt[c]*Sqrt[a + b*Sqr t[x] + c*x])])/(8*c^(5/2))
Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1680, 1160, 1087, 1092, 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.
\(\displaystyle \int \sqrt {a+b \sqrt {x}+c x} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 2 \int \sqrt {x} \sqrt {a+c x+b \sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}+c x\right )^{3/2}}{3 c}-\frac {b \int \sqrt {a+c x+b \sqrt {x}}d\sqrt {x}}{2 c}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}+c x\right )^{3/2}}{3 c}-\frac {b \left (\frac {\left (b+2 c \sqrt {x}\right ) \sqrt {a+b \sqrt {x}+c x}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {a+c x+b \sqrt {x}}}d\sqrt {x}}{8 c}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}+c x\right )^{3/2}}{3 c}-\frac {b \left (\frac {\left (b+2 c \sqrt {x}\right ) \sqrt {a+b \sqrt {x}+c x}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x}d\frac {b+2 c \sqrt {x}}{\sqrt {a+c x+b \sqrt {x}}}}{4 c}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}+c x\right )^{3/2}}{3 c}-\frac {b \left (\frac {\left (b+2 c \sqrt {x}\right ) \sqrt {a+b \sqrt {x}+c x}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{8 c^{3/2}}\right )}{2 c}\right )\) |
Input:
Int[Sqrt[a + b*Sqrt[x] + c*x],x]
Output:
2*((a + b*Sqrt[x] + c*x)^(3/2)/(3*c) - (b*(((b + 2*c*Sqrt[x])*Sqrt[a + b*S qrt[x] + c*x])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Sqrt[x])/(2*Sqrt[c] *Sqrt[a + b*Sqrt[x] + c*x])])/(8*c^(3/2))))/(2*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)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
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 = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 \left (a +b \sqrt {x}+c x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (b +2 c \sqrt {x}\right ) \sqrt {a +b \sqrt {x}+c x}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \sqrt {x}}{\sqrt {c}}+\sqrt {a +b \sqrt {x}+c x}\right )}{8 c^{\frac {3}{2}}}\right )}{c}\) | \(93\) |
default | \(\frac {2 \left (a +b \sqrt {x}+c x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (b +2 c \sqrt {x}\right ) \sqrt {a +b \sqrt {x}+c x}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \sqrt {x}}{\sqrt {c}}+\sqrt {a +b \sqrt {x}+c x}\right )}{8 c^{\frac {3}{2}}}\right )}{c}\) | \(93\) |
Input:
int((a+b*x^(1/2)+c*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*(a+b*x^(1/2)+c*x)^(3/2)/c-b/c*(1/4*(b+2*c*x^(1/2))/c*(a+b*x^(1/2)+c*x) ^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x^(1/2))/c^(1/2)+(a+b*x^(1/2)+c *x)^(1/2)))
Timed out. \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\text {Timed out} \] Input:
integrate((a+b*x^(1/2)+c*x)^(1/2),x, algorithm="fricas")
Output:
Timed out
Time = 0.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.71 \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=2 \left (\begin {cases} \left (- \frac {a b}{12 c} - \frac {b \left (\frac {a}{3} - \frac {b^{2}}{8 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b \sqrt {x} + c x} + 2 c \sqrt {x} \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + \sqrt {x}\right ) \log {\left (\frac {b}{2 c} + \sqrt {x} \right )}}{\sqrt {c \left (\frac {b}{2 c} + \sqrt {x}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a + b \sqrt {x} + c x} \left (\frac {b \sqrt {x}}{12 c} + \frac {x}{3} + \frac {\frac {a}{3} - \frac {b^{2}}{8 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (- \frac {a \left (a + b \sqrt {x}\right )^{\frac {3}{2}}}{3} + \frac {\left (a + b \sqrt {x}\right )^{\frac {5}{2}}}{5}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x}{2} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b*x**(1/2)+c*x)**(1/2),x)
Output:
2*Piecewise(((-a*b/(12*c) - b*(a/3 - b**2/(8*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*sqrt(x) + c*x) + 2*c*sqrt(x))/sqrt(c), Ne(a - b**2/( 4*c), 0)), ((b/(2*c) + sqrt(x))*log(b/(2*c) + sqrt(x))/sqrt(c*(b/(2*c) + s qrt(x))**2), True)) + sqrt(a + b*sqrt(x) + c*x)*(b*sqrt(x)/(12*c) + x/3 + (a/3 - b**2/(8*c))/c), Ne(c, 0)), (2*(-a*(a + b*sqrt(x))**(3/2)/3 + (a + b *sqrt(x))**(5/2)/5)/b**2, Ne(b, 0)), (sqrt(a)*x/2, True))
\[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\int { \sqrt {c x + b \sqrt {x} + a} \,d x } \] Input:
integrate((a+b*x^(1/2)+c*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x + b*sqrt(x) + a), x)
Exception generated. \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^(1/2)+c*x)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueDone
Timed out. \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\int \sqrt {a+c\,x+b\,\sqrt {x}} \,d x \] Input:
int((a + c*x + b*x^(1/2))^(1/2),x)
Output:
int((a + c*x + b*x^(1/2))^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41 \[ \int \sqrt {a+b \sqrt {x}+c x} \, dx=\frac {4 \sqrt {x}\, \sqrt {\sqrt {x}\, b +a +c x}\, b \,c^{2}+16 \sqrt {\sqrt {x}\, b +a +c x}\, a \,c^{2}-6 \sqrt {\sqrt {x}\, b +a +c x}\, b^{2} c +16 \sqrt {\sqrt {x}\, b +a +c x}\, c^{3} x -12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {\sqrt {x}\, b +a +c x}+2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) a b c +3 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {\sqrt {x}\, b +a +c x}+2 \sqrt {x}\, c +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3}}{24 c^{3}} \] Input:
int((a+b*x^(1/2)+c*x)^(1/2),x)
Output:
(4*sqrt(x)*sqrt(sqrt(x)*b + a + c*x)*b*c**2 + 16*sqrt(sqrt(x)*b + a + c*x) *a*c**2 - 6*sqrt(sqrt(x)*b + a + c*x)*b**2*c + 16*sqrt(sqrt(x)*b + a + c*x )*c**3*x - 12*sqrt(c)*log((2*sqrt(c)*sqrt(sqrt(x)*b + a + c*x) + 2*sqrt(x) *c + b)/sqrt(4*a*c - b**2))*a*b*c + 3*sqrt(c)*log((2*sqrt(c)*sqrt(sqrt(x)* b + a + c*x) + 2*sqrt(x)*c + b)/sqrt(4*a*c - b**2))*b**3)/(24*c**3)