Integrand size = 20, antiderivative size = 99 \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=-\frac {3 \left (3 b-2 c \sqrt [3]{x}\right ) \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{4 c^2}+\frac {3 \left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \sqrt [3]{x}}{2 \sqrt {c} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}\right )}{8 c^{5/2}} \] Output:
-3/4*(3*b-2*c*x^(1/3))*(a+b*x^(1/3)+c*x^(2/3))^(1/2)/c^2+3/8*(-4*a*c+3*b^2 )*arctanh(1/2*(b+2*c*x^(1/3))/c^(1/2)/(a+b*x^(1/3)+c*x^(2/3))^(1/2))/c^(5/ 2)
Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\frac {3 \left (-3 b+2 c \sqrt [3]{x}\right ) \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{4 c^2}+\frac {3 \left (-3 b^2+4 a c\right ) \log \left (b c^2-2 c^{5/2} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}+2 c^3 \sqrt [3]{x}\right )}{8 c^{5/2}} \] Input:
Integrate[1/Sqrt[a + b*x^(1/3) + c*x^(2/3)],x]
Output:
(3*(-3*b + 2*c*x^(1/3))*Sqrt[a + b*x^(1/3) + c*x^(2/3)])/(4*c^2) + (3*(-3* b^2 + 4*a*c)*Log[b*c^2 - 2*c^(5/2)*Sqrt[a + b*x^(1/3) + c*x^(2/3)] + 2*c^3 *x^(1/3)])/(8*c^(5/2))
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1680, 1166, 27, 1160, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 3 \int \frac {x^{2/3}}{\sqrt {a+c x^{2/3}+b \sqrt [3]{x}}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle 3 \left (\frac {\int -\frac {2 a+3 b \sqrt [3]{x}}{2 \sqrt {a+c x^{2/3}+b \sqrt [3]{x}}}d\sqrt [3]{x}}{2 c}+\frac {\sqrt [3]{x} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{2 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \left (\frac {\sqrt [3]{x} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{2 c}-\frac {\int \frac {2 a+3 b \sqrt [3]{x}}{\sqrt {a+c x^{2/3}+b \sqrt [3]{x}}}d\sqrt [3]{x}}{4 c}\right )\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 3 \left (\frac {\sqrt [3]{x} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{2 c}-\frac {\frac {3 b \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{c}-\frac {\left (3 b^2-4 a c\right ) \int \frac {1}{\sqrt {a+c x^{2/3}+b \sqrt [3]{x}}}d\sqrt [3]{x}}{2 c}}{4 c}\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle 3 \left (\frac {\sqrt [3]{x} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{2 c}-\frac {\frac {3 b \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{c}-\frac {\left (3 b^2-4 a c\right ) \int \frac {1}{4 c-x^{2/3}}d\frac {b+2 c \sqrt [3]{x}}{\sqrt {a+c x^{2/3}+b \sqrt [3]{x}}}}{c}}{4 c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 3 \left (\frac {\sqrt [3]{x} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{2 c}-\frac {\frac {3 b \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}{c}-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \sqrt [3]{x}}{2 \sqrt {c} \sqrt {a+b \sqrt [3]{x}+c x^{2/3}}}\right )}{2 c^{3/2}}}{4 c}\right )\) |
Input:
Int[1/Sqrt[a + b*x^(1/3) + c*x^(2/3)],x]
Output:
3*((Sqrt[a + b*x^(1/3) + c*x^(2/3)]*x^(1/3))/(2*c) - ((3*b*Sqrt[a + b*x^(1 /3) + c*x^(2/3)])/c - ((3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^(1/3))/(2*Sqrt[c ]*Sqrt[a + b*x^(1/3) + c*x^(2/3)])])/(2*c^(3/2)))/(4*c))
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[(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[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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
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 = 119, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {3 x^{\frac {1}{3}} \sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}}{2 c}-\frac {9 b \left (\frac {\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{\frac {1}{3}}}{\sqrt {c}}+\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {3 a \ln \left (\frac {\frac {b}{2}+c \,x^{\frac {1}{3}}}{\sqrt {c}}+\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}\right )}{2 c^{\frac {3}{2}}}\) | \(119\) |
default | \(\frac {3 x^{\frac {1}{3}} \sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}}{2 c}-\frac {9 b \left (\frac {\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{\frac {1}{3}}}{\sqrt {c}}+\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {3 a \ln \left (\frac {\frac {b}{2}+c \,x^{\frac {1}{3}}}{\sqrt {c}}+\sqrt {a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}}\right )}{2 c^{\frac {3}{2}}}\) | \(119\) |
Input:
int(1/(a+b*x^(1/3)+c*x^(2/3))^(1/2),x,method=_RETURNVERBOSE)
Output:
3/2*x^(1/3)/c*(a+b*x^(1/3)+c*x^(2/3))^(1/2)-9/4*b/c*(1/c*(a+b*x^(1/3)+c*x^ (2/3))^(1/2)-1/2*b/c^(3/2)*ln((1/2*b+c*x^(1/3))/c^(1/2)+(a+b*x^(1/3)+c*x^( 2/3))^(1/2)))-3/2*a/c^(3/2)*ln((1/2*b+c*x^(1/3))/c^(1/2)+(a+b*x^(1/3)+c*x^ (2/3))^(1/2))
Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*x^(1/3)+c*x^(2/3))^(1/2),x, algorithm="fricas")
Output:
Timed out
Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.01 \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=3 \left (\begin {cases} \left (- \frac {a}{2 c} + \frac {3 b^{2}}{8 c^{2}}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b \sqrt [3]{x} + c x^{\frac {2}{3}}} + 2 c \sqrt [3]{x} \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + \sqrt [3]{x}\right ) \log {\left (\frac {b}{2 c} + \sqrt [3]{x} \right )}}{\sqrt {c \left (\frac {b}{2 c} + \sqrt [3]{x}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (- \frac {3 b}{4 c^{2}} + \frac {\sqrt [3]{x}}{2 c}\right ) \sqrt {a + b \sqrt [3]{x} + c x^{\frac {2}{3}}} & \text {for}\: c \neq 0 \\\frac {2 \left (a^{2} \sqrt {a + b \sqrt [3]{x}} - \frac {2 a \left (a + b \sqrt [3]{x}\right )^{\frac {3}{2}}}{3} + \frac {\left (a + b \sqrt [3]{x}\right )^{\frac {5}{2}}}{5}\right )}{b^{3}} & \text {for}\: b \neq 0 \\\frac {x}{3 \sqrt {a}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(1/(a+b*x**(1/3)+c*x**(2/3))**(1/2),x)
Output:
3*Piecewise(((-a/(2*c) + 3*b**2/(8*c**2))*Piecewise((log(b + 2*sqrt(c)*sqr t(a + b*x**(1/3) + c*x**(2/3)) + 2*c*x**(1/3))/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x**(1/3))*log(b/(2*c) + x**(1/3))/sqrt(c*(b/(2*c) + x**( 1/3))**2), True)) + (-3*b/(4*c**2) + x**(1/3)/(2*c))*sqrt(a + b*x**(1/3) + c*x**(2/3)), Ne(c, 0)), (2*(a**2*sqrt(a + b*x**(1/3)) - 2*a*(a + b*x**(1/ 3))**(3/2)/3 + (a + b*x**(1/3))**(5/2)/5)/b**3, Ne(b, 0)), (x/(3*sqrt(a)), True))
Exception generated. \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+b*x^(1/3)+c*x^(2/3))^(1/2),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 = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\frac {3}{4} \, \sqrt {c x^{\frac {2}{3}} + b x^{\frac {1}{3}} + a} {\left (\frac {2 \, x^{\frac {1}{3}}}{c} - \frac {3 \, b}{c^{2}}\right )} - \frac {3 \, {\left (3 \, b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, \sqrt {c} {\left (\sqrt {c} x^{\frac {1}{3}} - \sqrt {c x^{\frac {2}{3}} + b x^{\frac {1}{3}} + a}\right )} + b \right |}\right )}{8 \, c^{\frac {5}{2}}} \] Input:
integrate(1/(a+b*x^(1/3)+c*x^(2/3))^(1/2),x, algorithm="giac")
Output:
3/4*sqrt(c*x^(2/3) + b*x^(1/3) + a)*(2*x^(1/3)/c - 3*b/c^2) - 3/8*(3*b^2 - 4*a*c)*log(abs(2*sqrt(c)*(sqrt(c)*x^(1/3) - sqrt(c*x^(2/3) + b*x^(1/3) + a)) + b))/c^(5/2)
Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\int \frac {1}{\sqrt {a+b\,x^{1/3}+c\,x^{2/3}}} \,d x \] Input:
int(1/(a + b*x^(1/3) + c*x^(2/3))^(1/2),x)
Output:
int(1/(a + b*x^(1/3) + c*x^(2/3))^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {a+b \sqrt [3]{x}+c x^{2/3}}} \, dx=\frac {\frac {3 x^{\frac {1}{3}} \sqrt {x^{\frac {2}{3}} c +x^{\frac {1}{3}} b +a}\, c^{2}}{2}-\frac {9 \sqrt {x^{\frac {2}{3}} c +x^{\frac {1}{3}} b +a}\, b c}{4}-\frac {3 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {x^{\frac {2}{3}} c +x^{\frac {1}{3}} b +a}+2 x^{\frac {1}{3}} c +b}{\sqrt {4 a c -b^{2}}}\right ) a c}{2}+\frac {9 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {x^{\frac {2}{3}} c +x^{\frac {1}{3}} b +a}+2 x^{\frac {1}{3}} c +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2}}{8}}{c^{3}} \] Input:
int(1/(a+b*x^(1/3)+c*x^(2/3))^(1/2),x)
Output:
(3*(4*x**(1/3)*sqrt(x**(2/3)*c + x**(1/3)*b + a)*c**2 - 6*sqrt(x**(2/3)*c + x**(1/3)*b + a)*b*c - 4*sqrt(c)*log((2*sqrt(c)*sqrt(x**(2/3)*c + x**(1/3 )*b + a) + 2*x**(1/3)*c + b)/sqrt(4*a*c - b**2))*a*c + 3*sqrt(c)*log((2*sq rt(c)*sqrt(x**(2/3)*c + x**(1/3)*b + a) + 2*x**(1/3)*c + b)/sqrt(4*a*c - b **2))*b**2))/(8*c**3)