Integrand size = 26, antiderivative size = 147 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=-\frac {3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \] Output:
-3*a*(a+b*x^(1/3))*x^(1/3)/b^2/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)+3/2*( a+b*x^(1/3))*x^(2/3)/b/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)+3*a^2*(a+b*x^ (1/3))*ln(a+b*x^(1/3))/b^3/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \left (a+b \sqrt [3]{x}\right ) \left (b \left (-2 a+b \sqrt [3]{x}\right ) \sqrt [3]{x}+2 a^2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \] Input:
Integrate[1/Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
Output:
(3*(a + b*x^(1/3))*(b*(-2*a + b*x^(1/3))*x^(1/3) + 2*a^2*Log[a + b*x^(1/3) ]))/(2*b^3*Sqrt[(a + b*x^(1/3))^2])
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1384, 774, 27, 49, 2009}
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.
\(\displaystyle \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\left (a b+b^2 \sqrt [3]{x}\right ) \int \frac {1}{\sqrt [3]{x} b^2+a b}dx}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\) |
\(\Big \downarrow \) 774 |
\(\displaystyle \frac {3 \left (a b+b^2 \sqrt [3]{x}\right ) \int \frac {x^{2/3}}{b \left (a+b \sqrt [3]{x}\right )}d\sqrt [3]{x}}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \left (a b+b^2 \sqrt [3]{x}\right ) \int \frac {x^{2/3}}{a+b \sqrt [3]{x}}d\sqrt [3]{x}}{b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {3 \left (a b+b^2 \sqrt [3]{x}\right ) \int \left (\frac {a^2}{b^2 \left (a+b \sqrt [3]{x}\right )}-\frac {a}{b^2}+\frac {\sqrt [3]{x}}{b}\right )d\sqrt [3]{x}}{b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \left (a b+b^2 \sqrt [3]{x}\right ) \left (\frac {a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {a \sqrt [3]{x}}{b^2}+\frac {x^{2/3}}{2 b}\right )}{b \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\) |
Input:
Int[1/Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
Output:
(3*(a*b + b^2*x^(1/3))*(-((a*x^(1/3))/b^2) + x^(2/3)/(2*b) + (a^2*Log[a + b*x^(1/3)])/b^3))/(b*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])
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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.35
method | result | size |
derivativedivides | \(\frac {3 \left (a +b \,x^{\frac {1}{3}}\right ) \left (b^{2} x^{\frac {2}{3}}+2 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )-2 a b \,x^{\frac {1}{3}}\right )}{2 \sqrt {\left (a +b \,x^{\frac {1}{3}}\right )^{2}}\, b^{3}}\) | \(52\) |
default | \(\frac {\left (a +b \,x^{\frac {1}{3}}\right ) \left (3 b^{2} x^{\frac {2}{3}}-6 a b \,x^{\frac {1}{3}}+2 a^{2} \ln \left (b^{3} x +a^{3}\right )-2 a^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )+4 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )\right )}{2 \sqrt {a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}}\, b^{3}}\) | \(101\) |
Input:
int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x,method=_RETURNVERBOSE)
Output:
3/2*(a+b*x^(1/3))*(b^2*x^(2/3)+2*a^2*ln(a+b*x^(1/3))-2*a*b*x^(1/3))/((a+b* x^(1/3))^2)^(1/2)/b^3
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, {\left (2 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right ) + b^{2} x^{\frac {2}{3}} - 2 \, a b x^{\frac {1}{3}}\right )}}{2 \, b^{3}} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x, algorithm="fricas")
Output:
3/2*(2*a^2*log(b*x^(1/3) + a) + b^2*x^(2/3) - 2*a*b*x^(1/3))/b^3
Time = 0.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=3 \left (\begin {cases} \frac {a^{2} \left (\frac {a}{b} + \sqrt [3]{x}\right ) \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{b^{2} \sqrt {b^{2} \left (\frac {a}{b} + \sqrt [3]{x}\right )^{2}}} + \left (- \frac {3 a}{2 b^{3}} + \frac {\sqrt [3]{x}}{2 b^{2}}\right ) \sqrt {a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}} & \text {for}\: b^{2} \neq 0 \\\frac {a^{4} \sqrt {a^{2} + 2 a b \sqrt [3]{x}} - \frac {2 a^{2} \left (a^{2} + 2 a b \sqrt [3]{x}\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b \sqrt [3]{x}\right )^{\frac {5}{2}}}{5}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x}{3 \sqrt {a^{2}}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(1/2),x)
Output:
3*Piecewise((a**2*(a/b + x**(1/3))*log(a/b + x**(1/3))/(b**2*sqrt(b**2*(a/ b + x**(1/3))**2)) + (-3*a/(2*b**3) + x**(1/3)/(2*b**2))*sqrt(a**2 + 2*a*b *x**(1/3) + b**2*x**(2/3)), Ne(b**2, 0)), ((a**4*sqrt(a**2 + 2*a*b*x**(1/3 )) - 2*a**2*(a**2 + 2*a*b*x**(1/3))**(3/2)/3 + (a**2 + 2*a*b*x**(1/3))**(5 /2)/5)/(4*a**3*b**3), Ne(a*b, 0)), (x/(3*sqrt(a**2)), True))
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, a^{2} \log \left (x^{\frac {1}{3}} + \frac {a}{b}\right )}{b^{3}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {3 \, a x^{\frac {1}{3}}}{b^{2}} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x, algorithm="maxima")
Output:
3*a^2*log(x^(1/3) + a/b)/b^3 + 3/2*x^(2/3)/b - 3*a*x^(1/3)/b^2
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {3 \, {\left (b x^{\frac {2}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) - 2 \, a x^{\frac {1}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )\right )}}{2 \, b^{2}} + \frac {3 \, a^{2} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \] Input:
integrate(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x, algorithm="giac")
Output:
3/2*(b*x^(2/3)*sgn(b*x^(1/3) + a) - 2*a*x^(1/3)*sgn(b*x^(1/3) + a))/b^2 + 3*a^2*log(abs(b*x^(1/3) + a))/(b^3*sgn(b*x^(1/3) + a))
Timed out. \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\int \frac {1}{\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}} \,d x \] Input:
int(1/(a^2 + b^2*x^(2/3) + 2*a*b*x^(1/3))^(1/2),x)
Output:
int(1/(a^2 + b^2*x^(2/3) + 2*a*b*x^(1/3))^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx=\frac {\frac {3 x^{\frac {2}{3}} b^{2}}{2}-3 x^{\frac {1}{3}} a b +3 \,\mathrm {log}\left (x^{\frac {1}{3}} b +a \right ) a^{2}}{b^{3}} \] Input:
int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x)
Output:
(3*(x**(2/3)*b**2 - 2*x**(1/3)*a*b + 2*log(x**(1/3)*b + a)*a**2))/(2*b**3)