Integrand size = 26, antiderivative size = 189 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {9 a b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {9 a^2 b \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {3 b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log \left (\sqrt [3]{x}\right )}{a+\frac {b}{\sqrt [3]{x}}} \]
9*a*b^2*x^(1/3)*(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))+9/2*a^ 2*b*x^(2/3)*(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))+a^3*x*(a^2 +b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))+b^3*ln(x)*(a^2+b^2/x^(2/3) +2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {\left (b+a \sqrt [3]{x}\right ) \left (18 a b^2 \sqrt [3]{x}+9 a^2 b x^{2/3}+2 a^3 x+2 b^3 \log (x)\right )}{2 \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} \sqrt [3]{x}} \]
((b + a*x^(1/3))*(18*a*b^2*x^(1/3) + 9*a^2*b*x^(2/3) + 2*a^3*x + 2*b^3*Log [x]))/(2*Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*x^(1/3))
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1384, 774, 27, 795, 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.
\(\displaystyle \int \left (a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \int \left (\frac {b^2}{\sqrt [3]{x}}+a b\right )^3dx}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
\(\Big \downarrow \) 774 |
\(\displaystyle \frac {3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \int b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^{2/3}d\sqrt [3]{x}}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^{2/3}d\sqrt [3]{x}}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \frac {3 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \int \frac {\left (\sqrt [3]{x} a+b\right )^3}{\sqrt [3]{x}}d\sqrt [3]{x}}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {3 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \int \left (x^{2/3} a^3+3 b \sqrt [3]{x} a^2+3 b^2 a+\frac {b^3}{\sqrt [3]{x}}\right )d\sqrt [3]{x}}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (\frac {a^3 x}{3}+\frac {3}{2} a^2 b x^{2/3}+3 a b^2 \sqrt [3]{x}+b^3 \log \left (\sqrt [3]{x}\right )\right )}{a b^3+\frac {b^4}{\sqrt [3]{x}}}\) |
(3*b^3*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(3*a*b^2*x^(1/3) + (3*a^2 *b*x^(2/3))/2 + (a^3*x)/3 + b^3*Log[x^(1/3)]))/(a*b^3 + b^4/x^(1/3))
3.5.84.3.1 Defintions of rubi rules used
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[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 = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.37
method | result | size |
derivativedivides | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (2 a^{3} x +9 a^{2} b \,x^{\frac {2}{3}}+2 b^{3} \ln \left (x \right )+18 b^{2} a \,x^{\frac {1}{3}}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{3}}\) | \(69\) |
default | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (2 a^{3} x +9 a^{2} b \,x^{\frac {2}{3}}+2 b^{3} \ln \left (x \right )+18 b^{2} a \,x^{\frac {1}{3}}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{3}}\) | \(69\) |
1/2*((x^(2/3)*a^2+2*a*b*x^(1/3)+b^2)/x^(2/3))^(3/2)*x*(2*a^3*x+9*a^2*b*x^( 2/3)+2*b^3*ln(x)+18*b^2*a*x^(1/3))/(b+a*x^(1/3))^3
Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\int \left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.16 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=a^{3} x + b^{3} \log \left (x\right ) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} + 9 \, a b^{2} x^{\frac {1}{3}} \]
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.42 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=a^{3} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 9 \, a b^{2} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) \]
a^3*x*sgn(a*x + b*x^(2/3))*sgn(x) + b^3*log(abs(x))*sgn(a*x + b*x^(2/3))*s gn(x) + 9/2*a^2*b*x^(2/3)*sgn(a*x + b*x^(2/3))*sgn(x) + 9*a*b^2*x^(1/3)*sg n(a*x + b*x^(2/3))*sgn(x)
Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\int {\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{3/2} \,d x \]