Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {3}{2} \sqrt [3]{(a+b x)^2}+\frac {a \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{\sqrt [3]{a^2}} \]
3/2*((b*x+a)^2)^(1/3)+a/(a^2)^(1/3)*(3/2*ln(((b*x+a)^(1/3)-a^(1/3))/x^(1/3 ))+3^(1/2)*arctan(3^(1/2)*(b*x+a)^(1/3)/((b*x+a)^(1/3)+2*a^(1/3))))
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {\sqrt [3]{(a+b x)^2} \left (3 (a+b x)^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{2 (a+b x)^{2/3}} \]
(((a + b*x)^2)^(1/3)*(3*(a + b*x)^(2/3) + 2*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2 *(a + b*x)^(1/3))/a^(1/3))/Sqrt[3]] + 2*a^(2/3)*Log[a^(1/3) - (a + b*x)^(1 /3)] - a^(2/3)*Log[a^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)]))/ (2*(a + b*x)^(2/3))
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2008, 60, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx\) |
| \(\Big \downarrow \) 2008 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \int \frac {(a+b x)^{2/3}}{x}dx}{(a+b x)^{2/3}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \left (a \int \frac {1}{x \sqrt [3]{a+b x}}dx+\frac {3}{2} (a+b x)^{2/3}\right )}{(a+b x)^{2/3}}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \left (a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\right )}{(a+b x)^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \left (a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\right )}{(a+b x)^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \left (a \left (-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\right )}{(a+b x)^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt [3]{(a+b x)^2} \left (a \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\right )}{(a+b x)^{2/3}}\) |
(((a + b*x)^2)^(1/3)*((3*(a + b*x)^(2/3))/2 + a*((Sqrt[3]*ArcTan[(1 + (2*( a + b*x)^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[x]/(2*a^(1/3)) + (3*Log[a ^(1/3) - (a + b*x)^(1/3)])/(2*a^(1/3)))))/(a + b*x)^(2/3)
3.2.58.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Simp[((a + b*x)^Exp on[Px, x])^p/(a + b*x)^(Expon[Px, x]*p) Int[u*(a + b*x)^(Expon[Px, x]*p), x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; !IntegerQ[p] && PolyQ[Px, x ] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
\[\int \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{x}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (73) = 146\).
Time = 0.25 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.51 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=-\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (a b x + a^{2}\right )} + 2 \, \sqrt {3} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (a b x + a^{2}\right )}}\right ) - \frac {1}{2} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) + \frac {3}{2} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} \]
-sqrt(3)*(a^2)^(1/3)*arctan(1/3*(sqrt(3)*(a*b*x + a^2) + 2*sqrt(3)*(b^2*x^ 2 + 2*a*b*x + a^2)^(1/3)*(a^2)^(2/3))/(a*b*x + a^2)) - 1/2*(a^2)^(1/3)*log (((b^2*x^2 + 2*a*b*x + a^2)^(2/3)*a^2 + (b^2*x^2 + 2*a*b*x + a^2)^(1/3)*(a *b*x + a^2)*(a^2)^(1/3) + (b^2*x^2 + 2*a*b*x + a^2)*(a^2)^(2/3))/(b^2*x^2 + 2*a*b*x + a^2)) + (a^2)^(1/3)*log(-((a^2)^(1/3)*(b*x + a) - (b^2*x^2 + 2 *a*b*x + a^2)^(1/3)*a)/(b*x + a)) + 3/2*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)
\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int \frac {\sqrt [3]{\left (a + b x\right )^{2}}}{x}\, dx \]
\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int { \frac {{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {1}{3}}}{x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (73) = 146\).
Time = 3.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.62 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {1}{2} \, {\left (\frac {2 \, \sqrt {3} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}\right )}}{3 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}}\right )}{\mathrm {sgn}\left (b x + a\right )} - \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \log \left ({\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} + {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}}\right )}{\mathrm {sgn}\left (b x + a\right )} + \frac {2 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \log \left ({\left | {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} - \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \right |}\right )}{\mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}}}{\mathrm {sgn}\left (b x + a\right )}\right )} \mathrm {sgn}\left (b x + a\right ) \]
1/2*(2*sqrt(3)*(a*sgn(b*x + a))^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x*sgn(b*x + a) + a*sgn(b*x + a))^(1/3) + (a*sgn(b*x + a))^(1/3))/(a*sgn(b*x + a))^(1/ 3))/sgn(b*x + a) - (a*sgn(b*x + a))^(2/3)*log((b*x*sgn(b*x + a) + a*sgn(b* x + a))^(2/3) + (b*x*sgn(b*x + a) + a*sgn(b*x + a))^(1/3)*(a*sgn(b*x + a)) ^(1/3) + (a*sgn(b*x + a))^(2/3))/sgn(b*x + a) + 2*(a*sgn(b*x + a))^(2/3)*l og(abs((b*x*sgn(b*x + a) + a*sgn(b*x + a))^(1/3) - (a*sgn(b*x + a))^(1/3)) )/sgn(b*x + a) + 3*(b*x*sgn(b*x + a) + a*sgn(b*x + a))^(2/3)/sgn(b*x + a)) *sgn(b*x + a)
Timed out. \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{1/3}}{x} \,d x \]