Integrand size = 15, antiderivative size = 79 \[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=\frac {-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )}{\sqrt [3]{a^2}} \]
1/(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 = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=-\frac {(a+b x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{2 a^{2/3} \sqrt [3]{(a+b x)^2}} \]
-1/2*((a + b*x)^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3))/ Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b *x)^(1/3) + (a + b*x)^(2/3)]))/(a^(2/3)*((a + b*x)^2)^(1/3))
Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2008, 69, 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.
| \(\displaystyle \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx\) |
| \(\Big \downarrow \) 2008 |
| \(\displaystyle \frac {(a+b x)^{2/3} \int \frac {1}{x (a+b x)^{2/3}}dx}{\sqrt [3]{(a+b x)^2}}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {(a+b x)^{2/3} \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}\right )}{\sqrt [3]{(a+b x)^2}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(a+b x)^{2/3} \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{\sqrt [3]{(a+b x)^2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(a+b x)^{2/3} \left (\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{\sqrt [3]{(a+b x)^2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(a+b x)^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{\sqrt [3]{(a+b x)^2}}\) |
((a + b*x)^(2/3)*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3))/Sqrt [3]])/a^(2/3)) - Log[x]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x)^(1/3)])/( 2*a^(2/3))))/((a + b*x)^2)^(1/3)
3.2.51.3.1 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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) 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]
\[\int \frac {1}{x \left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (60) = 120\).
Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.85 \[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a \arctan \left (\frac {\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} {\left ({\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a\right )}}{3 \, {\left (a b x + a^{2}\right )}}\right ) - {\left (a^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right )}{2 \, a^{2}} \]
1/2*(2*sqrt(3)*(a^2)^(1/6)*a*arctan(1/3*sqrt(3)*(a^2)^(1/6)*((a^2)^(1/3)*( b*x + a) + 2*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a)/(a*b*x + a^2)) - (a^2)^(2/ 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)) + 2*(a^2)^(2/3)*log(-((a^2)^(1/3)*(b*x + a) - (b^2 *x^2 + 2*a*b*x + a^2)^(1/3)*a)/(b*x + a)))/a^2
\[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=\int \frac {1}{x \sqrt [3]{\left (a + b x\right )^{2}}}\, dx \]
\[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=\int { \frac {1}{{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {1}{3}} x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (60) = 120\).
Time = 3.15 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.77 \[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=-\frac {\sqrt {3} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}\right )}}{3 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}}\right )}{a \mathrm {sgn}\left (b x + a\right )} - \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \log \left ({\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} + {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}}\right )}{2 \, a \mathrm {sgn}\left (b x + a\right )} + \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} - \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \right |}\right )}{a \mathrm {sgn}\left (b x + a\right )} \]
-sqrt(3)*(a*sgn(b*x + a))^(1/3)*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))/(a *sgn(b*x + a)) - 1/2*(a*sgn(b*x + a))^(1/3)*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))/(a*sgn(b*x + a)) + (a*sgn(b*x + a))^(1/ 3)*log(abs((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))
Timed out. \[ \int \frac {1}{x \sqrt [3]{(a+b x)^2}} \, dx=\int \frac {1}{x\,{\left ({\left (a+b\,x\right )}^2\right )}^{1/3}} \,d x \]