\(\int \frac {(a+b x)^{2/3}}{x} \, dx\) [382]

Optimal result

Integrand size = 13, antiderivative size = 92 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3}{2} (a+b x)^{2/3}+\sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]

[Out]

3/2*(b*x+a)^(2/3)-1/2*a^(2/3)*ln(x)+3/2*a^(2/3)*ln(a^(1/3)-(b*x+a)^(1/3))+a^(2/3)*arctan(1/3*(a^(1/3)+2*(b*x+a
)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 57, 631, 210, 31} \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} a^{2/3} \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )+\frac {3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{2} (a+b x)^{2/3} \]

[In]

Int[(a + b*x)^(2/3)/x,x]

[Out]

(3*(a + b*x)^(2/3))/2 + Sqrt[3]*a^(2/3)*ArcTan[(a^(1/3) + 2*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))] - (a^(2/3)*Log
[x])/2 + (3*a^(2/3)*Log[a^(1/3) - (a + b*x)^(1/3)])/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[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]

Rule 210

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])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} (a+b x)^{2/3}+a \int \frac {1}{x \sqrt [3]{a+b x}} \, dx \\ & = \frac {3}{2} (a+b x)^{2/3}-\frac {1}{2} a^{2/3} \log (x)-\frac {1}{2} \left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )+\frac {1}{2} (3 a) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right ) \\ & = \frac {3}{2} (a+b x)^{2/3}-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right ) \\ & = \frac {3}{2} (a+b x)^{2/3}+\sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3}{2} (a+b x)^{2/3}+\sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right ) \]

[In]

Integrate[(a + b*x)^(2/3)/x,x]

[Out]

(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]] + 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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+a^{\frac {2}{3}} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {a^{\frac {2}{3}} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}+a^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\)	\(86\)
derivativedivides	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right ) a\)	\(90\)
default	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right ) a\)	\(90\)

[In]

int((b*x+a)^(2/3)/x,x,method=_RETURNVERBOSE)

[Out]

3/2*(b*x+a)^(2/3)+a^(2/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/2*a^(2/3)*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3)
)+a^(2/3)*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{2} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) + {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]

[In]

integrate((b*x+a)^(2/3)/x,x, algorithm="fricas")

[Out]

sqrt(3)*(a^2)^(1/3)*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(a^2)^(1/3)*(b*x + a)^(1/3))/a) - 1/2*(a^2)^(1/3)*log((b
*x + a)^(2/3)*a + (a^2)^(1/3)*a + (a^2)^(2/3)*(b*x + a)^(1/3)) + (a^2)^(1/3)*log((b*x + a)^(1/3)*a - (a^2)^(2/
3)) + 3/2*(b*x + a)^(2/3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.74 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {5 a^{\frac {2}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 a^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 a^{\frac {2}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 b^{\frac {2}{3}} \left (\frac {a}{b} + x\right )^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )}{2 \Gamma \left (\frac {8}{3}\right )} \]

[In]

integrate((b*x+a)**(2/3)/x,x)

[Out]

5*a**(2/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(5/3)/(3*gamma(8/3)) + 5*a**(2/3)*exp(2*I*pi/3)*lo
g(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(5/3)/(3*gamma(8/3)) + 5*a**(2/3)*exp(-2*I*
pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(5/3)/(3*gamma(8/3)) + 5*b**(2/3)*(
a/b + x)**(2/3)*gamma(5/3)/(2*gamma(8/3))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]

[In]

integrate((b*x+a)^(2/3)/x,x, algorithm="maxima")

[Out]

sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(2/3)*log((b*x + a)^(2/3) +
(b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a^(2/3)*log((b*x + a)^(1/3) - a^(1/3)) + 3/2*(b*x + a)^(2/3)

Giac [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {2}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]

[In]

integrate((b*x+a)^(2/3)/x,x, algorithm="giac")

[Out]

sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(2/3)*log((b*x + a)^(2/3) +
(b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a^(2/3)*log(abs((b*x + a)^(1/3) - a^(1/3))) + 3/2*(b*x + a)^(2/3)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3\,{\left (a+b\,x\right )}^{2/3}}{2}+a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{7/3}\right )+\frac {a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{7/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{7/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]

[In]

int((a + b*x)^(2/3)/x,x)

[Out]

(3*(a + b*x)^(2/3))/2 + a^(2/3)*log(9*a^2*(a + b*x)^(1/3) - 9*a^(7/3)) + (a^(2/3)*log(9*a^2*(a + b*x)^(1/3) -
(9*a^(7/3)*(3^(1/2)*1i - 1)^2)/4)*(3^(1/2)*1i - 1))/2 - (a^(2/3)*log(9*a^2*(a + b*x)^(1/3) - (9*a^(7/3)*(3^(1/
2)*1i + 1)^2)/4)*(3^(1/2)*1i + 1))/2