\(\int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx\) [158]

Optimal result

Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {3}{2} \sqrt [3]{(a+b x)^2}+\frac {a \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{\sqrt [3]{a^2}} \]

[Out]

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))))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1973, 52, 57, 632, 210, 31} \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {\sqrt {3} \sqrt [3]{(a+b x)^2} \arctan \left (\frac {2 \sqrt [3]{\frac {b x}{a}+1}+1}{\sqrt {3}}\right )}{\left (\frac {b x}{a}+1\right )^{2/3}}+\frac {3}{2} \sqrt [3]{(a+b x)^2}-\frac {\log (x) \sqrt [3]{(a+b x)^2}}{2 \left (\frac {b x}{a}+1\right )^{2/3}}+\frac {3 \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{\frac {b x}{a}+1}\right )}{2 \left (\frac {b x}{a}+1\right )^{2/3}} \]

[In]

Int[((a + b*x)^2)^(1/3)/x,x]

[Out]

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

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 632

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

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {\sqrt [3]{(a+b x)^2} \left (3 (a+b x)^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{2 (a+b x)^{2/3}} \]

[In]

Integrate[((a + b*x)^2)^(1/3)/x,x]

[Out]

(((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))

Maple [F]

\[\int \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{x}d x\]

[In]

int(((b*x+a)^2)^(1/3)/x,x)

[Out]

int(((b*x+a)^2)^(1/3)/x,x)

Fricas [B] (verification not implemented)

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

[In]

integrate(((b*x+a)^2)^(1/3)/x,x, algorithm="fricas")

[Out]

-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)*lo
g(-((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
)

Sympy [F]

\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int \frac {\sqrt [3]{\left (a + b x\right )^{2}}}{x}\, dx \]

[In]

integrate(((b*x+a)**2)**(1/3)/x,x)

[Out]

Integral(((a + b*x)**2)**(1/3)/x, x)

Maxima [F]

\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int { \frac {{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {1}{3}}}{x} \,d x } \]

[In]

integrate(((b*x+a)^2)^(1/3)/x,x, algorithm="maxima")

[Out]

integrate(((b*x + a)^2)^(1/3)/x, x)

Giac [B] (verification not implemented)

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 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\frac {1}{2} \, {\left (\frac {2 \, \sqrt {3} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}\right )}}{3 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}}\right )}{\mathrm {sgn}\left (b x + a\right )} - \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \log \left ({\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} + {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}}\right )}{\mathrm {sgn}\left (b x + a\right )} + \frac {2 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} \log \left ({\left | {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} - \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \right |}\right )}{\mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}}}{\mathrm {sgn}\left (b x + a\right )}\right )} \mathrm {sgn}\left (b x + a\right ) \]

[In]

integrate(((b*x+a)^2)^(1/3)/x,x, algorithm="giac")

[Out]

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))/s
gn(b*x + a) + 2*(a*sgn(b*x + a))^(2/3)*log(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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x} \, dx=\int \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{1/3}}{x} \,d x \]

[In]

int(((a + b*x)^2)^(1/3)/x,x)

[Out]

int(((a + b*x)^2)^(1/3)/x, x)