\(\int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx\) [160]

Optimal result

Integrand size = 15, antiderivative size = 138 \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=\left (\frac {b}{6 a^3}-\frac {1}{2 a x^2}\right ) (a+b x)^{5/3}-\frac {b^2 \sqrt [3]{(a+b x)^2}}{6 a^2}-\frac {b^2 \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{9 a \sqrt [3]{a^2}} \]

[Out]

(-1/2/a/x^2+1/6*b/a^3)*(b*x+a)^(5/3)-1/6*b^2/a^2*((b*x+a)^2)^(1/3)-1/9*b^2/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 = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.34, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1973, 45} \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=\frac {b \log (x) \sqrt [3]{(a+b x)^3}}{a+b x}-\frac {a \sqrt [3]{(a+b x)^3}}{x (a+b x)} \]

[In]

Int[((a + b*x)^3)^(1/3)/x^2,x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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)^3} \int \frac {1+\frac {b x}{a}}{x^2} \, dx}{1+\frac {b x}{a}} \\ & = \frac {\sqrt [3]{(a+b x)^3} \int \left (\frac {1}{x^2}+\frac {b}{a x}\right ) \, dx}{1+\frac {b x}{a}} \\ & = -\frac {a \sqrt [3]{(a+b x)^3}}{x (a+b x)}+\frac {b \sqrt [3]{(a+b x)^3} \log (x)}{a+b x} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=\frac {\sqrt [3]{(a+b x)^3} (-a+b x \log (x))}{x (a+b x)} \]

[In]

Integrate[((a + b*x)^3)^(1/3)/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.32

method	result	size
risch	\(-\frac {\left (\left (b x +a \right )^{3}\right )^{\frac {1}{3}} a}{\left (b x +a \right ) x}+\frac {\left (\left (b x +a \right )^{3}\right )^{\frac {1}{3}} b \ln \left (x \right )}{b x +a}\)	\(44\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=\frac {b x \log \left (x\right ) - a}{x} \]

[In]

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

[Out]

(b*x*log(x) - a)/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=b \log \left (x\right ) - \frac {a}{x} \]

[In]

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

[Out]

b*log(x) - a/x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt [3]{(a+b x)^3}}{x^2} \, dx=b \log \left ({\left | x \right |}\right ) - \frac {a}{x} \]

[In]

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

[Out]

b*log(abs(x)) - a/x

Mupad [F(-1)]

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

[In]

int(((a + b*x)^3)^(1/3)/x^2,x)

[Out]

int(((a + b*x)^3)^(1/3)/x^2, x)