\(\int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx\) [159]

Optimal result

Integrand size = 15, antiderivative size = 115 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {b \sqrt [3]{(a+b x)^2}}{a}-\frac {\sqrt [3]{(a+b x)^5}}{a x}+\frac {b \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{\sqrt [3]{a^2}} \]

[Out]

-((b*x+a)^5)^(1/3)/a/x+b/a*((b*x+a)^2)^(1/3)+b/(a^2)^(1/3)*(3/2*ln(((b*x+a)^(1/3)-a^(1/3))/x^(1/3))+3^(1/2)*ar
ctan(3^(1/2)*(b*x+a)^(1/3)/((b*x+a)^(1/3)+2*a^(1/3))))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1973, 43, 57, 632, 210, 31} \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {2 b \sqrt [3]{(a+b x)^2} \arctan \left (\frac {2 \sqrt [3]{\frac {b x}{a}+1}+1}{\sqrt {3}}\right )}{\sqrt {3} a \left (\frac {b x}{a}+1\right )^{2/3}}-\frac {\sqrt [3]{(a+b x)^2}}{x}-\frac {b \log (x) \sqrt [3]{(a+b x)^2}}{3 a \left (\frac {b x}{a}+1\right )^{2/3}}+\frac {b \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{\frac {b x}{a}+1}\right )}{a \left (\frac {b x}{a}+1\right )^{2/3}} \]

[In]

Int[((a + b*x)^2)^(1/3)/x^2,x]

[Out]

-(((a + b*x)^2)^(1/3)/x) + (2*b*((a + b*x)^2)^(1/3)*ArcTan[(1 + 2*(1 + (b*x)/a)^(1/3))/Sqrt[3]])/(Sqrt[3]*a*(1
+ (b*x)/a)^(2/3)) - (b*((a + b*x)^2)^(1/3)*Log[x])/(3*a*(1 + (b*x)/a)^(2/3)) + (b*((a + b*x)^2)^(1/3)*Log[1 -
(1 + (b*x)/a)^(1/3)])/(a*(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 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=-\frac {\sqrt [3]{(a+b x)^2} \left (3 \sqrt [3]{a} (a+b x)^{2/3}-2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{3 \sqrt [3]{a} x (a+b x)^{2/3}} \]

[In]

Integrate[((a + b*x)^2)^(1/3)/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{x}+\frac {2 b \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{3 \left (b x +a \right )^{\frac {2}{3}}}\)	\(109\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (93) = 186\).

Time = 0.26 (sec) , antiderivative size = 560, normalized size of antiderivative = 4.87 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (-\frac {b^{2} x^{2} + 4 \, a b x + 3 \, a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {1}{3}}}{b x^{2} + a x}\right ) - a^{\frac {2}{3}} b x \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + 2 \, a^{\frac {2}{3}} b x \log \left (-\frac {{\left (b x + a\right )} a^{\frac {2}{3}} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{3 \, a x}, -\frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left ({\left (b x + a\right )} a^{\frac {1}{3}} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a^{\frac {2}{3}}\right )}}{{\left (b x + a\right )} a^{\frac {1}{3}}}\right ) + a^{\frac {2}{3}} b x \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - 2 \, a^{\frac {2}{3}} b x \log \left (-\frac {{\left (b x + a\right )} a^{\frac {2}{3}} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{3 \, a x}\right ] \]

[In]

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

[Out]

[1/3*(3*sqrt(1/3)*a*b*x*sqrt(-1/a^(2/3))*log(-(b^2*x^2 + 4*a*b*x + 3*a^2 + 3*sqrt(1/3)*((b^2*x^2 + 2*a*b*x + a
^2)^(1/3)*(b*x + a)*a^(2/3) - 2*(b^2*x^2 + 2*a*b*x + a^2)^(2/3)*a + (b^2*x^2 + 2*a*b*x + a^2)*a^(1/3))*sqrt(-1
/a^(2/3)) - 3*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)*(b*x + a)*a^(1/3))/(b*x^2 + a*x)) - a^(2/3)*b*x*log(((b^2*x^2 +
2*a*b*x + a^2)^(1/3)*(b*x + a)*a^(2/3) + (b^2*x^2 + 2*a*b*x + a^2)^(2/3)*a + (b^2*x^2 + 2*a*b*x + a^2)*a^(1/3)
)/(b^2*x^2 + 2*a*b*x + a^2)) + 2*a^(2/3)*b*x*log(-((b*x + a)*a^(2/3) - (b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a)/(b*x
+ a)) - 3*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a)/(a*x), -1/3*(6*sqrt(1/3)*a^(2/3)*b*x*arctan(sqrt(1/3)*((b*x + a)
*a^(1/3) + 2*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a^(2/3))/((b*x + a)*a^(1/3))) + a^(2/3)*b*x*log(((b^2*x^2 + 2*a*b
*x + a^2)^(1/3)*(b*x + a)*a^(2/3) + (b^2*x^2 + 2*a*b*x + a^2)^(2/3)*a + (b^2*x^2 + 2*a*b*x + a^2)*a^(1/3))/(b^
2*x^2 + 2*a*b*x + a^2)) - 2*a^(2/3)*b*x*log(-((b*x + a)*a^(2/3) - (b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a)/(b*x + a)
) + 3*(b^2*x^2 + 2*a*b*x + a^2)^(1/3)*a)/(a*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(((b*x + a)^2)^(1/3)/x^2, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (93) = 186\).

Time = 3.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {\frac {2 \, \sqrt {3} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}\right )}}{3 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}}\right )}{a} - \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \log \left ({\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} + {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}}\right )}{a} + \frac {2 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \log \left ({\left | {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} - \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \right |}\right )}{a} - \frac {3 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} b}{x}}{3 \, b} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(((a + b*x)^2)^(1/3)/x^2,x)

[Out]

int(((a + b*x)^2)^(1/3)/x^2, x)