\(\int \frac {(a+b x^3)^{2/3}}{x^4} \, dx\) [532]

Optimal result

Integrand size = 15, antiderivative size = 107 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {2 b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}} \]

[Out]

-1/3*(b*x^3+a)^(2/3)/x^3-1/3*b*ln(x)/a^(1/3)+1/3*b*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(1/3)+2/9*b*arctan(1/3*(a^(1/
3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 43, 57, 631, 210, 31} \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {2 b \arctan \left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a}}-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}} \]

[In]

Int[(a + b*x^3)^(2/3)/x^4,x]

[Out]

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 {1}{3} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {1}{9} (2 b) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {1}{3} b \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}} \\ & = -\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a}} \\ & = -\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {2 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {-3 \sqrt [3]{a} \left (a+b x^3\right )^{2/3}+2 \sqrt {3} b x^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{9 \sqrt [3]{a} x^3} \]

[In]

Integrate[(a + b*x^3)^(2/3)/x^4,x]

[Out]

(-3*a^(1/3)*(a + b*x^3)^(2/3) + 2*Sqrt[3]*b*x^3*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 2*b*x^3*
Log[-a^(1/3) + (a + b*x^3)^(1/3)] - b*x^3*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(9*a^(
1/3)*x^3)

Maple [A] (verified)

Time = 3.98 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b \,x^{3}+2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b \,x^{3}-\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b \,x^{3}-3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{\frac {1}{3}}}{9 x^{3} a^{\frac {1}{3}}}\)	\(112\)

[In]

int((b*x^3+a)^(2/3)/x^4,x,method=_RETURNVERBOSE)

[Out]

1/9*(2*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)*b*x^3+2*ln((b*x^3+a)^(1/3)-a^(1/3))*b*x
^3-ln((b*x^3+a)^(2/3)+a^(1/3)*(b*x^3+a)^(1/3)+a^(2/3))*b*x^3-3*(b*x^3+a)^(2/3)*a^(1/3))/x^3/a^(1/3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) - a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x^{3} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}\right ] \]

[In]

integrate((b*x^3+a)^(2/3)/x^4,x, algorithm="fricas")

[Out]

[1/9*(3*sqrt(1/3)*a*b*x^3*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*sqrt(1/3)*(2*(b*x^3 + a)^(2/3)*a^(2/3) - (b*x^3 +
a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^3 + a)^(1/3)*a^(2/3) + 3*a)/x^3) - a^(2/3)*b*x^3*log((b*x^3 +
a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)*b*x^3*log((b*x^3 + a)^(1/3) - a^(1/3)) - 3*(b*x^3
+ a)^(2/3)*a)/(a*x^3), 1/9*(6*sqrt(1/3)*a^(2/3)*b*x^3*arctan(sqrt(1/3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)
) - a^(2/3)*b*x^3*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)*b*x^3*log((b*x^3 +
a)^(1/3) - a^(1/3)) - 3*(b*x^3 + a)^(2/3)*a)/(a*x^3)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.36 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=- \frac {b^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \]

[In]

integrate((b*x**3+a)**(2/3)/x**4,x)

[Out]

-b**(2/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), a*exp_polar(I*pi)/(b*x**3))/(3*x*gamma(4/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {1}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {1}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{3 \, x^{3}} \]

[In]

integrate((b*x^3+a)^(2/3)/x^4,x, algorithm="maxima")

[Out]

2/9*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3) - 1/9*b*log((b*x^3 + a)^(2/3
) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) + 2/9*b*log((b*x^3 + a)^(1/3) - a^(1/3))/a^(1/3) - 1/3*(b*x^3
+ a)^(2/3)/x^3

Giac [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {b^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, b^{2} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {1}{3}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{3}}}{9 \, b} \]

[In]

integrate((b*x^3+a)^(2/3)/x^4,x, algorithm="giac")

[Out]

1/9*(2*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3) - b^2*log((b*x^3 + a)^(
2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) + 2*b^2*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/a^(1/3) - 3*
(b*x^3 + a)^(2/3)*b/x^3)/b

Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {2\,b\,\ln \left (\frac {4\,a^{1/3}\,b^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )}{9\,a^{1/3}}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{3\,x^3}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}} \]

[In]

int((a + b*x^3)^(2/3)/x^4,x)

[Out]

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