\(\int \frac {(a+b x^3)^{2/3}}{x^4} \, dx\) [377]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {-3 \sqrt [3]{a} \left (a+b x^3\right )^{2/3}+2 \sqrt {3} b x^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{9 \sqrt [3]{a} x^3} \] Input:

Integrate[(a + b*x^3)^(2/3)/x^4,x]
 

Output:

(-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)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 51, 67, 16, 1082, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{2/3}}{x^6}dx^3\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} b \int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3-\frac {\left (a+b x^3\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} b \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^3\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} b \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^3\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} b \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^3\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^3\right )^{2/3}}{x^3}\right )\)

Input:

Int[(a + b*x^3)^(2/3)/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 16

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

rule 51

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] - Simp[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 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[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 217

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 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05

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

Input:

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

Output:

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

Time = 0.08 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) - a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x^{3} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}\right ] \] Input:

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

Output:

[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 = 0.82 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.36 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=- \frac {b^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

-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)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {1}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {1}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{3 \, x^{3}} \] Input:

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

Output:

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)

Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {1}{9} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {1}{3}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{b x^{3}}\right )} b \] Input:

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

Output:

1/9*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)) 
/a^(1/3) - log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/a^ 
(1/3) + 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 = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {2\,b\,\ln \left (\frac {4\,a^{1/3}\,b^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )}{9\,a^{1/3}}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{3\,x^3}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}} \] Input:

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

Output:

(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))
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx=\frac {-\left (b \,x^{3}+a \right )^{\frac {2}{3}}+2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b \,x^{4}+a x}d x \right ) b \,x^{3}}{3 x^{3}} \] Input:

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

Output:

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