\(\int \frac {(1-x^3)^{2/3}}{a+b x} \, dx\) [108]

Optimal result

Integrand size = 19, antiderivative size = 384 \[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\frac {\left (1-x^3\right )^{2/3}}{2 b}-\frac {\left (a^3+b^3\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2 b^2}+\frac {a^2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}-\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a^3+b^3} x}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {\left (a^3+b^3\right )^{2/3} \arctan \left (\frac {1+\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^3}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}-\frac {\left (a^3+b^3\right )^{2/3} \log \left (a^3+b^3 x^3\right )}{3 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (-\frac {\sqrt [3]{a^3+b^3} x}{a}-\sqrt [3]{1-x^3}\right )}{2 b^3}-\frac {a^2 \log \left (x+\sqrt [3]{1-x^3}\right )}{2 b^3}+\frac {\left (a^3+b^3\right )^{2/3} \log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 b^3} \] Output:

1/2*(-x^3+1)^(2/3)/b-1/2*(a^3+b^3)*x^2*AppellF1(2/3,1/3,1,5/3,x^3,-b^3*x^3 
/a^3)/a^2/b^2+1/2*a*x^2*hypergeom([1/3, 2/3],[5/3],x^3)/b^2-1/3*(a^3+b^3)^ 
(2/3)*ln(b^3*x^3+a^3)/b^3+1/2*(a^3+b^3)^(2/3)*ln(-(a^3+b^3)^(1/3)*x/a-(-x^ 
3+1)^(1/3))/b^3-1/2*a^2*ln(x+(-x^3+1)^(1/3))/b^3+1/2*(a^3+b^3)^(2/3)*ln((a 
^3+b^3)^(1/3)-b*(-x^3+1)^(1/3))/b^3+1/3*a^2*arctan(1/3*(1-2*x/(-x^3+1)^(1/ 
3))*3^(1/2))/b^3*3^(1/2)-1/3*(a^3+b^3)^(2/3)*arctan(1/3*(1-2*(a^3+b^3)^(1/ 
3)*x/a/(-x^3+1)^(1/3))*3^(1/2))/b^3*3^(1/2)+1/3*(a^3+b^3)^(2/3)*arctan(1/3 
*(1+2*b*(-x^3+1)^(1/3)/(a^3+b^3)^(1/3))*3^(1/2))/b^3*3^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx \] Input:

Integrate[(1 - x^3)^(2/3)/(a + b*x),x]
 

Output:

Integrate[(1 - x^3)^(2/3)/(a + b*x), x]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2578, 888, 2577, 769, 2581, 2009}

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.

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

\(\Big \downarrow \) 2578

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {\int \frac {b^2-a^2 x}{(a+b x) \sqrt [3]{1-x^3}}dx}{b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b}\)

\(\Big \downarrow \) 2577

\(\displaystyle \frac {\frac {\left (a^3+b^3\right ) \int \frac {1}{(a+b x) \sqrt [3]{1-x^3}}dx}{b}-\frac {a^2 \int \frac {1}{\sqrt [3]{1-x^3}}dx}{b}}{b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b}\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {\frac {\left (a^3+b^3\right ) \int \frac {1}{(a+b x) \sqrt [3]{1-x^3}}dx}{b}-\frac {a^2 \left (\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{b}}{b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b}\)

\(\Big \downarrow \) 2581

\(\displaystyle \frac {\frac {\left (a^3+b^3\right ) \int \left (\frac {a^2}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}-\frac {b x a}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}+\frac {b^2 x^2}{\sqrt [3]{1-x^3} \left (a^3+b^3 x^3\right )}\right )dx}{b}-\frac {a^2 \left (\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{b}}{b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\left (a^3+b^3\right ) \left (-\frac {\arctan \left (\frac {1-\frac {2 x \sqrt [3]{a^3+b^3}}{a \sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a^3+b^3}}+\frac {\arctan \left (\frac {\frac {2 b \sqrt [3]{1-x^3}}{\sqrt [3]{a^3+b^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a^3+b^3}}-\frac {\log \left (a^3+b^3 x^3\right )}{3 \sqrt [3]{a^3+b^3}}+\frac {\log \left (-\frac {x \sqrt [3]{a^3+b^3}}{a}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{a^3+b^3}}+\frac {\log \left (\sqrt [3]{a^3+b^3}-b \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{a^3+b^3}}-\frac {b x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {b^3 x^3}{a^3}\right )}{2 a^2}\right )}{b}-\frac {a^2 \left (\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{b}}{b^2}+\frac {a x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{2 b^2}+\frac {\left (1-x^3\right )^{2/3}}{2 b}\)

Input:

Int[(1 - x^3)^(2/3)/(a + b*x),x]
 

Output:

(1 - x^3)^(2/3)/(2*b) + (a*x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/(2*b 
^2) + (-((a^2*(-(ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) + Lo 
g[x + (1 - x^3)^(1/3)]/2))/b) + ((a^3 + b^3)*(-1/2*(b*x^2*AppellF1[2/3, 1/ 
3, 1, 5/3, x^3, -((b^3*x^3)/a^3)])/a^2 - ArcTan[(1 - (2*(a^3 + b^3)^(1/3)* 
x)/(a*(1 - x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*(a^3 + b^3)^(1/3)) + ArcTan[(1 + 
 (2*b*(1 - x^3)^(1/3))/(a^3 + b^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*(a^3 + b^3)^(1 
/3)) - Log[a^3 + b^3*x^3]/(3*(a^3 + b^3)^(1/3)) + Log[-(((a^3 + b^3)^(1/3) 
*x)/a) - (1 - x^3)^(1/3)]/(2*(a^3 + b^3)^(1/3)) + Log[(a^3 + b^3)^(1/3) - 
b*(1 - x^3)^(1/3)]/(2*(a^3 + b^3)^(1/3))))/b)/b^2
 

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2577

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)) 
, x_Symbol] :> Simp[f/d   Int[1/(a + b*x^3)^(1/3), x], x] + Simp[(d*e - c*f 
)/d   Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, 
f}, x]
 

rule 2578

Int[((a_) + (b_.)*(x_)^3)^(2/3)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(a + 
 b*x^3)^(2/3)/(2*d), x] + (Simp[1/d^2   Int[(a*d^2 + b*c^2*x)/((c + d*x)*(a 
 + b*x^3)^(1/3)), x], x] - Simp[b*(c/d^2)   Int[x/(a + b*x^3)^(1/3), x], x] 
) /; FreeQ[{a, b, c, d}, x]
 

rule 2581

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d 
^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ[q, 0 
] && RationalQ[p] && EqQ[Denominator[p], 3]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{a + b x}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{a+b x} \, dx=\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{b \,x^{4}+a \,x^{3}-b x -a}d x \right ) b -2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x^{2}}{b \,x^{4}+a \,x^{3}-b x -a}d x \right ) a}{2 b} \] Input:

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

Output:

(( - x**3 + 1)**(2/3) - 2*int(( - x**3 + 1)**(2/3)/(a*x**3 - a + b*x**4 - 
b*x),x)*b - 2*int((( - x**3 + 1)**(2/3)*x**2)/(a*x**3 - a + b*x**4 - b*x), 
x)*a)/(2*b)