\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{4/3}} \, dx\) [82]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {4 a x}{\sqrt [3]{a+b x^3}}+\frac {1}{3} x \left (a+b x^3\right )^{2/3}-\frac {10 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \] Output:

4*a*x/(b*x^3+a)^(1/3)+1/3*x*(b*x^3+a)^(2/3)-10/9*a*arctan(1/3*(1+2*b^(1/3) 
*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(1/3)+5/3*a*ln(-b^(1/3)*x+(b*x^3+a) 
^(1/3))/b^(1/3)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \left (\frac {3 \left (13 a x+b x^4\right )}{\sqrt [3]{a+b x^3}}-\frac {10 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}+\frac {10 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b}}-\frac {5 a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b}}\right ) \] Input:

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

Output:

((3*(13*a*x + b*x^4))/(a + b*x^3)^(1/3) - (10*Sqrt[3]*a*ArcTan[(Sqrt[3]*b^ 
(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))])/b^(1/3) + (10*a*Log[-(b^(1/3) 
*x) + (a + b*x^3)^(1/3)])/b^(1/3) - (5*a*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + 
b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(1/3))/9
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {930, 25, 27, 913, 769}

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.

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

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int -\frac {a b \left (a-7 b x^3\right )}{\sqrt [3]{b x^3+a}}dx}{a b}+\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}-\frac {\int \frac {a b \left (a-7 b x^3\right )}{\sqrt [3]{b x^3+a}}dx}{a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}-\int \frac {a-7 b x^3}{\sqrt [3]{b x^3+a}}dx\)

\(\Big \downarrow \) 913

\(\displaystyle -\frac {10}{3} a \int \frac {1}{\sqrt [3]{b x^3+a}}dx+\frac {7}{3} x \left (a+b x^3\right )^{2/3}+\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}\)

\(\Big \downarrow \) 769

\(\displaystyle -\frac {10}{3} a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )+\frac {7}{3} x \left (a+b x^3\right )^{2/3}+\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}\)

Input:

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

Output:

(2*x*(a - b*x^3))/(a + b*x^3)^(1/3) + (7*x*(a + b*x^3)^(2/3))/3 - (10*a*(A 
rcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - L 
og[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3))))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 913

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( 
p + 1) + 1))/(b*(n*(p + 1) + 1))   Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b 
, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
 

rule 930

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 
1))), x] - Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q 
- 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( 
p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
 

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.25

method	result	size
pseudoelliptic	\(-\frac {a \left (-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{a}-\frac {10 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3 b^{\frac {1}{3}}}+\frac {5 \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{3 b^{\frac {1}{3}}}-\frac {10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )}{3 b^{\frac {1}{3}}}-\frac {12 x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}\right )}{3}\)	\(132\)

Input:

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

Output:

-1/3*a*(-(b*x^3+a)^(2/3)/a*x-10/3/b^(1/3)*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/ 
x)+5/3/b^(1/3)*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/ 
x^2)-10/3*3^(1/2)/b^(1/3)*arctan(1/3*3^(1/2)*(2*(b*x^3+a)^(1/3)/b^(1/3)+x) 
/x)-12*x/(b*x^3+a)^(1/3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (81) = 162\).

Time = 0.16 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.89 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {15 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}, \frac {10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {30 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}} + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}\right ] \] Input:

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

Output:

[1/9*(15*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b 
*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*(b^(4/3)*x^3 + (b*x^3 + a)^(1/3) 
*b*x^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) + 10*(a*b* 
x^3 + a^2)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5*(a*b*x^3 + 
a^2)*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^ 
(2/3))/x^2) + 3*(b^2*x^4 + 13*a*b*x)*(b*x^3 + a)^(2/3))/(b^2*x^3 + a*b), 1 
/9*(10*(a*b*x^3 + a^2)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5 
*(a*b*x^3 + a^2)*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + 
(b*x^3 + a)^(2/3))/x^2) + 30*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*arctan(sqrt(1/3 
)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3) + 3*(b^2*x^4 + 13 
*a*b*x)*(b*x^3 + a)^(2/3))/(b^2*x^3 + a*b)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (81) = 162\).

Time = 0.12 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \, b^{2} {\left (\frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (3 \, a b - \frac {4 \, {\left (b x^{3} + a\right )} a}{x^{3}}\right )}}{\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{3}}{x} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}}} - \frac {2 \, a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} + \frac {1}{3} \, a b {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {a x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \] Input:

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

Output:

1/9*b^2*(4*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/ 
b^(1/3))/b^(7/3) + 3*(3*a*b - 4*(b*x^3 + a)*a/x^3)/((b*x^3 + a)^(1/3)*b^3/ 
x - (b*x^3 + a)^(4/3)*b^2/x^4) - 2*a*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/ 
3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(7/3) + 4*a*log(-b^(1/3) + (b*x^3 + a)^(1/ 
3)/x)/b^(7/3)) + 1/3*a*b*(2*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 
 + a)^(1/3)/x)/b^(1/3))/b^(4/3) + 6*x/((b*x^3 + a)^(1/3)*b) - log(b^(2/3) 
+ (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(4/3) + 2*log(-b^ 
(1/3) + (b*x^3 + a)^(1/3)/x)/b^(4/3)) + a*x/(b*x^3 + a)^(1/3)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a +\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) b^{2}-2 \left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a +\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) a b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a +\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) a^{2} \] Input:

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

Output:

int(x**6/((a + b*x**3)**(1/3)*a + (a + b*x**3)**(1/3)*b*x**3),x)*b**2 - 2* 
int(x**3/((a + b*x**3)**(1/3)*a + (a + b*x**3)**(1/3)*b*x**3),x)*a*b + int 
(1/((a + b*x**3)**(1/3)*a + (a + b*x**3)**(1/3)*b*x**3),x)*a**2