\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{7/3}} \, dx\) [83]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\frac {a x}{\left (a+b x^3\right )^{4/3}}-\frac {x}{\sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\frac {-\frac {6 b^{4/3} x^4}{\left (a+b x^3\right )^{4/3}}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+\log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \] Input:

Integrate[(a - b*x^3)^2/(a + b*x^3)^(7/3),x]
 

Output:

((-6*b^(4/3)*x^4)/(a + b*x^3)^(4/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x) 
/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/ 
3)] + Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/ 
(6*b^(1/3))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {930, 27, 910, 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.

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

\(\Big \downarrow \) 930

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {2 b x^3+a}{\left (b x^3+a\right )^{4/3}}dx+\frac {x \left (a-b x^3\right )}{2 \left (a+b x^3\right )^{4/3}}\)

\(\Big \downarrow \) 910

\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{\sqrt [3]{b x^3+a}}dx-\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {x \left (a-b x^3\right )}{2 \left (a+b x^3\right )^{4/3}}\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {1}{2} \left (2 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )-\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {x \left (a-b x^3\right )}{2 \left (a+b x^3\right )^{4/3}}\)

Input:

Int[(a - b*x^3)^2/(a + b*x^3)^(7/3),x]
 

Output:

(x*(a - b*x^3))/(2*(a + b*x^3)^(4/3)) + (-(x/(a + b*x^3)^(1/3)) + 2*(ArcTa 
n[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[- 
(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3))))/2
 

Defintions of rubi rules used

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 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 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 = 1.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3 b^{\frac {1}{3}}}+\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 b^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )}{3 b^{\frac {1}{3}}}-\frac {b \,x^{4}}{\left (b \,x^{3}+a \right )^{\frac {4}{3}}}\)	\(117\)

Input:

int((-b*x^3+a)^2/(b*x^3+a)^(7/3),x,method=_RETURNVERBOSE)
 

Output:

-1/3/b^(1/3)*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)+1/6/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)-1/3*3^(1/2)/b^(1/3)*arct 
an(1/3*3^(1/2)*(2*(b*x^3+a)^(1/3)/b^(1/3)+x)/x)-b*x^4/(b*x^3+a)^(4/3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (78) = 156\).

Time = 0.14 (sec) , antiderivative size = 521, normalized size of antiderivative = 5.32 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx =\text {Too large to display} \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {7}{3}}}\, dx \] Input:

integrate((-b*x**3+a)**2/(b*x**3+a)**(7/3),x)
 

Output:

Integral((-a + b*x**3)**2/(a + b*x**3)**(7/3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (78) = 156\).

Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=-\frac {{\left (b - \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{4 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}}} - \frac {b x^{4}}{2 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}}} - \frac {1}{12} \, {\left (\frac {3 \, {\left (b + \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}} + \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} - \frac {2 \, \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} b^{2} \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="maxima")
 

Output:

-1/4*(b - 4*(b*x^3 + a)/x^3)*x^4/(b*x^3 + a)^(4/3) - 1/2*b*x^4/(b*x^3 + a) 
^(4/3) - 1/12*(3*(b + 4*(b*x^3 + a)/x^3)*x^4/((b*x^3 + a)^(4/3)*b^2) + 4*s 
qrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(7/ 
3) - 2*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*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(7/3))*b^2
 

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="giac")
 

Output:

integrate((b*x^3 - a)^2/(b*x^3 + a)^(7/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{7/3}} \,d x \] Input:

int((a - b*x^3)^2/(a + b*x^3)^(7/3),x)
 

Output:

int((a - b*x^3)^2/(a + b*x^3)^(7/3), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} x^{6}}d x \right ) b^{2}-2 \left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} x^{6}}d x \right ) a b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} x^{6}}d x \right ) a^{2} \] Input:

int((-b*x^3+a)^2/(b*x^3+a)^(7/3),x)
 

Output:

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