\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{11/3}} \, dx\) [93]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\frac {a x}{2 \left (a+b x^3\right )^{8/3}}-\frac {x}{4 \left (a+b x^3\right )^{5/3}}+\frac {3 x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {8}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{4 a \left (a+b x^3\right )^{2/3}} \] Output:

1/2*a*x/(b*x^3+a)^(8/3)-1/4*x/(b*x^3+a)^(5/3)+3/4*x*(1+b*x^3/a)^(2/3)*hype 
rgeom([1/3, 8/3],[4/3],-b*x^3/a)/a/(b*x^3+a)^(2/3)
 

Mathematica [A] (verified)

Time = 10.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\frac {7 a^2 x+5 a b x^4+3 b^2 x^7+3 x \left (a+b x^3\right )^2 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{10 a \left (a+b x^3\right )^{8/3}} \] Input:

Integrate[(a - b*x^3)^2/(a + b*x^3)^(11/3),x]
 

Output:

(7*a^2*x + 5*a*b*x^4 + 3*b^2*x^7 + 3*x*(a + b*x^3)^2*(1 + (b*x^3)/a)^(2/3) 
*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(10*a*(a + b*x^3)^(8/3))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {930, 27, 779, 778}

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.

\(\displaystyle \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {6 a^2 b}{\left (b x^3+a\right )^{8/3}}dx}{8 a b}+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} a \int \frac {1}{\left (b x^3+a\right )^{8/3}}dx+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}\)

\(\Big \downarrow \) 779

\(\displaystyle \frac {3 \left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{8/3}}dx}{4 a \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {8}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{4 a \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}\)

Input:

Int[(a - b*x^3)^2/(a + b*x^3)^(11/3),x]
 

Output:

(x*(a - b*x^3))/(4*(a + b*x^3)^(8/3)) + (3*x*(1 + (b*x^3)/a)^(2/3)*Hyperge 
ometric2F1[1/3, 8/3, 4/3, -((b*x^3)/a)])/(4*a*(a + b*x^3)^(2/3))
 

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 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 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 [F]

\[\int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {11}{3}}}d x\]

Input:

int((-b*x^3+a)^2/(b*x^3+a)^(11/3),x)
 

Output:

int((-b*x^3+a)^2/(b*x^3+a)^(11/3),x)
 

Fricas [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {11}{3}}} \,d x } \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(11/3),x, algorithm="fricas")
 

Output:

integral((b^2*x^6 - 2*a*b*x^3 + a^2)*(b*x^3 + a)^(1/3)/(b^4*x^12 + 4*a*b^3 
*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\text {Timed out} \] Input:

integrate((-b*x**3+a)**2/(b*x**3+a)**(11/3),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {11}{3}}} \,d x } \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(11/3),x, algorithm="maxima")
 

Output:

integrate((b*x^3 - a)^2/(b*x^3 + a)^(11/3), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {11}{3}}} \,d x } \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(11/3),x, algorithm="giac")
 

Output:

integrate((b*x^3 - a)^2/(b*x^3 + a)^(11/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{11/3}} \,d x \] Input:

int((a - b*x^3)^2/(a + b*x^3)^(11/3),x)
 

Output:

int((a - b*x^3)^2/(a + b*x^3)^(11/3), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,x^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{3} x^{9}}d x \right ) b^{2}-2 \left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,x^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{3} x^{9}}d x \right ) a b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,x^{3}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{3} x^{9}}d x \right ) a^{2} \] Input:

int((-b*x^3+a)^2/(b*x^3+a)^(11/3),x)
 

Output:

int(x**6/((a + b*x**3)**(2/3)*a**3 + 3*(a + b*x**3)**(2/3)*a**2*b*x**3 + 3 
*(a + b*x**3)**(2/3)*a*b**2*x**6 + (a + b*x**3)**(2/3)*b**3*x**9),x)*b**2 
- 2*int(x**3/((a + b*x**3)**(2/3)*a**3 + 3*(a + b*x**3)**(2/3)*a**2*b*x**3 
 + 3*(a + b*x**3)**(2/3)*a*b**2*x**6 + (a + b*x**3)**(2/3)*b**3*x**9),x)*a 
*b + int(1/((a + b*x**3)**(2/3)*a**3 + 3*(a + b*x**3)**(2/3)*a**2*b*x**3 + 
 3*(a + b*x**3)**(2/3)*a*b**2*x**6 + (a + b*x**3)**(2/3)*b**3*x**9),x)*a** 
2