\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{19/3}} \, dx\) [87]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\frac {a x}{4 \left (a+b x^3\right )^{16/3}}-\frac {x}{52 \left (a+b x^3\right )^{13/3}}+\frac {x}{13 a \left (a+b x^3\right )^{10/3}}+\frac {9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac {27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac {81 x}{182 a^4 \sqrt [3]{a+b x^3}} \] Output:

1/4*a*x/(b*x^3+a)^(16/3)-1/52*x/(b*x^3+a)^(13/3)+1/13*x/a/(b*x^3+a)^(10/3) 
+9/91*x/a^2/(b*x^3+a)^(7/3)+27/182*x/a^3/(b*x^3+a)^(4/3)+81/182*x/a^4/(b*x 
^3+a)^(1/3)
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\frac {364 a^5 x+1183 a^4 b x^4+2080 a^3 b^2 x^7+1872 a^2 b^3 x^{10}+864 a b^4 x^{13}+162 b^5 x^{16}}{364 a^4 \left (a+b x^3\right )^{16/3}} \] Input:

Integrate[(a - b*x^3)^2/(a + b*x^3)^(19/3),x]
 

Output:

(364*a^5*x + 1183*a^4*b*x^4 + 2080*a^3*b^2*x^7 + 1872*a^2*b^3*x^10 + 864*a 
*b^4*x^13 + 162*b^5*x^16)/(364*a^4*(a + b*x^3)^(16/3))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {930, 27, 910, 749, 749, 749, 746}

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.

\(\displaystyle \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {2 a b \left (7 a-4 b x^3\right )}{\left (b x^3+a\right )^{16/3}}dx}{16 a b}+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \frac {7 a-4 b x^3}{\left (b x^3+a\right )^{16/3}}dx+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 910

\(\displaystyle \frac {1}{8} \left (\frac {80}{13} \int \frac {1}{\left (b x^3+a\right )^{13/3}}dx+\frac {11 x}{13 \left (a+b x^3\right )^{13/3}}\right )+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {1}{8} \left (\frac {80}{13} \left (\frac {9 \int \frac {1}{\left (b x^3+a\right )^{10/3}}dx}{10 a}+\frac {x}{10 a \left (a+b x^3\right )^{10/3}}\right )+\frac {11 x}{13 \left (a+b x^3\right )^{13/3}}\right )+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {1}{8} \left (\frac {80}{13} \left (\frac {9 \left (\frac {6 \int \frac {1}{\left (b x^3+a\right )^{7/3}}dx}{7 a}+\frac {x}{7 a \left (a+b x^3\right )^{7/3}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^3\right )^{10/3}}\right )+\frac {11 x}{13 \left (a+b x^3\right )^{13/3}}\right )+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {1}{8} \left (\frac {80}{13} \left (\frac {9 \left (\frac {6 \left (\frac {3 \int \frac {1}{\left (b x^3+a\right )^{4/3}}dx}{4 a}+\frac {x}{4 a \left (a+b x^3\right )^{4/3}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^3\right )^{7/3}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^3\right )^{10/3}}\right )+\frac {11 x}{13 \left (a+b x^3\right )^{13/3}}\right )+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

\(\Big \downarrow \) 746

\(\displaystyle \frac {1}{8} \left (\frac {80}{13} \left (\frac {9 \left (\frac {6 \left (\frac {3 x}{4 a^2 \sqrt [3]{a+b x^3}}+\frac {x}{4 a \left (a+b x^3\right )^{4/3}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^3\right )^{7/3}}\right )}{10 a}+\frac {x}{10 a \left (a+b x^3\right )^{10/3}}\right )+\frac {11 x}{13 \left (a+b x^3\right )^{13/3}}\right )+\frac {x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}\)

Input:

Int[(a - b*x^3)^2/(a + b*x^3)^(19/3),x]
 

Output:

(x*(a - b*x^3))/(8*(a + b*x^3)^(16/3)) + ((11*x)/(13*(a + b*x^3)^(13/3)) + 
 (80*(x/(10*a*(a + b*x^3)^(10/3)) + (9*(x/(7*a*(a + b*x^3)^(7/3)) + (6*(x/ 
(4*a*(a + b*x^3)^(4/3)) + (3*x)/(4*a^2*(a + b*x^3)^(1/3))))/(7*a)))/(10*a) 
))/13)/8
 

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 746

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) 
/a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
 

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 = 0.98 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {x \left (162 b^{5} x^{15}+864 a \,b^{4} x^{12}+1872 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1183 a^{4} b \,x^{3}+364 a^{5}\right )}{364 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{4}}\)	\(70\)
trager	\(\frac {x \left (162 b^{5} x^{15}+864 a \,b^{4} x^{12}+1872 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1183 a^{4} b \,x^{3}+364 a^{5}\right )}{364 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{4}}\)	\(70\)
pseudoelliptic	\(\frac {x \left (162 b^{5} x^{15}+864 a \,b^{4} x^{12}+1872 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1183 a^{4} b \,x^{3}+364 a^{5}\right )}{364 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{4}}\)	\(70\)
orering	\(\frac {x \left (162 b^{5} x^{15}+864 a \,b^{4} x^{12}+1872 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1183 a^{4} b \,x^{3}+364 a^{5}\right )}{364 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{4}}\)	\(70\)

Input:

int((-b*x^3+a)^2/(b*x^3+a)^(19/3),x,method=_RETURNVERBOSE)
 

Output:

1/364*x*(162*b^5*x^15+864*a*b^4*x^12+1872*a^2*b^3*x^9+2080*a^3*b^2*x^6+118 
3*a^4*b*x^3+364*a^5)/(b*x^3+a)^(16/3)/a^4
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\frac {{\left (162 \, b^{5} x^{16} + 864 \, a b^{4} x^{13} + 1872 \, a^{2} b^{3} x^{10} + 2080 \, a^{3} b^{2} x^{7} + 1183 \, a^{4} b x^{4} + 364 \, a^{5} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{364 \, {\left (a^{4} b^{6} x^{18} + 6 \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{12} + 20 \, a^{7} b^{3} x^{9} + 15 \, a^{8} b^{2} x^{6} + 6 \, a^{9} b x^{3} + a^{10}\right )}} \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(19/3),x, algorithm="fricas")
 

Output:

1/364*(162*b^5*x^16 + 864*a*b^4*x^13 + 1872*a^2*b^3*x^10 + 2080*a^3*b^2*x^ 
7 + 1183*a^4*b*x^4 + 364*a^5*x)*(b*x^3 + a)^(2/3)/(a^4*b^6*x^18 + 6*a^5*b^ 
5*x^15 + 15*a^6*b^4*x^12 + 20*a^7*b^3*x^9 + 15*a^8*b^2*x^6 + 6*a^9*b*x^3 + 
 a^10)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\text {Timed out} \] Input:

integrate((-b*x**3+a)**2/(b*x**3+a)**(19/3),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (86) = 172\).

Time = 0.05 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=-\frac {{\left (455 \, b^{3} - \frac {1680 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {2184 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {1040 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b^{2} x^{16}}{7280 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} - \frac {{\left (455 \, b^{4} - \frac {2240 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {4368 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {4160 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} b x^{16}}{3640 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} - \frac {{\left (91 \, b^{5} - \frac {560 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {1456 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {2080 \, {\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac {1456 \, {\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} x^{16}}{1456 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(19/3),x, algorithm="maxima")
 

Output:

-1/7280*(455*b^3 - 1680*(b*x^3 + a)*b^2/x^3 + 2184*(b*x^3 + a)^2*b/x^6 - 1 
040*(b*x^3 + a)^3/x^9)*b^2*x^16/((b*x^3 + a)^(16/3)*a^4) - 1/3640*(455*b^4 
 - 2240*(b*x^3 + a)*b^3/x^3 + 4368*(b*x^3 + a)^2*b^2/x^6 - 4160*(b*x^3 + a 
)^3*b/x^9 + 1820*(b*x^3 + a)^4/x^12)*b*x^16/((b*x^3 + a)^(16/3)*a^4) - 1/1 
456*(91*b^5 - 560*(b*x^3 + a)*b^4/x^3 + 1456*(b*x^3 + a)^2*b^3/x^6 - 2080* 
(b*x^3 + a)^3*b^2/x^9 + 1820*(b*x^3 + a)^4*b/x^12 - 1456*(b*x^3 + a)^5/x^1 
5)*x^16/((b*x^3 + a)^(16/3)*a^4)
 

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {19}{3}}} \,d x } \] Input:

integrate((-b*x^3+a)^2/(b*x^3+a)^(19/3),x, algorithm="giac")
 

Output:

integrate((b*x^3 - a)^2/(b*x^3 + a)^(19/3), x)
 

Mupad [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\frac {81\,x}{182\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}-\frac {x}{52\,{\left (b\,x^3+a\right )}^{13/3}}+\frac {27\,x}{182\,a^3\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {9\,x}{91\,a^2\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {x}{13\,a\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {a\,x}{4\,{\left (b\,x^3+a\right )}^{16/3}} \] Input:

int((a - b*x^3)^2/(a + b*x^3)^(19/3),x)
 

Output:

(81*x)/(182*a^4*(a + b*x^3)^(1/3)) - x/(52*(a + b*x^3)^(13/3)) + (27*x)/(1 
82*a^3*(a + b*x^3)^(4/3)) + (9*x)/(91*a^2*(a + b*x^3)^(7/3)) + x/(13*a*(a 
+ b*x^3)^(10/3)) + (a*x)/(4*(a + b*x^3)^(16/3))
 

Reduce [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{6}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{5} b \,x^{3}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} b^{2} x^{6}+20 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b^{3} x^{9}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{4} x^{12}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{5} x^{15}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{6} x^{18}}d x \right ) b^{2}-2 \left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{6}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{5} b \,x^{3}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} b^{2} x^{6}+20 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b^{3} x^{9}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{4} x^{12}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{5} x^{15}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{6} x^{18}}d x \right ) a b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{6}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{5} b \,x^{3}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} b^{2} x^{6}+20 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b^{3} x^{9}+15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{4} x^{12}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{5} x^{15}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{6} x^{18}}d x \right ) a^{2} \] Input:

int((-b*x^3+a)^2/(b*x^3+a)^(19/3),x)
 

Output:

int(x**6/((a + b*x**3)**(1/3)*a**6 + 6*(a + b*x**3)**(1/3)*a**5*b*x**3 + 1 
5*(a + b*x**3)**(1/3)*a**4*b**2*x**6 + 20*(a + b*x**3)**(1/3)*a**3*b**3*x* 
*9 + 15*(a + b*x**3)**(1/3)*a**2*b**4*x**12 + 6*(a + b*x**3)**(1/3)*a*b**5 
*x**15 + (a + b*x**3)**(1/3)*b**6*x**18),x)*b**2 - 2*int(x**3/((a + b*x**3 
)**(1/3)*a**6 + 6*(a + b*x**3)**(1/3)*a**5*b*x**3 + 15*(a + b*x**3)**(1/3) 
*a**4*b**2*x**6 + 20*(a + b*x**3)**(1/3)*a**3*b**3*x**9 + 15*(a + b*x**3)* 
*(1/3)*a**2*b**4*x**12 + 6*(a + b*x**3)**(1/3)*a*b**5*x**15 + (a + b*x**3) 
**(1/3)*b**6*x**18),x)*a*b + int(1/((a + b*x**3)**(1/3)*a**6 + 6*(a + b*x* 
*3)**(1/3)*a**5*b*x**3 + 15*(a + b*x**3)**(1/3)*a**4*b**2*x**6 + 20*(a + b 
*x**3)**(1/3)*a**3*b**3*x**9 + 15*(a + b*x**3)**(1/3)*a**2*b**4*x**12 + 6* 
(a + b*x**3)**(1/3)*a*b**5*x**15 + (a + b*x**3)**(1/3)*b**6*x**18),x)*a**2