\(\int \frac {a-b x^3}{(a+b x^3)^{13/3}} \, dx\) [32]

Optimal result

Integrand size = 20, antiderivative size = 74 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\frac {x}{5 \left (a+b x^3\right )^{10/3}}+\frac {4 x}{35 a \left (a+b x^3\right )^{7/3}}+\frac {6 x}{35 a^2 \left (a+b x^3\right )^{4/3}}+\frac {18 x}{35 a^3 \sqrt [3]{a+b x^3}} \]

[Out]

1/5*x/(b*x^3+a)^(10/3)+4/35*x/a/(b*x^3+a)^(7/3)+6/35*x/a^2/(b*x^3+a)^(4/3)+18/35*x/a^3/(b*x^3+a)^(1/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {393, 198, 197} \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\frac {18 x}{35 a^3 \sqrt [3]{a+b x^3}}+\frac {6 x}{35 a^2 \left (a+b x^3\right )^{4/3}}+\frac {4 x}{35 a \left (a+b x^3\right )^{7/3}}+\frac {x}{5 \left (a+b x^3\right )^{10/3}} \]

[In]

Int[(a - b*x^3)/(a + b*x^3)^(13/3),x]

[Out]

x/(5*(a + b*x^3)^(10/3)) + (4*x)/(35*a*(a + b*x^3)^(7/3)) + (6*x)/(35*a^2*(a + b*x^3)^(4/3)) + (18*x)/(35*a^3*
(a + b*x^3)^(1/3))

Rule 197

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 198

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {x}{5 \left (a+b x^3\right )^{10/3}}+\frac {4}{5} \int \frac {1}{\left (a+b x^3\right )^{10/3}} \, dx \\ & = \frac {x}{5 \left (a+b x^3\right )^{10/3}}+\frac {4 x}{35 a \left (a+b x^3\right )^{7/3}}+\frac {24 \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{35 a} \\ & = \frac {x}{5 \left (a+b x^3\right )^{10/3}}+\frac {4 x}{35 a \left (a+b x^3\right )^{7/3}}+\frac {6 x}{35 a^2 \left (a+b x^3\right )^{4/3}}+\frac {18 \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{35 a^2} \\ & = \frac {x}{5 \left (a+b x^3\right )^{10/3}}+\frac {4 x}{35 a \left (a+b x^3\right )^{7/3}}+\frac {6 x}{35 a^2 \left (a+b x^3\right )^{4/3}}+\frac {18 x}{35 a^3 \sqrt [3]{a+b x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\frac {35 a^3 x+70 a^2 b x^4+60 a b^2 x^7+18 b^3 x^{10}}{35 a^3 \left (a+b x^3\right )^{10/3}} \]

[In]

Integrate[(a - b*x^3)/(a + b*x^3)^(13/3),x]

[Out]

(35*a^3*x + 70*a^2*b*x^4 + 60*a*b^2*x^7 + 18*b^3*x^10)/(35*a^3*(a + b*x^3)^(10/3))

Maple [A] (verified)

Time = 3.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {x \left (18 b^{3} x^{9}+60 a \,b^{2} x^{6}+70 a^{2} b \,x^{3}+35 a^{3}\right )}{35 \left (b \,x^{3}+a \right )^{\frac {10}{3}} a^{3}}\)	\(48\)
trager	\(\frac {x \left (18 b^{3} x^{9}+60 a \,b^{2} x^{6}+70 a^{2} b \,x^{3}+35 a^{3}\right )}{35 \left (b \,x^{3}+a \right )^{\frac {10}{3}} a^{3}}\)	\(48\)
pseudoelliptic	\(\frac {x \left (18 b^{3} x^{9}+60 a \,b^{2} x^{6}+70 a^{2} b \,x^{3}+35 a^{3}\right )}{35 \left (b \,x^{3}+a \right )^{\frac {10}{3}} a^{3}}\)	\(48\)

[In]

int((-b*x^3+a)/(b*x^3+a)^(13/3),x,method=_RETURNVERBOSE)

[Out]

1/35*x*(18*b^3*x^9+60*a*b^2*x^6+70*a^2*b*x^3+35*a^3)/(b*x^3+a)^(10/3)/a^3

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\frac {{\left (18 \, b^{3} x^{10} + 60 \, a b^{2} x^{7} + 70 \, a^{2} b x^{4} + 35 \, a^{3} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{35 \, {\left (a^{3} b^{4} x^{12} + 4 \, a^{4} b^{3} x^{9} + 6 \, a^{5} b^{2} x^{6} + 4 \, a^{6} b x^{3} + a^{7}\right )}} \]

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(13/3),x, algorithm="fricas")

[Out]

1/35*(18*b^3*x^10 + 60*a*b^2*x^7 + 70*a^2*b*x^4 + 35*a^3*x)*(b*x^3 + a)^(2/3)/(a^3*b^4*x^12 + 4*a^4*b^3*x^9 +
6*a^5*b^2*x^6 + 4*a^6*b*x^3 + a^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\text {Timed out} \]

[In]

integrate((-b*x**3+a)/(b*x**3+a)**(13/3),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (58) = 116\).

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.61 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=-\frac {{\left (14 \, b^{2} - \frac {40 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {35 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} b x^{10}}{140 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} a^{3}} - \frac {{\left (14 \, b^{3} - \frac {60 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {105 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {140 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} x^{10}}{140 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} a^{3}} \]

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(13/3),x, algorithm="maxima")

[Out]

-1/140*(14*b^2 - 40*(b*x^3 + a)*b/x^3 + 35*(b*x^3 + a)^2/x^6)*b*x^10/((b*x^3 + a)^(10/3)*a^3) - 1/140*(14*b^3
- 60*(b*x^3 + a)*b^2/x^3 + 105*(b*x^3 + a)^2*b/x^6 - 140*(b*x^3 + a)^3/x^9)*x^10/((b*x^3 + a)^(10/3)*a^3)

Giac [F]

\[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\int { -\frac {b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac {13}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(13/3),x, algorithm="giac")

[Out]

integrate(-(b*x^3 - a)/(b*x^3 + a)^(13/3), x)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{13/3}} \, dx=\frac {x}{5\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {18\,x}{35\,a^3\,{\left (b\,x^3+a\right )}^{1/3}}+\frac {6\,x}{35\,a^2\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {4\,x}{35\,a\,{\left (b\,x^3+a\right )}^{7/3}} \]

[In]

int((a - b*x^3)/(a + b*x^3)^(13/3),x)

[Out]

x/(5*(a + b*x^3)^(10/3)) + (18*x)/(35*a^3*(a + b*x^3)^(1/3)) + (6*x)/(35*a^2*(a + b*x^3)^(4/3)) + (4*x)/(35*a*
(a + b*x^3)^(7/3))