### 3.112 $$\int \frac{x^{-1+n} (b+2 c x^n)}{(a+b x^n+c x^{2 n})^8} \, dx$$

Optimal. Leaf size=23 $-\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7}$

[Out]

-1/(7*n*(a + b*x^n + c*x^(2*n))^7)

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Rubi [A]  time = 0.0266705, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {1468, 629} $-\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n))^8,x]

[Out]

-1/(7*n*(a + b*x^n + c*x^(2*n))^7)

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx,x,x^n\right )}{n}\\ &=-\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7}\\ \end{align*}

Mathematica [A]  time = 0.0621506, size = 22, normalized size = 0.96 $-\frac{1}{7 n \left (a+x^n \left (b+c x^n\right )\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n))^8,x]

[Out]

-1/(7*n*(a + x^n*(b + c*x^n))^7)

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Maple [A]  time = 0.059, size = 22, normalized size = 1. \begin{align*} -{\frac{1}{7\,n \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x)

[Out]

-1/7/n/(a+b*x^n+c*(x^n)^2)^7

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Maxima [B]  time = 3.20277, size = 562, normalized size = 24.43 \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} n + a c^{6} n\right )} x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} n + 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} n + 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} n + 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \,{\left (b^{6} c n + 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n + 5 \, a^{3} c^{4} n\right )} x^{8 \, n} +{\left (b^{7} n + 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n + 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} + 7 \,{\left (a b^{6} n + 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n + 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} n + 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} n + 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} n + 6 \, a^{5} b c n\right )} x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} n + a^{6} c n\right )} x^{2 \, n}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x, algorithm="maxima")

[Out]

-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n + a^7*n + 7*(3*b^2*c^5*n + a*c^6*n)*x^(12*n) + 7*(5*
b^3*c^4*n + 6*a*b*c^5*n)*x^(11*n) + 7*(5*b^4*c^3*n + 15*a*b^2*c^4*n + 3*a^2*c^5*n)*x^(10*n) + 7*(3*b^5*c^2*n +
20*a*b^3*c^3*n + 15*a^2*b*c^4*n)*x^(9*n) + 7*(b^6*c*n + 15*a*b^4*c^2*n + 30*a^2*b^2*c^3*n + 5*a^3*c^4*n)*x^(8
*n) + (b^7*n + 42*a*b^5*c*n + 210*a^2*b^3*c^2*n + 140*a^3*b*c^3*n)*x^(7*n) + 7*(a*b^6*n + 15*a^2*b^4*c*n + 30*
a^3*b^2*c^2*n + 5*a^4*c^3*n)*x^(6*n) + 7*(3*a^2*b^5*n + 20*a^3*b^3*c*n + 15*a^4*b*c^2*n)*x^(5*n) + 7*(5*a^3*b^
4*n + 15*a^4*b^2*c*n + 3*a^5*c^2*n)*x^(4*n) + 7*(5*a^4*b^3*n + 6*a^5*b*c*n)*x^(3*n) + 7*(3*a^5*b^2*n + a^6*c*n
)*x^(2*n))

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Fricas [B]  time = 1.35582, size = 851, normalized size = 37. \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} n x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} n x^{8 \, n} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} n x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} n x^{2 \, n}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x, algorithm="fricas")

[Out]

-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n + a^7*n + 7*(3*b^2*c^5 + a*c^6)*n*x^(12*n) + 7*(5*b^
3*c^4 + 6*a*b*c^5)*n*x^(11*n) + 7*(5*b^4*c^3 + 15*a*b^2*c^4 + 3*a^2*c^5)*n*x^(10*n) + 7*(3*b^5*c^2 + 20*a*b^3*
c^3 + 15*a^2*b*c^4)*n*x^(9*n) + 7*(b^6*c + 15*a*b^4*c^2 + 30*a^2*b^2*c^3 + 5*a^3*c^4)*n*x^(8*n) + (b^7 + 42*a*
b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*c^3)*n*x^(7*n) + 7*(a*b^6 + 15*a^2*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*n*x
^(6*n) + 7*(3*a^2*b^5 + 20*a^3*b^3*c + 15*a^4*b*c^2)*n*x^(5*n) + 7*(5*a^3*b^4 + 15*a^4*b^2*c + 3*a^5*c^2)*n*x^
(4*n) + 7*(5*a^4*b^3 + 6*a^5*b*c)*n*x^(3*n) + 7*(3*a^5*b^2 + a^6*c)*n*x^(2*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(b+2*c*x**n)/(a+b*x**n+c*x**(2*n))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.18251, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n} + a\right )}^{7} n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x, algorithm="giac")

[Out]

-1/7/((c*x^(2*n) + b*x^n + a)^7*n)