### 3.128 $$\int \frac{x^{-1+n} (b+2 c x^n)}{(b x^n+c x^{2 n})^8} \, dx$$

Optimal. Leaf size=21 $-\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7}$

[Out]

-1/(7*n*x^(7*n)*(b + c*x^n)^7)

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Rubi [A]  time = 0.0324075, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.103, Rules used = {1584, 446, 74} $-\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(7*n*x^(7*n)*(b + c*x^n)^7)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 446

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

Rule 74

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

Rubi steps

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

Mathematica [A]  time = 0.181327, size = 21, normalized size = 1. $-\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(7*n*x^(7*n)*(b + c*x^n)^7)

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Maple [B]  time = 0.05, size = 203, normalized size = 9.7 \begin{align*} -132\,{\frac{{c}^{6}}{{b}^{13}n{x}^{n}}}+66\,{\frac{{c}^{5}}{{b}^{12}n \left ({x}^{n} \right ) ^{2}}}-30\,{\frac{{c}^{4}}{{b}^{11}n \left ({x}^{n} \right ) ^{3}}}+12\,{\frac{{c}^{3}}{{b}^{10}n \left ({x}^{n} \right ) ^{4}}}-4\,{\frac{{c}^{2}}{{b}^{9}n \left ({x}^{n} \right ) ^{5}}}+{\frac{c}{{b}^{8}n \left ({x}^{n} \right ) ^{6}}}-{\frac{1}{7\,{b}^{7}n \left ({x}^{n} \right ) ^{7}}}+{\frac{{c}^{7} \left ( 924\, \left ({x}^{n} \right ) ^{6}{c}^{6}+6006\,b{c}^{5} \left ({x}^{n} \right ) ^{5}+16380\,{b}^{2}{c}^{4} \left ({x}^{n} \right ) ^{4}+24024\,{b}^{3}{c}^{3} \left ({x}^{n} \right ) ^{3}+20020\,{b}^{4}{c}^{2} \left ({x}^{n} \right ) ^{2}+9009\,{b}^{5}c{x}^{n}+1716\,{b}^{6} \right ) }{7\,{b}^{13}n \left ( b+c{x}^{n} \right ) ^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-132/b^13*c^6/n/(x^n)+66/b^12*c^5/n/(x^n)^2-30/b^11*c^4/n/(x^n)^3+12/b^10*c^3/n/(x^n)^4-4/b^9*c^2/n/(x^n)^5+1/
b^8*c/n/(x^n)^6-1/7/b^7/n/(x^n)^7+1/7*c^7*(924*(x^n)^6*c^6+6006*b*c^5*(x^n)^5+16380*b^2*c^4*(x^n)^4+24024*b^3*
c^3*(x^n)^3+20020*b^4*c^2*(x^n)^2+9009*b^5*c*x^n+1716*b^6)/b^13/n/(b+c*x^n)^7

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Maxima [B]  time = 1.21517, size = 826, normalized size = 39.33 \begin{align*} -\frac{1}{105} \, b{\left (\frac{360360 \, c^{13} x^{13 \, n} + 2342340 \, b c^{12} x^{12 \, n} + 6426420 \, b^{2} c^{11} x^{11 \, n} + 9579570 \, b^{3} c^{10} x^{10 \, n} + 8270262 \, b^{4} c^{9} x^{9 \, n} + 4018014 \, b^{5} c^{8} x^{8 \, n} + 934362 \, b^{6} c^{7} x^{7 \, n} + 45045 \, b^{7} c^{6} x^{6 \, n} - 5005 \, b^{8} c^{5} x^{5 \, n} + 1001 \, b^{9} c^{4} x^{4 \, n} - 273 \, b^{10} c^{3} x^{3 \, n} + 91 \, b^{11} c^{2} x^{2 \, n} - 35 \, b^{12} c x^{n} + 15 \, b^{13}}{b^{14} c^{7} n x^{14 \, n} + 7 \, b^{15} c^{6} n x^{13 \, n} + 21 \, b^{16} c^{5} n x^{12 \, n} + 35 \, b^{17} c^{4} n x^{11 \, n} + 35 \, b^{18} c^{3} n x^{10 \, n} + 21 \, b^{19} c^{2} n x^{9 \, n} + 7 \, b^{20} c n x^{8 \, n} + b^{21} n x^{7 \, n}} + \frac{360360 \, c^{7} \log \left (x\right )}{b^{15}} - \frac{360360 \, c^{7} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{15} n}\right )} + \frac{1}{105} \, c{\left (\frac{360360 \, c^{12} x^{12 \, n} + 2342340 \, b c^{11} x^{11 \, n} + 6426420 \, b^{2} c^{10} x^{10 \, n} + 9579570 \, b^{3} c^{9} x^{9 \, n} + 8270262 \, b^{4} c^{8} x^{8 \, n} + 4018014 \, b^{5} c^{7} x^{7 \, n} + 934362 \, b^{6} c^{6} x^{6 \, n} + 45045 \, b^{7} c^{5} x^{5 \, n} - 5005 \, b^{8} c^{4} x^{4 \, n} + 1001 \, b^{9} c^{3} x^{3 \, n} - 273 \, b^{10} c^{2} x^{2 \, n} + 91 \, b^{11} c x^{n} - 35 \, b^{12}}{b^{13} c^{7} n x^{13 \, n} + 7 \, b^{14} c^{6} n x^{12 \, n} + 21 \, b^{15} c^{5} n x^{11 \, n} + 35 \, b^{16} c^{4} n x^{10 \, n} + 35 \, b^{17} c^{3} n x^{9 \, n} + 21 \, b^{18} c^{2} n x^{8 \, n} + 7 \, b^{19} c n x^{7 \, n} + b^{20} n x^{6 \, n}} + \frac{360360 \, c^{6} \log \left (x\right )}{b^{14}} - \frac{360360 \, c^{6} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{14} n}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*b*((360360*c^13*x^(13*n) + 2342340*b*c^12*x^(12*n) + 6426420*b^2*c^11*x^(11*n) + 9579570*b^3*c^10*x^(10
*n) + 8270262*b^4*c^9*x^(9*n) + 4018014*b^5*c^8*x^(8*n) + 934362*b^6*c^7*x^(7*n) + 45045*b^7*c^6*x^(6*n) - 500
5*b^8*c^5*x^(5*n) + 1001*b^9*c^4*x^(4*n) - 273*b^10*c^3*x^(3*n) + 91*b^11*c^2*x^(2*n) - 35*b^12*c*x^n + 15*b^1
3)/(b^14*c^7*n*x^(14*n) + 7*b^15*c^6*n*x^(13*n) + 21*b^16*c^5*n*x^(12*n) + 35*b^17*c^4*n*x^(11*n) + 35*b^18*c^
3*n*x^(10*n) + 21*b^19*c^2*n*x^(9*n) + 7*b^20*c*n*x^(8*n) + b^21*n*x^(7*n)) + 360360*c^7*log(x)/b^15 - 360360*
c^7*log((c*x^n + b)/c)/(b^15*n)) + 1/105*c*((360360*c^12*x^(12*n) + 2342340*b*c^11*x^(11*n) + 6426420*b^2*c^10
*x^(10*n) + 9579570*b^3*c^9*x^(9*n) + 8270262*b^4*c^8*x^(8*n) + 4018014*b^5*c^7*x^(7*n) + 934362*b^6*c^6*x^(6*
n) + 45045*b^7*c^5*x^(5*n) - 5005*b^8*c^4*x^(4*n) + 1001*b^9*c^3*x^(3*n) - 273*b^10*c^2*x^(2*n) + 91*b^11*c*x^
n - 35*b^12)/(b^13*c^7*n*x^(13*n) + 7*b^14*c^6*n*x^(12*n) + 21*b^15*c^5*n*x^(11*n) + 35*b^16*c^4*n*x^(10*n) +
35*b^17*c^3*n*x^(9*n) + 21*b^18*c^2*n*x^(8*n) + 7*b^19*c*n*x^(7*n) + b^20*n*x^(6*n)) + 360360*c^6*log(x)/b^14
- 360360*c^6*log((c*x^n + b)/c)/(b^14*n))

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Fricas [B]  time = 1.31501, size = 236, normalized size = 11.24 \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 21 \, b^{2} c^{5} n x^{12 \, n} + 35 \, b^{3} c^{4} n x^{11 \, n} + 35 \, b^{4} c^{3} n x^{10 \, n} + 21 \, b^{5} c^{2} n x^{9 \, n} + 7 \, b^{6} c n x^{8 \, n} + b^{7} n x^{7 \, n}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b+2*c*x^n)/(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) + 21*b^2*c^5*n*x^(12*n) + 35*b^3*c^4*n*x^(11*n) + 35*b^4*c^3*n*x^(10
*n) + 21*b^5*c^2*n*x^(9*n) + 7*b^6*c*n*x^(8*n) + b^7*n*x^(7*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)/(b*x**n+c*x**(2*n))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.13892, size = 27, normalized size = 1.29 \begin{align*} -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n}\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)/(b*x^n+c*x^(2*n))^8,x, algorithm="giac")

[Out]

-1/7/((c*x^(2*n) + b*x^n)^7*n)