### 3.116 $$\int \frac{x^{-1+n} (b+2 c x^n)}{-a+b x^n+c x^{2 n}} \, dx$$

Optimal. Leaf size=21 $\frac{\log \left (a-b x^n-c x^{2 n}\right )}{n}$

[Out]

Log[a - b*x^n - c*x^(2*n)]/n

________________________________________________________________________________________

Rubi [A]  time = 0.0288613, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {1468, 628} $\frac{\log \left (a-b x^n-c x^{2 n}\right )}{n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[a - b*x^n - c*x^(2*n)]/n

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 628

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

Rubi steps

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

Mathematica [A]  time = 0.11436, size = 21, normalized size = 1. $\frac{\log \left (a-b x^n-c x^{2 n}\right )}{n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[a - b*x^n - c*x^(2*n)]/n

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 26, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( -c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) }{n}} \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)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.14785, size = 34, normalized size = 1.62 \begin{align*} \frac{\log \left (\frac{c x^{2 \, n} + b x^{n} - a}{c}\right )}{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)),x, algorithm="maxima")

[Out]

log((c*x^(2*n) + b*x^n - a)/c)/n

________________________________________________________________________________________

Fricas [A]  time = 1.20768, size = 41, normalized size = 1.95 \begin{align*} \frac{\log \left (c x^{2 \, n} + b x^{n} - a\right )}{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)),x, algorithm="fricas")

[Out]

log(c*x^(2*n) + b*x^n - a)/n

________________________________________________________________________________________

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)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.09546, size = 28, normalized size = 1.33 \begin{align*} \frac{\log \left (c x^{2 \, n} + b x^{n} - a\right )}{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)),x, algorithm="giac")

[Out]

log(c*x^(2*n) + b*x^n - a)/n