3.829 \(\int \frac{a+b x^{-1+n}}{c x+d x^n} \, dx\)

Optimal. Leaf size=43 \[ \frac{b \log (x)}{d}-\frac{(b c-a d) \log \left (c x^{1-n}+d\right )}{c d (1-n)} \]

[Out]

(b*Log[x])/d - ((b*c - a*d)*Log[d + c*x^(1 - n)])/(c*d*(1 - n))

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Rubi [A]  time = 0.0668804, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1593, 514, 446, 72} \[ \frac{b \log (x)}{d}-\frac{(b c-a d) \log \left (c x^{1-n}+d\right )}{c d (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Log[x])/d - ((b*c - a*d)*Log[d + c*x^(1 - n)])/(c*d*(1 - n))

Rule 1593

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

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0408059, size = 38, normalized size = 0.88 \[ \frac{\frac{(b c-a d) \log \left (c x^{1-n}+d\right )}{c (n-1)}+b \log (x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Log[x] + ((b*c - a*d)*Log[d + c*x^(1 - n)])/(c*(-1 + n)))/d

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Maple [A]  time = 0.015, size = 73, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( x \right ) an}{c \left ( -1+n \right ) }}-{\frac{\ln \left ( x \right ) b}{d \left ( -1+n \right ) }}-{\frac{\ln \left ( cx+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a}{c \left ( -1+n \right ) }}+{\frac{\ln \left ( cx+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) b}{d \left ( -1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01548, size = 115, normalized size = 2.67 \begin{align*} b{\left (\frac{\log \left (x\right )}{d} - \frac{n \log \left (x\right )}{d{\left (n - 1\right )}} + \frac{\log \left (\frac{c x + d x^{n}}{d}\right )}{d{\left (n - 1\right )}}\right )} + a{\left (\frac{n \log \left (x\right )}{c{\left (n - 1\right )}} - \frac{\log \left (\frac{c x + d x^{n}}{d}\right )}{c{\left (n - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(log(x)/d - n*log(x)/(d*(n - 1)) + log((c*x + d*x^n)/d)/(d*(n - 1))) + a*(n*log(x)/(c*(n - 1)) - log((c*x +
d*x^n)/d)/(c*(n - 1)))

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Fricas [A]  time = 1.74653, size = 93, normalized size = 2.16 \begin{align*} \frac{{\left (b c - a d\right )} \log \left (c x + d x^{n}\right ) +{\left (a d n - b c\right )} \log \left (x\right )}{c d n - c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c - a*d)*log(c*x + d*x^n) + (a*d*n - b*c)*log(x))/(c*d*n - c*d)

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Sympy [A]  time = 9.88533, size = 212, normalized size = 4.93 \begin{align*} \begin{cases} \tilde{\infty } \left (a + b\right ) \log{\left (x \right )} & \text{for}\: c = 0 \wedge d = 0 \wedge n = 1 \\\frac{- \frac{a n x}{n^{2} x^{n} - n x^{n}} + \frac{b n^{2} x^{n} \log{\left (x \right )}}{n^{2} x^{n} - n x^{n}} - \frac{b n x^{n} \log{\left (x \right )}}{n^{2} x^{n} - n x^{n}} - \frac{b n x^{n}}{n^{2} x^{n} - n x^{n}}}{d} & \text{for}\: c = 0 \\\frac{\frac{a n x \log{\left (x \right )}}{n x - x} - \frac{a x \log{\left (x \right )}}{n x - x} + \frac{b x^{n}}{n x - x}}{c} & \text{for}\: d = 0 \\\frac{\left (a + b\right ) \log{\left (x \right )}}{c + d} & \text{for}\: n = 1 \\\frac{a d n \log{\left (x \right )}}{c d n - c d} - \frac{a d \log{\left (x + \frac{d x^{n}}{c} \right )}}{c d n - c d} - \frac{b c \log{\left (x \right )}}{c d n - c d} + \frac{b c \log{\left (x + \frac{d x^{n}}{c} \right )}}{c d n - c d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(a + b)*log(x), Eq(c, 0) & Eq(d, 0) & Eq(n, 1)), ((-a*n*x/(n**2*x**n - n*x**n) + b*n**2*x**n*lo
g(x)/(n**2*x**n - n*x**n) - b*n*x**n*log(x)/(n**2*x**n - n*x**n) - b*n*x**n/(n**2*x**n - n*x**n))/d, Eq(c, 0))
, ((a*n*x*log(x)/(n*x - x) - a*x*log(x)/(n*x - x) + b*x**n/(n*x - x))/c, Eq(d, 0)), ((a + b)*log(x)/(c + d), E
q(n, 1)), (a*d*n*log(x)/(c*d*n - c*d) - a*d*log(x + d*x**n/c)/(c*d*n - c*d) - b*c*log(x)/(c*d*n - c*d) + b*c*l
og(x + d*x**n/c)/(c*d*n - c*d), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{n - 1} + a}{c x + d x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^(n - 1) + a)/(c*x + d*x^n), x)