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

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 = 0.167116, 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 \[ \frac{b \log (x)}{d}-\frac{(b c-a d) \log \left (c x^{1-n}+d\right )}{c d (1-n)} \]

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))

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Rubi in Sympy [A]  time = 9.51961, size = 36, normalized size = 0.84 \[ \frac{b \log{\left (x^{- n + 1} \right )}}{d \left (- n + 1\right )} + \frac{\left (a d - b c\right ) \log{\left (c x^{- n + 1} + d \right )}}{c d \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0670188, size = 44, normalized size = 1.02 \[ \frac{(b c-a d) \log \left (c x+d x^n\right )+\log (x) (a d n-b c)}{c d (n-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.025, size = 73, normalized size = 1.7 \[{\frac{\ln \left ( x \right ) an}{c \left ( -1+n \right ) }}-{\frac{\ln \left ( x \right ) b}{d \left ( -1+n \right ) }}-{\frac{\ln \left ( cx+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a}{c \left ( -1+n \right ) }}+{\frac{\ln \left ( cx+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) b}{d \left ( -1+n \right ) }} \]

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 [A]  time = 0.740299, size = 115, normalized size = 2.67 \[ b{\left (\frac{\log \left (x\right )}{d} - \frac{n \log \left (x\right )}{d{\left (n - 1\right )}} + \frac{\log \left (\frac{c x + d x^{n}}{d}\right )}{d{\left (n - 1\right )}}\right )} + a{\left (\frac{n \log \left (x\right )}{c{\left (n - 1\right )}} - \frac{\log \left (\frac{c x + d x^{n}}{d}\right )}{c{\left (n - 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(n - 1) + a)/(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*lo
g(x)/(c*(n - 1)) - log((c*x + d*x^n)/d)/(c*(n - 1)))

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Fricas [A]  time = 0.297474, size = 59, normalized size = 1.37 \[ \frac{{\left (b c - a d\right )} \log \left (c x + d x^{n}\right ) +{\left (a d n - b c\right )} \log \left (x\right )}{c d n - c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(n - 1) + a)/(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 = 40.4087, size = 206, normalized size = 4.79 \[ \begin{cases} \tilde{\infty } \left (a + b\right ) \log{\left (x \right )} & \text{for}\: c = 0 \wedge d = 0 \wedge n = 1 \\\frac{- \frac{a n x}{n^{2} x^{n} - n x^{n}} + \frac{b n^{2} x^{n} \log{\left (x \right )}}{n^{2} x^{n} - n x^{n}} - \frac{b n x^{n} \log{\left (x \right )}}{n^{2} x^{n} - n x^{n}} - \frac{b n x^{n}}{n^{2} x^{n} - n x^{n}}}{d} & \text{for}\: c = 0 \\\frac{\frac{a n \log{\left (x \right )}}{n - 1} - \frac{a \log{\left (x \right )}}{n - 1} + \frac{b x^{n}}{n x - x}}{c} & \text{for}\: d = 0 \\\frac{\left (a + b\right ) \log{\left (x \right )}}{c + d} & \text{for}\: n = 1 \\\frac{a d n \log{\left (x \right )}}{c d n - c d} - \frac{a d \log{\left (x + \frac{d x^{n}}{c} \right )}}{c d n - c d} - \frac{b c \log{\left (x \right )}}{c d n - c d} + \frac{b c \log{\left (x + \frac{d x^{n}}{c} \right )}}{c d n - c d} & \text{otherwise} \end{cases} \]

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*log(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*log(x)/(n -
1) - a*log(x)/(n - 1) + b*x**n/(n*x - x))/c, Eq(d, 0)), ((a + b)*log(x)/(c + d),
Eq(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*log(x + d*x**n/c)/(c*d*n - c*d), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{n - 1} + a}{c x + d x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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