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

Optimal. Leaf size=42 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]

[Out]

(c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a + (d*Log[a + b*x^
n])/(b*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0630661, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a + (d*Log[a + b*x^
n])/(b*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7596, size = 32, normalized size = 0.76 \[ \frac{d \log{\left (a + b x^{n} \right )}}{b n} + \frac{c x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0704356, size = 48, normalized size = 1.14 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \left (\log \left (a-a x^n\right )+n \log (x)\right )}{b n} \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a + (d*(n*Log[x] +
Log[a - a*x^n]))/(b*n)

_______________________________________________________________________________________

Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{-1+n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^(-1+n))/(a+b*x^n),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d \log \left (x\right )}{b} + \int \frac{b c x - a d}{b^{2} x x^{n} + a b x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(x)/b + integrate((b*c*x - a*d)/(b^2*x*x^n + a*b*x), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{n - 1} + c}{b x^{n} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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