3.1231 \(\int \frac{(a c-b c x)^n}{a+b x} \, dx\)

Optimal. Leaf size=52 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a-b x}{2 a}\right )}{2 a b c (n+1)} \]

[Out]

-((a*c - b*c*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a - b*x)/(2*a)])/(2*
a*b*c*(1 + n))

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Rubi [A]  time = 0.0390882, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a-b x}{2 a}\right )}{2 a b c (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((a*c - b*c*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a - b*x)/(2*a)])/(2*
a*b*c*(1 + n))

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Rubi in Sympy [A]  time = 7.82872, size = 37, normalized size = 0.71 \[ - \frac{\left (a c - b c x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{2 a b c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*c - b*c*x)**(n + 1)*hyper((1, n + 1), (n + 2,), (a/2 - b*x/2)/a)/(2*a*b*c*(n
+ 1))

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Mathematica [A]  time = 0.0362592, size = 58, normalized size = 1.12 \[ \frac{\left (\frac{b x-a}{a+b x}\right )^{-n} (c (a-b x))^n \, _2F_1\left (-n,-n;1-n;\frac{2 a}{a+b x}\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

((c*(a - b*x))^n*Hypergeometric2F1[-n, -n, 1 - n, (2*a)/(a + b*x)])/(b*n*((-a +
b*x)/(a + b*x))^n)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -bcx+ac \right ) ^{n}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*c*x+a*c)^n/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-c*(-a + b*x))**n/(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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