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

Optimal. Leaf size=72 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0638382, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/(c + d*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.1665, size = 54, normalized size = 0.75 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n}{{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0692971, size = 80, normalized size = 1.11 \[ -\frac{(a+b x)^n (c+d x)^{1-n} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;\frac{b (c+d x)}{b c-a d}\right )}{d (n-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.001, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/((d*x+c)^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n/(d*x + c)^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/(d*x + c)^n, x)