3.1864 \(\int (a+b x)^{1+n} (c+d x)^{-n} \, dx\)

Optimal. Leaf size=72 \[ \frac{(a+b x)^{n+2} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+2;n+3;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+2)} \]

[Out]

((a + b*x)^(2 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 2 + n, 3 +
n, -((d*(a + b*x))/(b*c - a*d))])/(b*(2 + n)*(c + d*x)^n)

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Rubi [A]  time = 0.0646945, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(a+b x)^{n+2} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+2;n+3;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+2)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(2 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 2 + n, 3 +
n, -((d*(a + b*x))/(b*c - a*d))])/(b*(2 + n)*(c + d*x)^n)

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Rubi in Sympy [A]  time = 16.4481, size = 66, normalized size = 0.92 \[ \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{n} \left (c + d x\right )^{- n + 1} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - n - 1, - n + 1 \\ - n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{2} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.620688, size = 200, normalized size = 2.78 \[ a (a+b x)^n (c+d x)^{-n} \left (\frac{3 b c x^2 F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 n x \left (b c F_1\left (3;1-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-n,n+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{(c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;\frac{b (c+d x)}{b c-a d}\right )}{d (n-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*(a + b*x)^n*((3*b*c*x^2*AppellF1[2, -n, n, 3, -((b*x)/a), -((d*x)/c)])/(6*a*c
*AppellF1[2, -n, n, 3, -((b*x)/a), -((d*x)/c)] + 2*n*x*(b*c*AppellF1[3, 1 - n, n
, 4, -((b*x)/a), -((d*x)/c)] - a*d*AppellF1[3, -n, 1 + n, 4, -((b*x)/a), -((d*x)
/c)])) - ((c + d*x)*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)))/(c + d*x)^n

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{1+n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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