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]

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

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {70, 69} \[ \frac {(a+b x)^{n+2} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+2;n+3;-\frac {d (a+b x)}{b c-a d}\right )}{b (n+2)} \]

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)

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rubi steps

\begin {align*} \int (a+b x)^{1+n} (c+d x)^{-n} \, dx &=\left ((c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{1+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=\frac {(a+b x)^{2+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {d (a+b x)}{b c-a d}\right )}{b (2+n)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 89, normalized size = 1.24 \[ \frac {(b c-a d) (a+b x)^n (c+d x)^{1-n} \left (\frac {d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n-1,1-n;2-n;\frac {b (c+d x)}{b c-a d}\right )}{d^2 (n-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n + 1}}{{\left (d x + c\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (d x + c\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{n +1} \left (d x +c \right )^{-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (d x + c\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{n+1}}{{\left (c+d\,x\right )}^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(n + 1)/(c + d*x)^n,x)

[Out]

int((a + b*x)^(n + 1)/(c + d*x)^n, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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