3.1866 \(\int (a+b x)^{-n} (c+d x)^n \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {70, 69} \[ \frac {(a+b x)^{-n} (c+d x)^{n+1} \left (-\frac {d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^n/(a + b*x)^n,x]

[Out]

((-((d*(a + b*x))/(b*c - a*d)))^n*(c + d*x)^(1 + n)*Hypergeometric2F1[n, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*
d)])/(d*(1 + n)*(a + b*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)^{-n} (c+d x)^n \, dx &=\left ((a+b x)^{-n} \left (\frac {d (a+b x)}{-b c+a d}\right )^n\right ) \int (c+d x)^n \left (-\frac {a d}{b c-a d}-\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=\frac {(a+b x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^n (c+d x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.99 \[ \frac {(a+b x)^{-n} (c+d x)^{n+1} \left (\frac {d (a+b x)}{a d-b c}\right )^n \, _2F_1\left (n,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^n/(a + b*x)^n,x]

[Out]

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

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)^n/(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n/(a + b*x)^n,x)

[Out]

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

[Out]

Exception raised: HeuristicGCDFailed

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