3.1890 \(\int (\frac{d (a+b x)}{-b c+a d})^m (c+d x)^n \, dx\)

Optimal. Leaf size=45 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.01852, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {186, 69} \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 186

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

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]))

Rubi steps

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

Mathematica [A]  time = 0.0327576, size = 88, normalized size = 1.96 \[ \frac{(a+b x) (c+d x)^n \left (\frac{d (a+b x)}{a d-b c}\right )^m \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{d \left ( bx+a \right ) }{ad-bc}} \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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