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]

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

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{n} \left (-\frac {b d x + a d}{b c - a d}\right )^{m}, x\right ) \]

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

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)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (\frac {\left (b x +a \right ) d}{a d -b c}\right )^{m} \left (d x +c \right )^{n}\, dx \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (c+d\,x\right )}^n\,{\left (\frac {d\,\left (a+b\,x\right )}{a\,d-b\,c}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: HeuristicGCDFailed

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