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

Optimal. Leaf size=124 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac{(a+b x)^{n+1} (c+d x)^{-n} (a d (1-n)+b c (n+1)) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (n+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0511325, antiderivative size = 120, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 70, 69} \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac{(a+b x)^{n+1} (c+d x)^{-n} \left (\frac{a (1-n)}{n+1}+\frac{b c}{d}\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/(c + d*x)^n,x]

[Out]

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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])

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

Mathematica [A]  time = 0.0643206, size = 101, normalized size = 0.81 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n} \left (b (c+d x)-\frac{(b c (n+1)-a d (n-1)) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac{d (a+b x)}{a d-b c}\right )}{n+1}\right )}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(b*(c + d*x) - ((-(a*d*(-1 + n)) + b*c*(1 + n))*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometri
c2F1[n, 1 + n, 2 + n, (d*(a + b*x))/(-(b*c) + a*d)])/(1 + n)))/(2*b^2*d*(c + d*x)^n)

________________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}x}{ \left ( dx+c \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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