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

Optimal. Leaf size=131 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0971801, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {514, 375, 103, 156, 65, 68} \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x)^n*x)/(c + d*x),x]

[Out]

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

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 65

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

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.0632969, size = 119, normalized size = 0.91 \[ \frac{(a x+b) \left (a+\frac{b}{x}\right )^n \left (a^2 c^2 \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )+(a c-b d) \left ((b d n-a c) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )+a d (n+1) x\right )\right )}{a^2 d^2 (n+1) x (a c-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.495, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{dx+c} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x)^n*x/(d*x + c), x)