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

Optimal. Leaf size=79 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]

[Out]

(d*(a + b/x)^(1 + m)*x^2)/(2*a) - (b*(2*a*c - b*d*(1 - m))*(a + b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m
, 1 + b/(a*x)])/(2*a^3*(1 + m))

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Rubi [A]  time = 0.0378231, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {434, 446, 78, 65} \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^m*(c + d*x),x]

[Out]

(d*(a + b/x)^(1 + m)*x^2)/(2*a) - (b*(2*a*c - b*d*(1 - m))*(a + b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m
, 1 + b/(a*x)])/(2*a^3*(1 + m))

Rule 434

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

Rule 446

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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])

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^m (c+d x) \, dx &=\int \left (a+\frac{b}{x}\right )^m \left (d+\frac{c}{x}\right ) x \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^m (d+c x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{d \left (a+\frac{b}{x}\right )^{1+m} x^2}{2 a}-\frac{(2 a c+b d (-1+m)) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{x^2} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{d \left (a+\frac{b}{x}\right )^{1+m} x^2}{2 a}-\frac{b (2 a c-b d (1-m)) \left (a+\frac{b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;1+\frac{b}{a x}\right )}{2 a^3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0309613, size = 73, normalized size = 0.92 \[ \frac{(a x+b) \left (a+\frac{b}{x}\right )^m \left (a^2 d (m+1) x^2+b (-2 a c-b d (m-1)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )\right )}{2 a^3 (m+1) x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^m*(c + d*x),x]

[Out]

((a + b/x)^m*(b + a*x)*(a^2*d*(1 + m)*x^2 + b*(-2*a*c - b*d*(-1 + m))*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b
/(a*x)]))/(2*a^3*(1 + m)*x)

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*(a + b/x)^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 )} \left (\frac{a x + b}{x}\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 3.42879, size = 75, normalized size = 0.95 \begin{align*} \frac{b^{m} c x x^{- m} \Gamma \left (1 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 1 - m \\ 2 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac{b^{m} d x^{2} x^{- m} \Gamma \left (2 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 2 - m \\ 3 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**m*c*x*x**(-m)*gamma(1 - m)*hyper((-m, 1 - m), (2 - m,), a*x*exp_polar(I*pi)/b)/gamma(2 - m) + b**m*d*x**2*x
**(-m)*gamma(2 - m)*hyper((-m, 2 - m), (3 - m,), a*x*exp_polar(I*pi)/b)/gamma(3 - m)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*(a + b/x)^m, x)