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

Optimal. Leaf size=95 \[ -\frac{d (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0270198, antiderivative size = 95, normalized size of antiderivative = 1., 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 = {86, 65, 68} \[ -\frac{d (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x*(c + d*x)),x]

[Out]

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

Rule 86

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

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

Mathematica [A]  time = 0.0262878, size = 85, normalized size = 0.89 \[ \frac{(a+b x)^{n+1} \left (a d \, _2F_1\left (1,n+1;n+2;\frac{d (a+b x)}{a d-b c}\right )+(b c-a d) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )\right )}{a c (n+1) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(a*d*Hypergeometric2F1[1, 1 + n, 2 + n, (d*(a + b*x))/(-(b*c) + a*d)] + (b*c - a*d)*Hyperge
ometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a*c*(-(b*c) + a*d)*(1 + n))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{d x^{2} + c x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n/(d*x^2 + c*x), 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((b*x+a)**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 (b x + a\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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