3.2927 \(\int \frac{(a+b (c x)^n)^p}{x} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\left (a+b (c x)^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c x)^n}{a}+1\right )}{a n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0406841, antiderivative size = 46, 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 = {367, 12, 266, 65} \[ -\frac{\left (a+b (c x)^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c x)^n}{a}+1\right )}{a n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x)^n)^p/x,x]

[Out]

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

Rule 367

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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 \frac{\left (a+b (c x)^n\right )^p}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c \left (a+b x^n\right )^p}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{\left (a+b x^n\right )^p}{x} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,(c x)^n\right )}{n}\\ &=-\frac{\left (a+b (c x)^n\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac{b (c x)^n}{a}\right )}{a n (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0244869, size = 46, normalized size = 1. \[ -\frac{\left (a+b (c x)^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c x)^n}{a}+1\right )}{a n (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x)^n)^p/x,x]

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*(c*x)**n)**p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p/x, x)