3.3050 \(\int \frac {(a+b (c x^q)^n)^p}{x} \, dx\)

Optimal. Leaf size=53 \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {370, 266, 65} \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^q)^n)^p/x,x]

[Out]

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

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 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 370

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 53, normalized size = 1.00 \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^q)^n)^p/x,x]

[Out]

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

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x^q)^n*b + a)^p/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x, x)

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \left (c \,x^{q}\right )^{n}+a \right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^q)^n)^p/x,x)

[Out]

int((a + b*(c*x^q)^n)^p/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*(c*x**q)**n)**p/x, x)

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