3.3051 \(\int \frac {(a+b (c x^q)^n)^p}{x^2} \, dx\)

Optimal. Leaf size=71 \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac {1}{n q};1-\frac {1}{n q};-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 71, 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, 365, 364} \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac {1}{n q};1-\frac {1}{n q};-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^q)^n)^p/x^2,x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

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

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 71, normalized size = 1.00 \[ -\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac {1}{n q};1-\frac {1}{n q};-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^q)^n)^p/x^2,x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="fricas")

[Out]

integral(((c*x^q)^n*b + a)^p/x^2, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="giac")

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)

________________________________________________________________________________________

maple [F]  time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \left (c \,x^{q}\right )^{n}+a \right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*(c*x^q)^n+a)^p/x^2,x)

[Out]

int((b*(c*x^q)^n+a)^p/x^2,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^q)^n)^p/x^2,x, algorithm="maxima")

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^q)^n)^p/x^2,x)

[Out]

int((a + b*(c*x^q)^n)^p/x^2, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x**q)**n)**p/x**2,x)

[Out]

Integral((a + b*(c*x**q)**n)**p/x**2, x)

________________________________________________________________________________________