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]
_______________________________________________________________________________________
Rubi [A] time = 0.0841715, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x^q)^n)^p/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**q)**n)**p/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.104769, size = 70, normalized size = 1.32 \[ \frac{\left (\frac{a \left (c x^q\right )^{-n}}{b}+1\right )^{-p} \left (a+b \left (c x^q\right )^n\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a \left (c x^q\right )^{-n}}{b}\right )}{n p q} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c*x^q)^n)^p/x,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.472, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^q)^n)^p/x,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^q)^n*b + a)^p/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^q)^n*b + a)^p/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**q)**n)**p/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^q)^n*b + a)^p/x,x, algorithm="giac")
[Out]