Optimal. Leaf size=53 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]
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Rubi [A] time = 0.0335463, 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, 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.
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Rule 370
Rule 266
Rule 65
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*}
Mathematica [A] time = 0.055117, size = 53, normalized size = 1. \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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