Optimal. Leaf size=51 \[ -\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}+1\right )}{a (p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 65} \[ -\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}+1\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 368
Rubi steps
\begin {align*} \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx &=\operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}\right )}{a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ -\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}+1\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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