Optimal. Leaf size=83 \[ \frac{x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
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Rubi [A] time = 0.0303635, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {368, 43} \[ \frac{x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{1+p}}{b^2 (1+p)}+\frac{x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{2+p}}{b^2 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0284865, size = 63, normalized size = 0.76 \[ \frac{x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \left (b (p+1) \left (c x^n\right )^{\frac{1}{n}}-a\right )}{b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.291, size = 1007, normalized size = 12.1 \begin{align*} \text{result too large to display} \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{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56547, size = 150, normalized size = 1.81 \begin{align*} \frac{{\left (a b c^{\left (\frac{1}{n}\right )} p x +{\left (b^{2} p + b^{2}\right )} c^{\frac{2}{n}} x^{2} - a^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\frac{2}{n}}} \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 x \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.5671, size = 184, normalized size = 2.22 \begin{align*} \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac{2}{n}} p x^{2} +{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} a b c^{\left (\frac{1}{n}\right )} p x +{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac{2}{n}} x^{2} -{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} a^{2}}{b^{2} c^{\frac{2}{n}} p^{2} + 3 \, b^{2} c^{\frac{2}{n}} p + 2 \, b^{2} c^{\frac{2}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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