Optimal. Leaf size=70 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]
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Rubi [A] time = 0.0306298, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 43} \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 368
Rule 43
Rubi steps
\begin{align*} \int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2}+\frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0268751, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a-4 b \left (c x^n\right )^{\frac{1}{n}}\right ) \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{20 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.247, size = 0, normalized size = 0. \begin{align*} \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{3}\, 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{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3} \left (c x^{n}\right )^{\left (\frac{1}{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62265, size = 117, normalized size = 1.67 \begin{align*} \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.34525, size = 76, normalized size = 1.09 \begin{align*} \frac{a^{3} c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + a^{2} b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{b^{3} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15871, size = 81, normalized size = 1.16 \begin{align*} \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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