Optimal. Leaf size=62 \[ a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac {3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac {1}{7} b^3 x \left (c x^n\right )^{6/n} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {254, 194} \begin {gather*} a^2 b x \left (c x^n\right )^{2/n}+a^3 x+\frac {3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac {1}{7} b^3 x \left (c x^n\right )^{6/n} \end {gather*}
Antiderivative was successfully verified.
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Rule 194
Rule 254
Rubi steps
\begin {align*} \int \left (a+b \left (c x^n\right )^{2/n}\right )^3 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (a+b x^2\right )^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 (a^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac {3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac {1}{7} b^3 x \left (c x^n\right )^{6/n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 1.00 \begin {gather*} a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac {3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac {1}{7} b^3 x \left (c x^n\right )^{6/n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (c x^n\right )^{2/n}\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.26, size = 52, normalized size = 0.84 \begin {gather*} \frac {1}{7} \, b^{3} c^{\frac {6}{n}} x^{7} + \frac {3}{5} \, a b^{2} c^{\frac {4}{n}} x^{5} + a^{2} b c^{\frac {2}{n}} x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 52, normalized size = 0.84 \begin {gather*} \frac {1}{7} \, b^{3} c^{\frac {6}{n}} x^{7} + \frac {3}{5} \, a b^{2} c^{\frac {4}{n}} x^{5} + a^{2} b c^{\frac {2}{n}} x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \left (c \,x^{n}\right )^{\frac {2}{n}}+a \right )^{3}\, dx \end {gather*}
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 \begin {gather*} b^{3} c^{\frac {6}{n}} \int {\left (x^{n}\right )}^{\frac {6}{n}}\,{d x} + 3 \, a b^{2} c^{\frac {4}{n}} \int {\left (x^{n}\right )}^{\frac {4}{n}}\,{d x} + 3 \, a^{2} b c^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 58, normalized size = 0.94 \begin {gather*} a^3\,x+\frac {b^3\,x\,{\left (c\,x^n\right )}^{6/n}}{7}+a^2\,b\,x\,{\left (c\,x^n\right )}^{2/n}+\frac {3\,a\,b^2\,x\,{\left (c\,x^n\right )}^{4/n}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 63, normalized size = 1.02 \begin {gather*} a^{3} x + a^{2} b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}} + \frac {3 a b^{2} c^{\frac {4}{n}} x \left (x^{n}\right )^{\frac {4}{n}}}{5} + \frac {b^{3} c^{\frac {6}{n}} x \left (x^{n}\right )^{\frac {6}{n}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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