Optimal. Leaf size=65 \[ a^3 x+\frac {3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac {3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac {1}{10} b^3 x \left (c x^n\right )^{9/n} \]
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Rubi [A] time = 0.02, antiderivative size = 65, 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} \[ \frac {3}{4} a^2 b x \left (c x^n\right )^{3/n}+a^3 x+\frac {3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac {1}{10} b^3 x \left (c x^n\right )^{9/n} \]
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 )^{3/n}\right )^3 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (a+b x^3\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^3+3 a b^2 x^6+b^3 x^9\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=a^3 x+\frac {3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac {3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac {1}{10} b^3 x \left (c x^n\right )^{9/n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 1.00 \[ a^3 x+\frac {3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac {3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac {1}{10} b^3 x \left (c x^n\right )^{9/n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 53, normalized size = 0.82 \[ \frac {1}{10} \, b^{3} c^{\frac {9}{n}} x^{10} + \frac {3}{7} \, a b^{2} c^{\frac {6}{n}} x^{7} + \frac {3}{4} \, a^{2} b c^{\frac {3}{n}} x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 53, normalized size = 0.82 \[ \frac {1}{10} \, b^{3} c^{\frac {9}{n}} x^{10} + \frac {3}{7} \, a b^{2} c^{\frac {6}{n}} x^{7} + \frac {3}{4} \, a^{2} b c^{\frac {3}{n}} x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {3}{n}}+a \right )^{3}\, dx \]
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 \[ b^{3} c^{\frac {9}{n}} \int {\left (x^{n}\right )}^{\frac {9}{n}}\,{d x} + 3 \, a b^{2} c^{\frac {6}{n}} \int {\left (x^{n}\right )}^{\frac {6}{n}}\,{d x} + 3 \, a^{2} b c^{\frac {3}{n}} \int {\left (x^{n}\right )}^{\frac {3}{n}}\,{d x} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 59, normalized size = 0.91 \[ a^3\,x+\frac {b^3\,x\,{\left (c\,x^n\right )}^{9/n}}{10}+\frac {3\,a^2\,b\,x\,{\left (c\,x^n\right )}^{3/n}}{4}+\frac {3\,a\,b^2\,x\,{\left (c\,x^n\right )}^{6/n}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 66, normalized size = 1.02 \[ a^{3} x + \frac {3 a^{2} b c^{\frac {3}{n}} x \left (x^{n}\right )^{\frac {3}{n}}}{4} + \frac {3 a b^{2} c^{\frac {6}{n}} x \left (x^{n}\right )^{\frac {6}{n}}}{7} + \frac {b^{3} c^{\frac {9}{n}} x \left (x^{n}\right )^{\frac {9}{n}}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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