Optimal. Leaf size=43 \[ a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]
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Rubi [A] time = 0.01, antiderivative size = 43, 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} \[ a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]
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 )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (a+b x^3\right )^2 \, 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^2+2 a b x^3+b^2 x^6\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 35, normalized size = 0.81 \[ \frac {1}{7} \, b^{2} c^{\frac {6}{n}} x^{7} + \frac {1}{2} \, a b c^{\frac {3}{n}} x^{4} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 35, normalized size = 0.81 \[ \frac {1}{7} \, b^{2} c^{\frac {6}{n}} x^{7} + \frac {1}{2} \, a b c^{\frac {3}{n}} x^{4} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {3}{n}}+a \right )^{2}\, dx \]
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 \[ b^{2} c^{\frac {6}{n}} \int {\left (x^{n}\right )}^{\frac {6}{n}}\,{d x} + 2 \, a b c^{\frac {3}{n}} \int {\left (x^{n}\right )}^{\frac {3}{n}}\,{d x} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 39, normalized size = 0.91 \[ a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{6/n}}{7}+\frac {a\,b\,x\,{\left (c\,x^n\right )}^{3/n}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 41, normalized size = 0.95 \[ a^{2} x + \frac {a b c^{\frac {3}{n}} x \left (x^{n}\right )^{\frac {3}{n}}}{2} + \frac {b^{2} c^{\frac {6}{n}} x \left (x^{n}\right )^{\frac {6}{n}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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