Optimal. Leaf size=55 \[ \frac {1}{2} a^2 x \left (c x^n\right )^{\frac {1}{n}}+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{4} b^2 x \left (c x^n\right )^{3/n} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 43} \[ \frac {1}{2} a^2 x \left (c x^n\right )^{\frac {1}{n}}+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{4} b^2 x \left (c x^n\right )^{3/n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rule 368
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 )^2 \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x (a+b x)^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 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {1}{2} a^2 x \left (c x^n\right )^{\frac {1}{n}}+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{4} b^2 x \left (c x^n\right )^{3/n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.89 \[ \frac {1}{12} x \left (c x^n\right )^{\frac {1}{n}} \left (6 a^2+8 a b \left (c x^n\right )^{\frac {1}{n}}+3 b^2 \left (c x^n\right )^{2/n}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 43, normalized size = 0.78 \[ \frac {1}{4} \, b^{2} c^{\frac {3}{n}} x^{4} + \frac {2}{3} \, a b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a^{2} c^{\left (\frac {1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 43, normalized size = 0.78 \[ \frac {1}{4} \, b^{2} c^{\frac {3}{n}} x^{4} + \frac {2}{3} \, a b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a^{2} c^{\left (\frac {1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{2} \left (c \,x^{n}\right )^{\frac {1}{n}}\, dx \]
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 \[ \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} \left (c x^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 47, normalized size = 0.85 \[ \frac {x\,{\left (c\,x^n\right )}^{1/n}\,\left (3\,b^2\,{\left (c\,x^n\right )}^{2/n}+6\,a^2+8\,a\,b\,{\left (c\,x^n\right )}^{1/n}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 56, normalized size = 1.02 \[ \frac {a^{2} c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2} + \frac {2 a b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} + \frac {b^{2} c^{\frac {3}{n}} x \left (x^{n}\right )^{\frac {3}{n}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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