Optimal. Leaf size=33 \[ \frac {1}{2} a x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b x \left (c x^n\right )^{2/n} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {15, 14, 30} \[ \frac {1}{2} a x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 15
Rule 30
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 ) \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right ) \, dx}{x}\\ &=\frac {\left (c x^n\right )^{\frac {1}{n}} \int \left (a x+b x \left (c x^n\right )^{\frac {1}{n}}\right ) \, dx}{x}\\ &=\frac {1}{2} a x \left (c x^n\right )^{\frac {1}{n}}+\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}\right ) \int x \left (c x^n\right )^{\frac {1}{n}} \, dx}{x}\\ &=\frac {1}{2} a x \left (c x^n\right )^{\frac {1}{n}}+\frac {\left (b \left (c x^n\right )^{2/n}\right ) \int x^2 \, dx}{x^2}\\ &=\frac {1}{2} a x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b x \left (c x^n\right )^{2/n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.91 \[ \frac {1}{6} x \left (c x^n\right )^{\frac {1}{n}} \left (3 a+2 b \left (c x^n\right )^{\frac {1}{n}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 25, normalized size = 0.76 \[ \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a c^{\left (\frac {1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 25, normalized size = 0.76 \[ \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a c^{\left (\frac {1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right ) \left (c \,x^{n}\right )^{\frac {1}{n}}\, dx \]
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 \[ \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} \left (c x^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 28, normalized size = 0.85 \[ \frac {x\,{\left (c\,x^n\right )}^{1/n}\,\left (3\,a+2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 32, normalized size = 0.97 \[ \frac {a c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2} + \frac {b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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