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}
\begin {gather*} \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 {gather*}
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 = 0.06, size = 30, normalized size = 0.91 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (c \,x^{n}\right )^{\frac {1}{n}} \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a c^{\left (\frac {1}{n}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 26, normalized size = 0.79 \begin {gather*} \frac {a x \left (c x^{n}\right )^{\frac {1}{n}}}{2} + \frac {b x \left (c x^{n}\right )^{\frac {2}{n}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a c^{\left (\frac {1}{n}\right )} x^{2} \end {gather*}
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 \begin {gather*} \frac {x\,{\left (c\,x^n\right )}^{1/n}\,\left (3\,a+2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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