Optimal. Leaf size=19 \[ a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30}
\begin {gather*} a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right ) \, dx &=a x+b \int \left (c x^n\right )^{\frac {1}{n}} \, dx\\ &=a x+\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}\right ) \int x \, dx}{x}\\ &=a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 22, normalized size = 1.16
method | result | size |
default | \(a x +\frac {x b \,{\mathrm e}^{\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}}}{2}\) | \(22\) |
norman | \(a x +\frac {x b \,{\mathrm e}^{\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}}}{2}\) | \(22\) |
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 = 0.37, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, b c^{\left (\frac {1}{n}\right )} x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 15, normalized size = 0.79 \begin {gather*} a x + \frac {b x \left (c x^{n}\right )^{\frac {1}{n}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.53, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, b c^{\left (\frac {1}{n}\right )} x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 17, normalized size = 0.89 \begin {gather*} a\,x+\frac {b\,x\,{\left (c\,x^n\right )}^{1/n}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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