Optimal. Leaf size=19 \[ a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \]
________________________________________________________________________________________
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.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, 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.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.15, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.46, 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.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, 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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 22, normalized size = 1.16 \begin {gather*} \frac {b x \,{\mathrm e}^{\frac {\ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{n}}}{2}+a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} b c^{\left (\frac {1}{n}\right )} \int {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}\,{d x} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 19, normalized size = 1.00 \begin {gather*} a x + \frac {b c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________