Optimal. Leaf size=21 \[ a x+\frac {1}{3} b x \left (c x^n\right )^{2/n} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 30} \begin {gather*} a x+\frac {1}{3} b x \left (c x^n\right )^{2/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 )^{2/n}\right ) \, dx &=a x+b \int \left (c x^n\right )^{2/n} \, dx\\ &=a x+\frac {\left (b \left (c x^n\right )^{2/n}\right ) \int x^2 \, dx}{x^2}\\ &=a x+\frac {1}{3} b x \left (c x^n\right )^{2/n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} a x+\frac {1}{3} b x \left (c x^n\right )^{2/n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 21, normalized size = 1.00 \begin {gather*} a x+\frac {1}{3} b x \left (c x^n\right )^{2/n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.22, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, b c^{\frac {2}{n}} x^{3} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 23, normalized size = 1.10 \begin {gather*} \frac {b x \,{\mathrm e}^{\frac {2 \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{n}}}{3}+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^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.17, size = 19, normalized size = 0.90 \begin {gather*} a\,x+\frac {b\,x\,{\left (c\,x^n\right )}^{2/n}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.31, size = 19, normalized size = 0.90 \begin {gather*} a x + \frac {b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________