Optimal. Leaf size=43 \[ a^2 x+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{5} b^2 x \left (c x^n\right )^{4/n} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {254, 194} \begin {gather*} a^2 x+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{5} b^2 x \left (c x^n\right )^{4/n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 194
Rule 254
Rubi steps
\begin {align*} \int \left (a+b \left (c x^n\right )^{2/n}\right )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (a+b x^2\right )^2 \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=a^2 x+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{5} b^2 x \left (c x^n\right )^{4/n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} a^2 x+\frac {2}{3} a b x \left (c x^n\right )^{2/n}+\frac {1}{5} b^2 x \left (c x^n\right )^{4/n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (c x^n\right )^{2/n}\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{5} \, b^{2} c^{\frac {4}{n}} x^{5} + \frac {2}{3} \, a b c^{\frac {2}{n}} x^{3} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{5} \, b^{2} c^{\frac {4}{n}} x^{5} + \frac {2}{3} \, a b c^{\frac {2}{n}} x^{3} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \left (c \,x^{n}\right )^{\frac {2}{n}}+a \right )^{2}\, dx \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^{2} c^{\frac {4}{n}} \int {\left (x^{n}\right )}^{\frac {4}{n}}\,{d x} + 2 \, a b c^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.22, size = 39, normalized size = 0.91 \begin {gather*} a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{4/n}}{5}+\frac {2\,a\,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 = 0.57, size = 42, normalized size = 0.98 \begin {gather*} a^{2} x + \frac {2 a b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} + \frac {b^{2} c^{\frac {4}{n}} x \left (x^{n}\right )^{\frac {4}{n}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________