Optimal. Leaf size=67 \[ \frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 67, 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 = {368, 43} \begin {gather*} \frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 50, normalized size = 0.75 \begin {gather*} \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.11, size = 52, normalized size = 0.78 \begin {gather*} \frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac {3}{n}} x + a b^{2} c^{\frac {2}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.11, size = 310, normalized size = 4.63 \begin {gather*} \frac {x^{2} \left (c^{-\frac {1}{n}}\right )^{2} \left (\left (x^{n}\right )^{-\frac {1}{n}}\right )^{2} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{b^{2}}-\frac {x^{2} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{a b}+\frac {x^{2}}{\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right ) a} \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*} \frac {x^{2}}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} - \int \frac {x}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________