Optimal. Leaf size=90 \[ -\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}+\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 43} \begin {gather*} -\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}+\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2}-\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.74 \begin {gather*} \frac {x^3 \left (c x^n\right )^{-3/n} \left (-\frac {a^2}{a+b \left (c x^n\right )^{\frac {1}{n}}}-2 a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 83, normalized size = 0.92 \begin {gather*} \frac {b^{2} c^{\frac {2}{n}} x^{2} + a b c^{\left (\frac {1}{n}\right )} x - a^{2} - 2 \, {\left (a b c^{\left (\frac {1}{n}\right )} x + a^{2}\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{4} c^{\frac {4}{n}} x + a b^{3} c^{\frac {3}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\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.
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maple [C] time = 0.13, size = 386, normalized size = 4.29 \begin {gather*} -\frac {2 a \,x^{3} c^{-\frac {2}{n}} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {3 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}} \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^{3}}-\frac {x^{3} 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 {2 x^{3} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{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 )}{n}}}{b^{2}}+\frac {x^{3}}{\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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x^{3}}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} - 2 \, \int \frac {x^{2}}{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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\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.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\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.
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