Optimal. Leaf size=77 \[ \frac {a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}-\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 77, 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} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}-\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x} \, 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 {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b}+\frac {a^2 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 = 0.04, size = 67, normalized size = 0.87 \begin {gather*} \frac {x^3 \left (c x^n\right )^{-3/n} \left (2 a^2 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )+b \left (c x^n\right )^{\frac {1}{n}} \left (b \left (c x^n\right )^{\frac {1}{n}}-2 a\right )\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.20, size = 55, normalized size = 0.71 \begin {gather*} \frac {b^{2} c^{\frac {2}{n}} x^{2} - 2 \, a b c^{\left (\frac {1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{2 \, 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 (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 310, normalized size = 4.03 \begin {gather*} \frac {a^{2} 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 {a \,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} 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}}}{2 b} \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*} \int \frac {x^{2}}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a}\,{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}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,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}}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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