Optimal. Leaf size=66 \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {b c n x^{3+n} \, _2F_1\left (1,\frac {3+n}{2 n};\frac {3 (1+n)}{2 n};c^2 x^{2 n}\right )}{3 (3+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6037, 371}
\begin {gather*} \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {b c n x^{n+3} \, _2F_1\left (1,\frac {n+3}{2 n};\frac {3 (n+1)}{2 n};c^2 x^{2 n}\right )}{3 (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 6037
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {1}{3} (b c n) \int \frac {x^{2+n}}{1-c^2 x^{2 n}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {b c n x^{3+n} \, _2F_1\left (1,\frac {3+n}{2 n};\frac {3 (1+n)}{2 n};c^2 x^{2 n}\right )}{3 (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 73, normalized size = 1.11 \begin {gather*} \frac {a x^3}{3}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c x^n\right )-\frac {b c n x^{3+n} \, _2F_1\left (1,\frac {3+n}{2 n};1+\frac {3+n}{2 n};c^2 x^{2 n}\right )}{3 (3+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \arctanh \left (c \,x^{n}\right )\right )\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 x^{2} \left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )\, dx \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*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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