Optimal. Leaf size=27 \[ \frac {\left (a+b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1482, 643}
\begin {gather*} \frac {\left (a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 1482
Rubi steps
\begin {align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx &=\frac {\text {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 26, normalized size = 0.96 \begin {gather*} \frac {\left (a+x^n \left (b+c x^n\right )\right )^{1+p}}{n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 40, normalized size = 1.48
method | result | size |
risch | \(\frac {\left (a +b \,x^{n}+c \,x^{2 n}\right ) \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p}}{n \left (1+p \right )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 39, normalized size = 1.44 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 1.41 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{n p + n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.49, size = 27, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p + 1}}{n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.57, size = 56, normalized size = 2.07 \begin {gather*} {\left (a+b\,x^n+c\,x^{2\,n}\right )}^p\,\left (\frac {a}{n\,\left (p+1\right )}+\frac {b\,x^n}{n\,\left (p+1\right )}+\frac {c\,x^{2\,n}}{n\,\left (p+1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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