Optimal. Leaf size=20 \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]
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Rubi [A] time = 0.023506, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {1775} \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]
Antiderivative was successfully verified.
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Rule 1775
Rubi steps
\begin{align*} \int \left (a+b x^n+c x^{2 n}\right )^p \left (a+b (1+n+n p) x^n+c (1+2 n (1+p)) x^{2 n}\right ) \, dx &=x \left (a+b x^n+c x^{2 n}\right )^{1+p}\\ \end{align*}
Mathematica [A] time = 0.354304, size = 19, normalized size = 0.95 \[ x \left (a+x^n \left (b+c x^n\right )\right )^{p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 33, normalized size = 1.7 \begin{align*} x \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4414, size = 47, normalized size = 2.35 \begin{align*}{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78385, size = 77, normalized size = 3.85 \begin{align*}{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15811, size = 89, normalized size = 4.45 \begin{align*}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} c x x^{2 \, n} +{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} b x x^{n} +{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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