Optimal. Leaf size=20 \[ x \left (a+b x^n+c x^{2 n}\right )^{1+p} \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {1789}
\begin {gather*} x \left (a+b x^n+c x^{2 n}\right )^{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 1789
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*}
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Mathematica [A]
time = 0.47, size = 19, normalized size = 0.95 \begin {gather*} x \left (a+x^n \left (b+c x^n\right )\right )^{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 33, normalized size = 1.65
method | result | size |
risch | \(x \left (a +b \,x^{n}+c \,x^{2 n}\right ) \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 35, normalized size = 1.75 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 1.75 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (17) = 34\).
time = 30.11, size = 63, normalized size = 3.15 \begin {gather*} a x \left (a + b x^{n} + c x^{2 n}\right )^{p} + b x x^{n} \left (a + b x^{n} + c x^{2 n}\right )^{p} + c x x^{2 n} \left (a + b x^{n} + c x^{2 n}\right )^{p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (20) = 40\).
time = 2.99, size = 66, normalized size = 3.30 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.18, size = 35, normalized size = 1.75 \begin {gather*} {\left (a+b\,x^n+c\,x^{2\,n}\right )}^p\,\left (a\,x+b\,x\,x^n+c\,x\,x^{2\,n}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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