Optimal. Leaf size=19 \[ \frac {\left (a+b x^n\right )^4}{4 b n} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267}
\begin {gather*} \frac {\left (a+b x^n\right )^4}{4 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int x^{-1+n} \left (a+b x^n\right )^3 \, dx &=\frac {\left (a+b x^n\right )^4}{4 b n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(19)=38\).
time = 0.02, size = 45, normalized size = 2.37 \begin {gather*} \frac {x^n \left (4 a^3+6 a^2 b x^n+4 a b^2 x^{2 n}+b^3 x^{3 n}\right )}{4 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs.
\(2(17)=34\).
time = 0.23, size = 52, normalized size = 2.74
method | result | size |
risch | \(\frac {a^{3} x^{n}}{n}+\frac {a \,b^{2} x^{3 n}}{n}+\frac {b^{3} x^{4 n}}{4 n}+\frac {3 a^{2} b \,x^{2 n}}{2 n}\) | \(52\) |
norman | \(\frac {a^{3} {\mathrm e}^{n \ln \left (x \right )}}{n}+\frac {a \,b^{2} {\mathrm e}^{3 n \ln \left (x \right )}}{n}+\frac {b^{3} {\mathrm e}^{4 n \ln \left (x \right )}}{4 n}+\frac {3 a^{2} b \,{\mathrm e}^{2 n \ln \left (x \right )}}{2 n}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.89 \begin {gather*} \frac {{\left (b x^{n} + a\right )}^{4}}{4 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (17) = 34\).
time = 0.45, size = 45, normalized size = 2.37 \begin {gather*} \frac {b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \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. 54 vs.
\(2 (12) = 24\).
time = 0.59, size = 54, normalized size = 2.84 \begin {gather*} \begin {cases} \frac {a^{3} x^{n}}{n} + \frac {3 a^{2} b x^{2 n}}{2 n} + \frac {a b^{2} x^{3 n}}{n} + \frac {b^{3} x^{4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\left (x \right )} & \text {otherwise} \end {cases} \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. 45 vs.
\(2 (17) = 34\).
time = 2.17, size = 45, normalized size = 2.37 \begin {gather*} \frac {b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 51, normalized size = 2.68 \begin {gather*} \frac {a^3\,x^n}{n}+\frac {b^3\,x^{4\,n}}{4\,n}+\frac {3\,a^2\,b\,x^{2\,n}}{2\,n}+\frac {a\,b^2\,x^{3\,n}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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