Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^7}{14 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {28, 267}
\begin {gather*} \frac {\left (a+b x^2\right )^7}{14 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 267
Rubi steps
\begin {align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\left (a+b x^2\right )^7}{14 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^7}{14 b} \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. \(68\) vs.
\(2(14)=28\).
time = 0.05, size = 69, normalized size = 4.31
method | result | size |
default | \(\frac {1}{2} a^{6} x^{2}+\frac {3}{2} a^{5} b \,x^{4}+\frac {5}{2} a^{4} b^{2} x^{6}+\frac {5}{2} a^{3} b^{3} x^{8}+\frac {3}{2} a^{2} b^{4} x^{10}+\frac {1}{2} a \,b^{5} x^{12}+\frac {1}{14} b^{6} x^{14}\) | \(69\) |
norman | \(\frac {1}{2} a^{6} x^{2}+\frac {3}{2} a^{5} b \,x^{4}+\frac {5}{2} a^{4} b^{2} x^{6}+\frac {5}{2} a^{3} b^{3} x^{8}+\frac {3}{2} a^{2} b^{4} x^{10}+\frac {1}{2} a \,b^{5} x^{12}+\frac {1}{14} b^{6} x^{14}\) | \(69\) |
risch | \(\frac {1}{2} a^{6} x^{2}+\frac {3}{2} a^{5} b \,x^{4}+\frac {5}{2} a^{4} b^{2} x^{6}+\frac {5}{2} a^{3} b^{3} x^{8}+\frac {3}{2} a^{2} b^{4} x^{10}+\frac {1}{2} a \,b^{5} x^{12}+\frac {1}{14} b^{6} x^{14}\) | \(69\) |
gosper | \(\frac {x^{2} \left (b^{6} x^{12}+7 a \,b^{5} x^{10}+21 a^{2} b^{4} x^{8}+35 a^{3} b^{3} x^{6}+35 a^{4} b^{2} x^{4}+21 a^{5} b \,x^{2}+7 a^{6}\right )}{14}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (14) = 28\).
time = 0.30, size = 68, normalized size = 4.25 \begin {gather*} \frac {1}{14} \, b^{6} x^{14} + \frac {1}{2} \, a b^{5} x^{12} + \frac {3}{2} \, a^{2} b^{4} x^{10} + \frac {5}{2} \, a^{3} b^{3} x^{8} + \frac {5}{2} \, a^{4} b^{2} x^{6} + \frac {3}{2} \, a^{5} b x^{4} + \frac {1}{2} \, a^{6} x^{2} \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. 68 vs.
\(2 (14) = 28\).
time = 0.33, size = 68, normalized size = 4.25 \begin {gather*} \frac {1}{14} \, b^{6} x^{14} + \frac {1}{2} \, a b^{5} x^{12} + \frac {3}{2} \, a^{2} b^{4} x^{10} + \frac {5}{2} \, a^{3} b^{3} x^{8} + \frac {5}{2} \, a^{4} b^{2} x^{6} + \frac {3}{2} \, a^{5} b x^{4} + \frac {1}{2} \, a^{6} x^{2} \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. 78 vs.
\(2 (10) = 20\).
time = 0.02, size = 78, normalized size = 4.88 \begin {gather*} \frac {a^{6} x^{2}}{2} + \frac {3 a^{5} b x^{4}}{2} + \frac {5 a^{4} b^{2} x^{6}}{2} + \frac {5 a^{3} b^{3} x^{8}}{2} + \frac {3 a^{2} b^{4} x^{10}}{2} + \frac {a b^{5} x^{12}}{2} + \frac {b^{6} x^{14}}{14} \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. 68 vs.
\(2 (14) = 28\).
time = 4.93, size = 68, normalized size = 4.25 \begin {gather*} \frac {1}{14} \, b^{6} x^{14} + \frac {1}{2} \, a b^{5} x^{12} + \frac {3}{2} \, a^{2} b^{4} x^{10} + \frac {5}{2} \, a^{3} b^{3} x^{8} + \frac {5}{2} \, a^{4} b^{2} x^{6} + \frac {3}{2} \, a^{5} b x^{4} + \frac {1}{2} \, a^{6} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 68, normalized size = 4.25 \begin {gather*} \frac {a^6\,x^2}{2}+\frac {3\,a^5\,b\,x^4}{2}+\frac {5\,a^4\,b^2\,x^6}{2}+\frac {5\,a^3\,b^3\,x^8}{2}+\frac {3\,a^2\,b^4\,x^{10}}{2}+\frac {a\,b^5\,x^{12}}{2}+\frac {b^6\,x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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