Optimal. Leaf size=14 \[ \frac {(a+b x)^7}{7 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 32}
\begin {gather*} \frac {(a+b x)^7}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 32
Rubi steps
\begin {align*} \int \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \, dx\\ &=\frac {(a+b x)^7}{7 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^7}{7 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. \(64\) vs.
\(2(12)=24\).
time = 0.64, size = 65, normalized size = 4.64
method | result | size |
default | \(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\) | \(65\) |
norman | \(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\) | \(65\) |
risch | \(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\) | \(65\) |
gosper | \(\frac {x \left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right )}{7}\) | \(66\) |
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. 97 vs.
\(2 (12) = 24\).
time = 0.28, size = 97, normalized size = 6.93 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac {12}{5} \, a^{2} b^{4} x^{5} + 2 \, a^{3} b^{3} x^{4} + a^{6} x + {\left (b^{2} x^{3} + 3 \, a b x^{2}\right )} a^{4} + \frac {1}{5} \, {\left (3 \, b^{4} x^{5} + 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3}\right )} a^{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. 64 vs.
\(2 (12) = 24\).
time = 2.36, size = 64, normalized size = 4.57 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \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. 66 vs.
\(2 (8) = 16\).
time = 0.01, size = 66, normalized size = 4.71 \begin {gather*} a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7} \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. 64 vs.
\(2 (12) = 24\).
time = 1.53, size = 64, normalized size = 4.57 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 64, normalized size = 4.57 \begin {gather*} a^6\,x+3\,a^5\,b\,x^2+5\,a^4\,b^2\,x^3+5\,a^3\,b^3\,x^4+3\,a^2\,b^4\,x^5+a\,b^5\,x^6+\frac {b^6\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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