Optimal. Leaf size=14 \[ \frac {(a+b x)^5}{5 b} \]
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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)^5}{5 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 )^2 \, dx &=\int (a+b x)^4 \, dx\\ &=\frac {(a+b x)^5}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^5}{5 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. \(42\) vs.
\(2(12)=24\).
time = 0.58, size = 43, normalized size = 3.07
method | result | size |
default | \(\frac {1}{5} b^{4} x^{5}+a \,b^{3} x^{4}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(43\) |
norman | \(\frac {1}{5} b^{4} x^{5}+a \,b^{3} x^{4}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(43\) |
risch | \(\frac {1}{5} b^{4} x^{5}+a \,b^{3} x^{4}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(43\) |
gosper | \(\frac {x \left (b^{4} x^{4}+5 a \,b^{3} x^{3}+10 a^{2} b^{2} x^{2}+10 a^{3} b x +5 a^{4}\right )}{5}\) | \(44\) |
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. 53 vs.
\(2 (12) = 24\).
time = 0.28, size = 53, normalized size = 3.79 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + \frac {4}{3} \, a^{2} b^{2} x^{3} + a^{4} x + \frac {2}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2}\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. 42 vs.
\(2 (12) = 24\).
time = 1.87, size = 42, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} 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. 42 vs.
\(2 (8) = 16\).
time = 0.01, size = 42, normalized size = 3.00 \begin {gather*} a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5} \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. 42 vs.
\(2 (12) = 24\).
time = 1.41, size = 42, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 42, normalized size = 3.00 \begin {gather*} a^4\,x+2\,a^3\,b\,x^2+2\,a^2\,b^2\,x^3+a\,b^3\,x^4+\frac {b^4\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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