Optimal. Leaf size=43 \[ \frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{5+m}}{5+m}+\frac {b^2 x^{9+m}}{9+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} \frac {a^2 x^{m+1}}{m+1}+\frac {2 a b x^{m+5}}{m+5}+\frac {b^2 x^{m+9}}{m+9} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int x^m \left (a+b x^4\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{4+m}+b^2 x^{8+m}\right ) \, dx\\ &=\frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{5+m}}{5+m}+\frac {b^2 x^{9+m}}{9+m}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 0.93 \begin {gather*} x^{1+m} \left (\frac {a^2}{1+m}+\frac {2 a b x^4}{5+m}+\frac {b^2 x^8}{9+m}\right ) \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. \(91\) vs.
\(2(43)=86\).
time = 0.13, size = 92, normalized size = 2.14
method | result | size |
risch | \(\frac {x \left (b^{2} m^{2} x^{8}+6 m \,x^{8} b^{2}+5 b^{2} x^{8}+2 a b \,m^{2} x^{4}+20 m \,x^{4} a b +18 a b \,x^{4}+a^{2} m^{2}+14 m \,a^{2}+45 a^{2}\right ) x^{m}}{\left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(92\) |
gosper | \(\frac {x^{1+m} \left (b^{2} m^{2} x^{8}+6 m \,x^{8} b^{2}+5 b^{2} x^{8}+2 a b \,m^{2} x^{4}+20 m \,x^{4} a b +18 a b \,x^{4}+a^{2} m^{2}+14 m \,a^{2}+45 a^{2}\right )}{\left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{m + 9}}{m + 9} + \frac {2 \, a b x^{m + 5}}{m + 5} + \frac {a^{2} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 85, normalized size = 1.98 \begin {gather*} \frac {{\left ({\left (b^{2} m^{2} + 6 \, b^{2} m + 5 \, b^{2}\right )} x^{9} + 2 \, {\left (a b m^{2} + 10 \, a b m + 9 \, a b\right )} x^{5} + {\left (a^{2} m^{2} + 14 \, a^{2} m + 45 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 15 \, m^{2} + 59 \, m + 45} \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. 309 vs.
\(2 (36) = 72\).
time = 0.40, size = 309, normalized size = 7.19 \begin {gather*} \begin {cases} - \frac {a^{2}}{8 x^{8}} - \frac {a b}{2 x^{4}} + b^{2} \log {\left (x \right )} & \text {for}\: m = -9 \\- \frac {a^{2}}{4 x^{4}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{4}}{4} & \text {for}\: m = -5 \\a^{2} \log {\left (x \right )} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{8}}{8} & \text {for}\: m = -1 \\\frac {a^{2} m^{2} x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {14 a^{2} m x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {45 a^{2} x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {2 a b m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {20 a b m x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {18 a b x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {b^{2} m^{2} x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {6 b^{2} m x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac {5 b^{2} x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} & \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. 117 vs.
\(2 (43) = 86\).
time = 1.41, size = 117, normalized size = 2.72 \begin {gather*} \frac {b^{2} m^{2} x^{9} x^{m} + 6 \, b^{2} m x^{9} x^{m} + 5 \, b^{2} x^{9} x^{m} + 2 \, a b m^{2} x^{5} x^{m} + 20 \, a b m x^{5} x^{m} + 18 \, a b x^{5} x^{m} + a^{2} m^{2} x x^{m} + 14 \, a^{2} m x x^{m} + 45 \, a^{2} x x^{m}}{m^{3} + 15 \, m^{2} + 59 \, m + 45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 93, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {a^2\,x\,\left (m^2+14\,m+45\right )}{m^3+15\,m^2+59\,m+45}+\frac {b^2\,x^9\,\left (m^2+6\,m+5\right )}{m^3+15\,m^2+59\,m+45}+\frac {2\,a\,b\,x^5\,\left (m^2+10\,m+9\right )}{m^3+15\,m^2+59\,m+45}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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