Optimal. Leaf size=61 \[ \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{5+m}}{5+m}+\frac {3 a b^2 x^{9+m}}{9+m}+\frac {b^3 x^{13+m}}{13+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, 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^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+5}}{m+5}+\frac {3 a b^2 x^{m+9}}{m+9}+\frac {b^3 x^{m+13}}{m+13} \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 )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{4+m}+3 a b^2 x^{8+m}+b^3 x^{12+m}\right ) \, dx\\ &=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{5+m}}{5+m}+\frac {3 a b^2 x^{9+m}}{9+m}+\frac {b^3 x^{13+m}}{13+m}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.92 \begin {gather*} x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^4}{5+m}+\frac {3 a b^2 x^8}{9+m}+\frac {b^3 x^{12}}{13+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. \(176\) vs.
\(2(61)=122\).
time = 0.14, size = 177, normalized size = 2.90
method | result | size |
risch | \(\frac {x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right ) x^{m}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(177\) |
gosper | \(\frac {x^{1+m} \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right )}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 61, normalized size = 1.00 \begin {gather*} \frac {b^{3} x^{m + 13}}{m + 13} + \frac {3 \, a b^{2} x^{m + 9}}{m + 9} + \frac {3 \, a^{2} b x^{m + 5}}{m + 5} + \frac {a^{3} x^{m + 1}}{m + 1} \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. 157 vs.
\(2 (61) = 122\).
time = 0.37, size = 157, normalized size = 2.57 \begin {gather*} \frac {{\left ({\left (b^{3} m^{3} + 15 \, b^{3} m^{2} + 59 \, b^{3} m + 45 \, b^{3}\right )} x^{13} + 3 \, {\left (a b^{2} m^{3} + 19 \, a b^{2} m^{2} + 83 \, a b^{2} m + 65 \, a b^{2}\right )} x^{9} + 3 \, {\left (a^{2} b m^{3} + 23 \, a^{2} b m^{2} + 139 \, a^{2} b m + 117 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{3} + 27 \, a^{3} m^{2} + 227 \, a^{3} m + 585 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \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. 683 vs.
\(2 (53) = 106\).
time = 0.74, size = 683, normalized size = 11.20 \begin {gather*} \begin {cases} - \frac {a^{3}}{12 x^{12}} - \frac {3 a^{2} b}{8 x^{8}} - \frac {3 a b^{2}}{4 x^{4}} + b^{3} \log {\left (x \right )} & \text {for}\: m = -13 \\- \frac {a^{3}}{8 x^{8}} - \frac {3 a^{2} b}{4 x^{4}} + 3 a b^{2} \log {\left (x \right )} + \frac {b^{3} x^{4}}{4} & \text {for}\: m = -9 \\- \frac {a^{3}}{4 x^{4}} + 3 a^{2} b \log {\left (x \right )} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{8}}{8} & \text {for}\: m = -5 \\a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{12}}{12} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {27 a^{3} m^{2} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {227 a^{3} m x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {585 a^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a^{2} b m^{3} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {69 a^{2} b m^{2} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {417 a^{2} b m x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {351 a^{2} b x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a b^{2} m^{3} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {57 a b^{2} m^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {249 a b^{2} m x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {195 a b^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {b^{3} m^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {15 b^{3} m^{2} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {59 b^{3} m x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {45 b^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} & \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. 224 vs.
\(2 (61) = 122\).
time = 0.53, size = 224, normalized size = 3.67 \begin {gather*} \frac {b^{3} m^{3} x^{13} x^{m} + 15 \, b^{3} m^{2} x^{13} x^{m} + 59 \, b^{3} m x^{13} x^{m} + 45 \, b^{3} x^{13} x^{m} + 3 \, a b^{2} m^{3} x^{9} x^{m} + 57 \, a b^{2} m^{2} x^{9} x^{m} + 249 \, a b^{2} m x^{9} x^{m} + 195 \, a b^{2} x^{9} x^{m} + 3 \, a^{2} b m^{3} x^{5} x^{m} + 69 \, a^{2} b m^{2} x^{5} x^{m} + 417 \, a^{2} b m x^{5} x^{m} + 351 \, a^{2} b x^{5} x^{m} + a^{3} m^{3} x x^{m} + 27 \, a^{3} m^{2} x x^{m} + 227 \, a^{3} m x x^{m} + 585 \, a^{3} x x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 167, normalized size = 2.74 \begin {gather*} x^m\,\left (\frac {a^3\,x\,\left (m^3+27\,m^2+227\,m+585\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {b^3\,x^{13}\,\left (m^3+15\,m^2+59\,m+45\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a\,b^2\,x^9\,\left (m^3+19\,m^2+83\,m+65\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a^2\,b\,x^5\,\left (m^3+23\,m^2+139\,m+117\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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