Optimal. Leaf size=61 \[ \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{2+m}}{2+m}+\frac {3 a b^2 x^{3+m}}{3+m}+\frac {b^3 x^{4+m}}{4+m} \]
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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+2}}{m+2}+\frac {3 a b^2 x^{m+3}}{m+3}+\frac {b^3 x^{m+4}}{m+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^m (a+b x)^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{1+m}+3 a b^2 x^{2+m}+b^3 x^{3+m}\right ) \, dx\\ &=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{2+m}}{2+m}+\frac {3 a b^2 x^{3+m}}{3+m}+\frac {b^3 x^{4+m}}{4+m}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.89 \begin {gather*} x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x}{2+m}+\frac {3 a b^2 x^2}{3+m}+\frac {b^3 x^3}{4+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.47, size = 662, normalized size = 10.85 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {a^3}{3 x^3}-\frac {3 a^2 b}{2 x^2}-\frac {3 a b^2}{x}+b^3 \text {Log}\left [x\right ],m\text {==}-4\right \},\left \{-\frac {a^3}{2 x^2}-\frac {3 a^2 b}{x}+3 a b^2 \text {Log}\left [x\right ]+b^3 x,m\text {==}-3\right \},\left \{\frac {-a^3+\frac {b x \left (6 a^2 \text {Log}\left [x\right ]+6 a b x+b^2 x^2\right )}{2}}{x},m\text {==}-2\right \},\left \{a^3 \text {Log}\left [x\right ]+3 a^2 b x+\frac {3 a b^2 x^2}{2}+\frac {b^3 x^3}{3},m\text {==}-1\right \}\right \},\frac {24 a^3 x x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {26 a^3 m x x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {9 a^3 m^2 x x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {a^3 m^3 x x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {36 a^2 b x^2 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {57 a^2 b m x^2 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {24 a^2 b m^2 x^2 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {3 a^2 b m^3 x^2 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {24 a b^2 x^3 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {42 a b^2 m x^3 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {21 a b^2 m^2 x^3 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {3 a b^2 m^3 x^3 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {6 b^3 x^4 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {11 b^3 m x^4 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {6 b^3 m^2 x^4 x^m}{24+50 m+35 m^2+10 m^3+m^4}+\frac {b^3 m^3 x^4 x^m}{24+50 m+35 m^2+10 m^3+m^4}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 72, normalized size = 1.18
method | result | size |
norman | \(\frac {a^{3} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {3 a \,b^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {3 a^{2} b \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}\) | \(72\) |
risch | \(\frac {x \left (b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}+3 a^{2} b \,m^{3} x +21 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}+24 a^{2} b \,m^{2} x +42 m \,x^{2} a \,b^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}+57 m x \,a^{2} b +24 a \,b^{2} x^{2}+26 m \,a^{3}+36 a^{2} b x +24 a^{3}\right ) x^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(169\) |
gosper | \(\frac {x^{1+m} \left (b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}+3 a^{2} b \,m^{3} x +21 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}+24 a^{2} b \,m^{2} x +42 m \,x^{2} a \,b^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}+57 m x \,a^{2} b +24 a \,b^{2} x^{2}+26 m \,a^{3}+36 a^{2} b x +24 a^{3}\right )}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 61, normalized size = 1.00 \begin {gather*} \frac {b^{3} x^{m + 4}}{m + 4} + \frac {3 \, a b^{2} x^{m + 3}}{m + 3} + \frac {3 \, a^{2} b x^{m + 2}}{m + 2} + \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 = 0.31, size = 157, normalized size = 2.57 \begin {gather*} \frac {{\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \, {\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \, {\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} + {\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 663, normalized size = 10.87 \begin {gather*} \begin {cases} - \frac {a^{3}}{3 x^{3}} - \frac {3 a^{2} b}{2 x^{2}} - \frac {3 a b^{2}}{x} + b^{3} \log {\left (x \right )} & \text {for}\: m = -4 \\- \frac {a^{3}}{2 x^{2}} - \frac {3 a^{2} b}{x} + 3 a b^{2} \log {\left (x \right )} + b^{3} x & \text {for}\: m = -3 \\- \frac {a^{3}}{x} + 3 a^{2} b \log {\left (x \right )} + 3 a b^{2} x + \frac {b^{3} x^{2}}{2} & \text {for}\: m = -2 \\a^{3} \log {\left (x \right )} + 3 a^{2} b x + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{3}}{3} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {3 a^{2} b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{2} b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {57 a^{2} b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {36 a^{2} b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {3 a b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {21 a b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {42 a b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {b^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 b^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \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 = 0.00, size = 255, normalized size = 4.18 \begin {gather*} \frac {a^{3} m^{3} x \mathrm {e}^{m \ln x}+9 a^{3} m^{2} x \mathrm {e}^{m \ln x}+26 a^{3} m x \mathrm {e}^{m \ln x}+24 a^{3} x \mathrm {e}^{m \ln x}+3 a^{2} b m^{3} x^{2} \mathrm {e}^{m \ln x}+24 a^{2} b m^{2} x^{2} \mathrm {e}^{m \ln x}+57 a^{2} b m x^{2} \mathrm {e}^{m \ln x}+36 a^{2} b x^{2} \mathrm {e}^{m \ln x}+3 a b^{2} m^{3} x^{3} \mathrm {e}^{m \ln x}+21 a b^{2} m^{2} x^{3} \mathrm {e}^{m \ln x}+42 a b^{2} m x^{3} \mathrm {e}^{m \ln x}+24 a b^{2} x^{3} \mathrm {e}^{m \ln x}+b^{3} m^{3} x^{4} \mathrm {e}^{m \ln x}+6 b^{3} m^{2} x^{4} \mathrm {e}^{m \ln x}+11 b^{3} m x^{4} \mathrm {e}^{m \ln x}+6 b^{3} x^{4} \mathrm {e}^{m \ln x}}{m^{4}+10 m^{3}+35 m^{2}+50 m+24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 167, normalized size = 2.74 \begin {gather*} x^m\,\left (\frac {a^3\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b^3\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {3\,a\,b^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {3\,a^2\,b\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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