Optimal. Leaf size=71 \[ \frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{2+m}}{2+m}+\frac {b (A b+2 a B) x^{3+m}}{3+m}+\frac {b^2 B x^{4+m}}{4+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+2} (a B+2 A b)}{m+2}+\frac {b x^{m+3} (2 a B+A b)}{m+3}+\frac {b^2 B x^{m+4}}{m+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^m (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x^m+a (2 A b+a B) x^{1+m}+b (A b+2 a B) x^{2+m}+b^2 B x^{3+m}\right ) \, dx\\ &=\frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{2+m}}{2+m}+\frac {b (A b+2 a B) x^{3+m}}{3+m}+\frac {b^2 B x^{4+m}}{4+m}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 71, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (B (a+b x)^3+(-a B (1+m)+A b (4+m)) \left (\frac {a^2}{1+m}+\frac {2 a b x}{2+m}+\frac {b^2 x^2}{3+m}\right )\right )}{b (4+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 82, normalized size = 1.15
method | result | size |
norman | \(\frac {a \left (2 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {a^{2} A x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \left (A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {b^{2} B \,x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\) | \(82\) |
risch | \(\frac {x \left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 m \,x^{3} b^{2} B +A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} x^{2} m +8 B \,a^{2} m^{2} x +28 B a b \,x^{2} m +6 b^{2} B \,x^{3}+9 A \,a^{2} m^{2}+38 a A b x m +8 A \,b^{2} x^{2}+19 a^{2} B x m +16 B a b \,x^{2}+26 a^{2} A m +24 a A b x +12 a^{2} B x +24 a^{2} A \right ) x^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(245\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 m \,x^{3} b^{2} B +A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} x^{2} m +8 B \,a^{2} m^{2} x +28 B a b \,x^{2} m +6 b^{2} B \,x^{3}+9 A \,a^{2} m^{2}+38 a A b x m +8 A \,b^{2} x^{2}+19 a^{2} B x m +16 B a b \,x^{2}+26 a^{2} A m +24 a A b x +12 a^{2} B x +24 a^{2} A \right )}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 91, normalized size = 1.28 \begin {gather*} \frac {B b^{2} x^{m + 4}}{m + 4} + \frac {2 \, B a b x^{m + 3}}{m + 3} + \frac {A b^{2} x^{m + 3}}{m + 3} + \frac {B a^{2} x^{m + 2}}{m + 2} + \frac {2 \, A a b x^{m + 2}}{m + 2} + \frac {A a^{2} 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. 215 vs.
\(2 (71) = 142\).
time = 1.64, size = 215, normalized size = 3.03 \begin {gather*} \frac {{\left ({\left (B b^{2} m^{3} + 6 \, B b^{2} m^{2} + 11 \, B b^{2} m + 6 \, B b^{2}\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 16 \, B a b + 8 \, A b^{2} + 7 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 14 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 12 \, B a^{2} + 24 \, A a b + 8 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{3} + 9 \, A a^{2} m^{2} + 26 \, A a^{2} m + 24 \, A a^{2}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1020 vs.
\(2 (63) = 126\).
time = 0.29, size = 1020, normalized size = 14.37 \begin {gather*} \begin {cases} - \frac {A a^{2}}{3 x^{3}} - \frac {A a b}{x^{2}} - \frac {A b^{2}}{x} - \frac {B a^{2}}{2 x^{2}} - \frac {2 B a b}{x} + B b^{2} \log {\left (x \right )} & \text {for}\: m = -4 \\- \frac {A a^{2}}{2 x^{2}} - \frac {2 A a b}{x} + A b^{2} \log {\left (x \right )} - \frac {B a^{2}}{x} + 2 B a b \log {\left (x \right )} + B b^{2} x & \text {for}\: m = -3 \\- \frac {A a^{2}}{x} + 2 A a b \log {\left (x \right )} + A b^{2} x + B a^{2} \log {\left (x \right )} + 2 B a b x + \frac {B b^{2} x^{2}}{2} & \text {for}\: m = -2 \\A a^{2} \log {\left (x \right )} + 2 A a b x + \frac {A b^{2} x^{2}}{2} + B a^{2} x + B a b x^{2} + \frac {B b^{2} x^{3}}{3} & \text {for}\: m = -1 \\\frac {A a^{2} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 A a^{2} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 A a^{2} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 A a^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {2 A a b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {16 A a b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {38 A a b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 A a b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {A b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {7 A b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 A b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 A b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B a^{2} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 B a^{2} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 B a^{2} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 B a^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {2 B a b m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 B a b m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {28 B a b m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {16 B a b x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B b^{2} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B b^{2} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 B b^{2} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B b^{2} 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. 332 vs.
\(2 (71) = 142\).
time = 1.79, size = 332, normalized size = 4.68 \begin {gather*} \frac {B b^{2} m^{3} x^{4} x^{m} + 2 \, B a b m^{3} x^{3} x^{m} + A b^{2} m^{3} x^{3} x^{m} + 6 \, B b^{2} m^{2} x^{4} x^{m} + B a^{2} m^{3} x^{2} x^{m} + 2 \, A a b m^{3} x^{2} x^{m} + 14 \, B a b m^{2} x^{3} x^{m} + 7 \, A b^{2} m^{2} x^{3} x^{m} + 11 \, B b^{2} m x^{4} x^{m} + A a^{2} m^{3} x x^{m} + 8 \, B a^{2} m^{2} x^{2} x^{m} + 16 \, A a b m^{2} x^{2} x^{m} + 28 \, B a b m x^{3} x^{m} + 14 \, A b^{2} m x^{3} x^{m} + 6 \, B b^{2} x^{4} x^{m} + 9 \, A a^{2} m^{2} x x^{m} + 19 \, B a^{2} m x^{2} x^{m} + 38 \, A a b m x^{2} x^{m} + 16 \, B a b x^{3} x^{m} + 8 \, A b^{2} x^{3} x^{m} + 26 \, A a^{2} m x x^{m} + 12 \, B a^{2} x^{2} x^{m} + 24 \, A a b x^{2} x^{m} + 24 \, A a^{2} x 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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Mupad [B]
time = 0.51, size = 177, normalized size = 2.49 \begin {gather*} x^m\,\left (\frac {B\,b^2\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {A\,a^2\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,x^2\,\left (2\,A\,b+B\,a\right )\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b\,x^3\,\left (A\,b+2\,B\,a\right )\,\left (m^3+7\,m^2+14\,m+8\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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