Integrand size = 16, antiderivative size = 96 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac {3 a b (A b+a B) x^{3+m}}{3+m}+\frac {b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 B x^{5+m}}{5+m} \]
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Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {a^3 A x^{m+1}}{m+1}+\frac {a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac {b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac {3 a b x^{m+3} (a B+A b)}{m+3}+\frac {b^3 B x^{m+5}}{m+5} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^m+a^2 (3 A b+a B) x^{1+m}+3 a b (A b+a B) x^{2+m}+b^2 (A b+3 a B) x^{3+m}+b^3 B x^{4+m}\right ) \, dx \\ & = \frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac {3 a b (A b+a B) x^{3+m}}{3+m}+\frac {b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 B x^{5+m}}{5+m} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {x^{1+m} \left (B (a+b x)^4+(-a B (1+m)+A b (5+m)) \left (\frac {a^3}{1+m}+\frac {3 a^2 b x}{2+m}+\frac {3 a b^2 x^2}{3+m}+\frac {b^3 x^3}{4+m}\right )\right )}{b (5+m)} \]
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Time = 0.46 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15
method | result | size |
norman | \(\frac {a^{2} \left (3 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {a^{3} A x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{2} \left (A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {b^{3} B \,x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {3 a b \left (A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\) | \(110\) |
risch | \(\frac {x \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right ) x^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(453\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right )}{\left (1+m \right ) \left (2+m \right ) \left (3+m \right ) \left (4+m \right ) \left (5+m \right )}\) | \(454\) |
parallelrisch | \(\frac {60 B \,x^{2} x^{m} a^{3}+120 A x \,x^{m} a^{3}+24 B \,x^{5} x^{m} b^{3}+30 A \,x^{4} x^{m} b^{3}+123 B \,x^{4} x^{m} a \,b^{2} m^{2}+36 B \,x^{3} x^{m} a^{2} b \,m^{3}+147 A \,x^{3} x^{m} a \,b^{2} m^{2}+39 A \,x^{2} x^{m} a^{2} b \,m^{3}+183 B \,x^{4} x^{m} a \,b^{2} m +147 B \,x^{3} x^{m} a^{2} b \,m^{2}+234 A \,x^{3} x^{m} a \,b^{2} m +177 A \,x^{2} x^{m} a^{2} b \,m^{2}+234 B \,x^{3} x^{m} a^{2} b m +321 A \,x^{2} x^{m} a^{2} b m +3 B \,x^{4} x^{m} a \,b^{2} m^{4}+3 A \,x^{3} x^{m} a \,b^{2} m^{4}+33 B \,x^{4} x^{m} a \,b^{2} m^{3}+3 B \,x^{3} x^{m} a^{2} b \,m^{4}+36 A \,x^{3} x^{m} a \,b^{2} m^{3}+3 A \,x^{2} x^{m} a^{2} b \,m^{4}+B \,x^{5} x^{m} b^{3} m^{4}+A \,x^{4} x^{m} b^{3} m^{4}+10 B \,x^{5} x^{m} b^{3} m^{3}+11 A \,x^{4} x^{m} b^{3} m^{3}+35 B \,x^{5} x^{m} b^{3} m^{2}+41 A \,x^{4} x^{m} b^{3} m^{2}+50 B \,x^{5} x^{m} b^{3} m +B \,x^{2} x^{m} a^{3} m^{4}+61 A \,x^{4} x^{m} b^{3} m +A x \,x^{m} a^{3} m^{4}+13 B \,x^{2} x^{m} a^{3} m^{3}+14 A x \,x^{m} a^{3} m^{3}+90 B \,x^{4} x^{m} a \,b^{2}+59 B \,x^{2} x^{m} a^{3} m^{2}+120 A \,x^{3} x^{m} a \,b^{2}+71 A x \,x^{m} a^{3} m^{2}+120 B \,x^{3} x^{m} a^{2} b +107 B \,x^{2} x^{m} a^{3} m +180 A \,x^{2} x^{m} a^{2} b +154 A x \,x^{m} a^{3} m}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(594\) |
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (96) = 192\).
Time = 0.23 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.95 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {{\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} + {\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2018 vs. \(2 (87) = 174\).
Time = 0.42 (sec) , antiderivative size = 2018, normalized size of antiderivative = 21.02 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\text {Too large to display} \]
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Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {B b^{3} x^{m + 5}}{m + 5} + \frac {3 \, B a b^{2} x^{m + 4}}{m + 4} + \frac {A b^{3} x^{m + 4}}{m + 4} + \frac {3 \, B a^{2} b x^{m + 3}}{m + 3} + \frac {3 \, A a b^{2} x^{m + 3}}{m + 3} + \frac {B a^{3} x^{m + 2}}{m + 2} + \frac {3 \, A a^{2} b x^{m + 2}}{m + 2} + \frac {A a^{3} x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (96) = 192\).
Time = 0.29 (sec) , antiderivative size = 593, normalized size of antiderivative = 6.18 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {B b^{3} m^{4} x^{5} x^{m} + 3 \, B a b^{2} m^{4} x^{4} x^{m} + A b^{3} m^{4} x^{4} x^{m} + 10 \, B b^{3} m^{3} x^{5} x^{m} + 3 \, B a^{2} b m^{4} x^{3} x^{m} + 3 \, A a b^{2} m^{4} x^{3} x^{m} + 33 \, B a b^{2} m^{3} x^{4} x^{m} + 11 \, A b^{3} m^{3} x^{4} x^{m} + 35 \, B b^{3} m^{2} x^{5} x^{m} + B a^{3} m^{4} x^{2} x^{m} + 3 \, A a^{2} b m^{4} x^{2} x^{m} + 36 \, B a^{2} b m^{3} x^{3} x^{m} + 36 \, A a b^{2} m^{3} x^{3} x^{m} + 123 \, B a b^{2} m^{2} x^{4} x^{m} + 41 \, A b^{3} m^{2} x^{4} x^{m} + 50 \, B b^{3} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 13 \, B a^{3} m^{3} x^{2} x^{m} + 39 \, A a^{2} b m^{3} x^{2} x^{m} + 147 \, B a^{2} b m^{2} x^{3} x^{m} + 147 \, A a b^{2} m^{2} x^{3} x^{m} + 183 \, B a b^{2} m x^{4} x^{m} + 61 \, A b^{3} m x^{4} x^{m} + 24 \, B b^{3} x^{5} x^{m} + 14 \, A a^{3} m^{3} x x^{m} + 59 \, B a^{3} m^{2} x^{2} x^{m} + 177 \, A a^{2} b m^{2} x^{2} x^{m} + 234 \, B a^{2} b m x^{3} x^{m} + 234 \, A a b^{2} m x^{3} x^{m} + 90 \, B a b^{2} x^{4} x^{m} + 30 \, A b^{3} x^{4} x^{m} + 71 \, A a^{3} m^{2} x x^{m} + 107 \, B a^{3} m x^{2} x^{m} + 321 \, A a^{2} b m x^{2} x^{m} + 120 \, B a^{2} b x^{3} x^{m} + 120 \, A a b^{2} x^{3} x^{m} + 154 \, A a^{3} m x x^{m} + 60 \, B a^{3} x^{2} x^{m} + 180 \, A a^{2} b x^{2} x^{m} + 120 \, A a^{3} x x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]
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Time = 0.59 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.01 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {A\,a^3\,x\,x^m\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {B\,b^3\,x^m\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {a^2\,x^m\,x^2\,\left (3\,A\,b+B\,a\right )\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b^2\,x^m\,x^4\,\left (A\,b+3\,B\,a\right )\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {3\,a\,b\,x^m\,x^3\,\left (A\,b+B\,a\right )\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120} \]
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