Integrand size = 15, antiderivative size = 78 \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m}}{b^3 (1+m)}+\frac {2 d (b c-a d) (a+b x)^{2+m}}{b^3 (2+m)}+\frac {d^2 (a+b x)^{3+m}}{b^3 (3+m)} \]
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Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {(b c-a d)^2 (a+b x)^{m+1}}{b^3 (m+1)}+\frac {2 d (b c-a d) (a+b x)^{m+2}}{b^3 (m+2)}+\frac {d^2 (a+b x)^{m+3}}{b^3 (m+3)} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^2 (a+b x)^m}{b^2}+\frac {2 d (b c-a d) (a+b x)^{1+m}}{b^2}+\frac {d^2 (a+b x)^{2+m}}{b^2}\right ) \, dx \\ & = \frac {(b c-a d)^2 (a+b x)^{1+m}}{b^3 (1+m)}+\frac {2 d (b c-a d) (a+b x)^{2+m}}{b^3 (2+m)}+\frac {d^2 (a+b x)^{3+m}}{b^3 (3+m)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86 \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {(a+b x)^{1+m} \left (\frac {(b c-a d)^2}{1+m}+\frac {2 d (b c-a d) (a+b x)}{2+m}+\frac {d^2 (a+b x)^2}{3+m}\right )}{b^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(78)=156\).
Time = 0.39 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.04
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1+m} \left (b^{2} d^{2} m^{2} x^{2}+2 b^{2} c d \,m^{2} x +3 b^{2} d^{2} m \,x^{2}-2 a b \,d^{2} m x +b^{2} c^{2} m^{2}+8 b^{2} c d m x +2 d^{2} x^{2} b^{2}-2 a b c d m -2 x a b \,d^{2}+5 b^{2} c^{2} m +6 x \,b^{2} c d +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right )}{b^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(159\) |
norman | \(\frac {d^{2} x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{3+m}+\frac {a \left (b^{2} c^{2} m^{2}-2 a b c d m +5 b^{2} c^{2} m +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (a d m +2 b c m +6 b c \right ) d \,x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{2}+5 m +6\right )}-\frac {\left (-2 a b c d \,m^{2}-b^{2} c^{2} m^{2}+2 a^{2} d^{2} m -6 a b c d m -5 b^{2} c^{2} m -6 b^{2} c^{2}\right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{b^{2} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(226\) |
risch | \(\frac {\left (b^{3} d^{2} m^{2} x^{3}+a \,b^{2} d^{2} m^{2} x^{2}+2 b^{3} c d \,m^{2} x^{2}+3 b^{3} d^{2} m \,x^{3}+2 a \,b^{2} c d \,m^{2} x +a \,b^{2} d^{2} m \,x^{2}+b^{3} c^{2} m^{2} x +8 b^{3} c d m \,x^{2}+2 d^{2} x^{3} b^{3}-2 a^{2} b \,d^{2} m x +a \,b^{2} c^{2} m^{2}+6 a \,b^{2} c d m x +5 b^{3} c^{2} m x +6 b^{3} c d \,x^{2}-2 a^{2} b c d m +5 a \,b^{2} c^{2} m +6 b^{3} c^{2} x +2 a^{3} d^{2}-6 a^{2} b c d +6 a \,b^{2} c^{2}\right ) \left (b x +a \right )^{m}}{\left (2+m \right ) \left (3+m \right ) \left (1+m \right ) b^{3}}\) | \(242\) |
parallelrisch | \(\frac {x^{3} \left (b x +a \right )^{m} a \,b^{3} d^{2} m^{2}+3 x^{3} \left (b x +a \right )^{m} a \,b^{3} d^{2} m +x^{2} \left (b x +a \right )^{m} a^{2} b^{2} d^{2} m^{2}+2 x^{2} \left (b x +a \right )^{m} a \,b^{3} c d \,m^{2}+2 x^{3} \left (b x +a \right )^{m} a \,b^{3} d^{2}+x^{2} \left (b x +a \right )^{m} a^{2} b^{2} d^{2} m +8 x^{2} \left (b x +a \right )^{m} a \,b^{3} c d m +2 x \left (b x +a \right )^{m} a^{2} b^{2} c d \,m^{2}+x \left (b x +a \right )^{m} a \,b^{3} c^{2} m^{2}+6 x^{2} \left (b x +a \right )^{m} a \,b^{3} c d -2 x \left (b x +a \right )^{m} a^{3} b \,d^{2} m +6 x \left (b x +a \right )^{m} a^{2} b^{2} c d m +5 x \left (b x +a \right )^{m} a \,b^{3} c^{2} m +\left (b x +a \right )^{m} a^{2} b^{2} c^{2} m^{2}+6 x \left (b x +a \right )^{m} a \,b^{3} c^{2}-2 \left (b x +a \right )^{m} a^{3} b c d m +5 \left (b x +a \right )^{m} a^{2} b^{2} c^{2} m +2 \left (b x +a \right )^{m} a^{4} d^{2}-6 \left (b x +a \right )^{m} a^{3} b c d +6 \left (b x +a \right )^{m} a^{2} b^{2} c^{2}}{\left (3+m \right ) \left (2+m \right ) \left (1+m \right ) b^{3} a}\) | \(401\) |
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (78) = 156\).
Time = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.01 \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {{\left (a b^{2} c^{2} m^{2} + 6 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (b^{3} d^{2} m^{2} + 3 \, b^{3} d^{2} m + 2 \, b^{3} d^{2}\right )} x^{3} + {\left (6 \, b^{3} c d + {\left (2 \, b^{3} c d + a b^{2} d^{2}\right )} m^{2} + {\left (8 \, b^{3} c d + a b^{2} d^{2}\right )} m\right )} x^{2} + {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d\right )} m + {\left (6 \, b^{3} c^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d\right )} m^{2} + {\left (5 \, b^{3} c^{2} + 6 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (66) = 132\).
Time = 0.60 (sec) , antiderivative size = 1506, normalized size of antiderivative = 19.31 \[ \int (a+b x)^m (c+d x)^2 \, dx=\text {Too large to display} \]
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Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.77 \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {2 \, {\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} c^{2}}{b {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (78) = 156\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 4.94 \[ \int (a+b x)^m (c+d x)^2 \, dx=\frac {{\left (b x + a\right )}^{m} b^{3} d^{2} m^{2} x^{3} + 2 \, {\left (b x + a\right )}^{m} b^{3} c d m^{2} x^{2} + {\left (b x + a\right )}^{m} a b^{2} d^{2} m^{2} x^{2} + 3 \, {\left (b x + a\right )}^{m} b^{3} d^{2} m x^{3} + {\left (b x + a\right )}^{m} b^{3} c^{2} m^{2} x + 2 \, {\left (b x + a\right )}^{m} a b^{2} c d m^{2} x + 8 \, {\left (b x + a\right )}^{m} b^{3} c d m x^{2} + {\left (b x + a\right )}^{m} a b^{2} d^{2} m x^{2} + 2 \, {\left (b x + a\right )}^{m} b^{3} d^{2} x^{3} + {\left (b x + a\right )}^{m} a b^{2} c^{2} m^{2} + 5 \, {\left (b x + a\right )}^{m} b^{3} c^{2} m x + 6 \, {\left (b x + a\right )}^{m} a b^{2} c d m x - 2 \, {\left (b x + a\right )}^{m} a^{2} b d^{2} m x + 6 \, {\left (b x + a\right )}^{m} b^{3} c d x^{2} + 5 \, {\left (b x + a\right )}^{m} a b^{2} c^{2} m - 2 \, {\left (b x + a\right )}^{m} a^{2} b c d m + 6 \, {\left (b x + a\right )}^{m} b^{3} c^{2} x + 6 \, {\left (b x + a\right )}^{m} a b^{2} c^{2} - 6 \, {\left (b x + a\right )}^{m} a^{2} b c d + 2 \, {\left (b x + a\right )}^{m} a^{3} d^{2}}{b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}} \]
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Time = 0.79 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.90 \[ \int (a+b x)^m (c+d x)^2 \, dx={\left (a+b\,x\right )}^m\,\left (\frac {a\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d\,m-6\,a\,b\,c\,d+b^2\,c^2\,m^2+5\,b^2\,c^2\,m+6\,b^2\,c^2\right )}{b^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d^2\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x\,\left (-2\,a^2\,b\,d^2\,m+2\,a\,b^2\,c\,d\,m^2+6\,a\,b^2\,c\,d\,m+b^3\,c^2\,m^2+5\,b^3\,c^2\,m+6\,b^3\,c^2\right )}{b^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,x^2\,\left (m+1\right )\,\left (6\,b\,c+a\,d\,m+2\,b\,c\,m\right )}{b\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \]
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