Integrand size = 22, antiderivative size = 38 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(A b-a B) (a+b x)^3}{3 b^2}+\frac {B (a+b x)^4}{4 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 45} \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(a+b x)^3 (A b-a B)}{3 b^2}+\frac {B (a+b x)^4}{4 b^2} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^2 (A+B x) \, dx \\ & = \int \left (\frac {(A b-a B) (a+b x)^2}{b}+\frac {B (a+b x)^3}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^3}{3 b^2}+\frac {B (a+b x)^4}{4 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{12} x \left (6 a^2 (2 A+B x)+4 a b x (3 A+2 B x)+b^2 x^2 (4 A+3 B x)\right ) \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26
method | result | size |
norman | \(\frac {x^{4} B \,b^{2}}{4}+\left (\frac {1}{3} A \,b^{2}+\frac {2}{3} a b B \right ) x^{3}+\left (A b a +\frac {1}{2} B \,a^{2}\right ) x^{2}+a^{2} A x\) | \(48\) |
default | \(\frac {x^{4} B \,b^{2}}{4}+\frac {\left (A \,b^{2}+2 a b B \right ) x^{3}}{3}+\frac {\left (2 A b a +B \,a^{2}\right ) x^{2}}{2}+a^{2} A x\) | \(49\) |
gosper | \(\frac {x \left (3 B \,b^{2} x^{3}+4 A \,b^{2} x^{2}+8 B a b \,x^{2}+12 a A b x +6 a^{2} B x +12 A \,a^{2}\right )}{12}\) | \(50\) |
risch | \(\frac {1}{4} x^{4} B \,b^{2}+\frac {1}{3} A \,b^{2} x^{3}+\frac {2}{3} x^{3} a b B +x^{2} A b a +\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(50\) |
parallelrisch | \(\frac {1}{4} x^{4} B \,b^{2}+\frac {1}{3} A \,b^{2} x^{3}+\frac {2}{3} x^{3} a b B +x^{2} A b a +\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(50\) |
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Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, B b^{2} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=A a^{2} x + \frac {B b^{2} x^{4}}{4} + x^{3} \left (\frac {A b^{2}}{3} + \frac {2 B a b}{3}\right ) + x^{2} \left (A a b + \frac {B a^{2}}{2}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, B b^{2} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, B b^{2} x^{4} + \frac {2}{3} \, B a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \]
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Time = 10.73 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}\right )+\frac {B\,b^2\,x^4}{4}+A\,a^2\,x \]
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