Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^3 (A+B x) \, dx=\frac {(A b-a B) (a+b x)^4}{4 b^2}+\frac {B (a+b x)^5}{5 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x)^3 (A+B x) \, dx=\frac {(a+b x)^4 (A b-a B)}{4 b^2}+\frac {B (a+b x)^5}{5 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^3}{b}+\frac {B (a+b x)^4}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^4}{4 b^2}+\frac {B (a+b x)^5}{5 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int (a+b x)^3 (A+B x) \, dx=a^3 A x+\frac {1}{2} a^2 (3 A b+a B) x^2+a b (A b+a B) x^3+\frac {1}{4} b^2 (A b+3 a B) x^4+\frac {1}{5} b^3 B x^5 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(34)=68\).
Time = 0.66 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.84
method | result | size |
norman | \(\frac {b^{3} B \,x^{5}}{5}+\left (\frac {1}{4} b^{3} A +\frac {3}{4} a \,b^{2} B \right ) x^{4}+\left (a \,b^{2} A +a^{2} b B \right ) x^{3}+\left (\frac {3}{2} a^{2} b A +\frac {1}{2} a^{3} B \right ) x^{2}+a^{3} A x\) | \(70\) |
gosper | \(\frac {1}{5} b^{3} B \,x^{5}+\frac {1}{4} x^{4} b^{3} A +\frac {3}{4} x^{4} a \,b^{2} B +A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} a^{2} b A +\frac {1}{2} x^{2} a^{3} B +a^{3} A x\) | \(73\) |
default | \(\frac {b^{3} B \,x^{5}}{5}+\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{4}}{4}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{3}}{3}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{2}}{2}+a^{3} A x\) | \(73\) |
risch | \(\frac {1}{5} b^{3} B \,x^{5}+\frac {1}{4} x^{4} b^{3} A +\frac {3}{4} x^{4} a \,b^{2} B +A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} a^{2} b A +\frac {1}{2} x^{2} a^{3} B +a^{3} A x\) | \(73\) |
parallelrisch | \(\frac {1}{5} b^{3} B \,x^{5}+\frac {1}{4} x^{4} b^{3} A +\frac {3}{4} x^{4} a \,b^{2} B +A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} a^{2} b A +\frac {1}{2} x^{2} a^{3} B +a^{3} A x\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.82 \[ \int (a+b x)^3 (A+B x) \, dx=\frac {1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (32) = 64\).
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92 \[ \int (a+b x)^3 (A+B x) \, dx=A a^{3} x + \frac {B b^{3} x^{5}}{5} + x^{4} \left (\frac {A b^{3}}{4} + \frac {3 B a b^{2}}{4}\right ) + x^{3} \left (A a b^{2} + B a^{2} b\right ) + x^{2} \cdot \left (\frac {3 A a^{2} b}{2} + \frac {B a^{3}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.82 \[ \int (a+b x)^3 (A+B x) \, dx=\frac {1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int (a+b x)^3 (A+B x) \, dx=\frac {1}{5} \, B b^{3} x^{5} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{2} \, A a^{2} b x^{2} + A a^{3} x \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int (a+b x)^3 (A+B x) \, dx=x^2\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,b\,a^2}{2}\right )+x^4\,\left (\frac {A\,b^3}{4}+\frac {3\,B\,a\,b^2}{4}\right )+\frac {B\,b^3\,x^5}{5}+A\,a^3\,x+a\,b\,x^3\,\left (A\,b+B\,a\right ) \]
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