Integrand size = 22, antiderivative size = 128 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
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Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=-\frac {2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e) \sqrt {d+e x}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^3}+\frac {b^2 B (d+e x)^{7/2}}{e^3}\right ) \, dx \\ & = -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\frac {2 (d+e x)^{3/2} \left (21 a^2 e^2 (-2 B d+5 A e+3 B e x)+6 a b e \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^2 \left (3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{315 e^4} \]
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Time = 0.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {3 x^{2} \left (\frac {7 B x}{9}+A \right ) b^{2}}{7}+\frac {6 x \left (\frac {5 B x}{7}+A \right ) a b}{5}+a^{2} \left (\frac {3 B x}{5}+A \right )\right ) e^{3}-\frac {4 d \left (\frac {3 x \left (\frac {5 B x}{6}+A \right ) b^{2}}{7}+a \left (\frac {6 B x}{7}+A \right ) b +\frac {a^{2} B}{2}\right ) e^{2}}{5}+\frac {8 d^{2} b \left (\left (B x +A \right ) b +2 B a \right ) e}{35}-\frac {16 b^{2} B \,d^{3}}{105}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{4}}\) | \(116\) |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(122\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(122\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{2} B \,x^{3} e^{3}+45 A \,b^{2} e^{3} x^{2}+90 B a b \,e^{3} x^{2}-30 B \,b^{2} d \,e^{2} x^{2}+126 A a b \,e^{3} x -36 A \,b^{2} d \,e^{2} x +63 B \,a^{2} e^{3} x -72 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +105 a^{2} A \,e^{3}-84 A a b d \,e^{2}+24 A \,b^{2} d^{2} e -42 B \,a^{2} d \,e^{2}+48 B a b \,d^{2} e -16 b^{2} B \,d^{3}\right )}{315 e^{4}}\) | \(169\) |
trager | \(\frac {2 \left (35 b^{2} B \,e^{4} x^{4}+45 A \,b^{2} e^{4} x^{3}+90 B a b \,e^{4} x^{3}+5 b^{2} B d \,e^{3} x^{3}+126 A a b \,e^{4} x^{2}+9 A \,b^{2} d \,e^{3} x^{2}+63 B \,a^{2} e^{4} x^{2}+18 B a b d \,e^{3} x^{2}-6 B \,b^{2} d^{2} e^{2} x^{2}+105 a^{2} A \,e^{4} x +42 A a b d \,e^{3} x -12 A \,b^{2} d^{2} e^{2} x +21 B \,a^{2} d \,e^{3} x -24 B a b \,d^{2} e^{2} x +8 b^{2} B \,d^{3} e x +105 a^{2} A d \,e^{3}-84 A a b \,d^{2} e^{2}+24 A \,b^{2} d^{3} e -42 B \,a^{2} d^{2} e^{2}+48 B a b \,d^{3} e -16 b^{2} B \,d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(253\) |
risch | \(\frac {2 \left (35 b^{2} B \,e^{4} x^{4}+45 A \,b^{2} e^{4} x^{3}+90 B a b \,e^{4} x^{3}+5 b^{2} B d \,e^{3} x^{3}+126 A a b \,e^{4} x^{2}+9 A \,b^{2} d \,e^{3} x^{2}+63 B \,a^{2} e^{4} x^{2}+18 B a b d \,e^{3} x^{2}-6 B \,b^{2} d^{2} e^{2} x^{2}+105 a^{2} A \,e^{4} x +42 A a b d \,e^{3} x -12 A \,b^{2} d^{2} e^{2} x +21 B \,a^{2} d \,e^{3} x -24 B a b \,d^{2} e^{2} x +8 b^{2} B \,d^{3} e x +105 a^{2} A d \,e^{3}-84 A a b \,d^{2} e^{2}+24 A \,b^{2} d^{3} e -42 B \,a^{2} d^{2} e^{2}+48 B a b \,d^{3} e -16 b^{2} B \,d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(253\) |
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Time = 0.23 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.72 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \, {\left (35 \, B b^{2} e^{4} x^{4} - 16 \, B b^{2} d^{4} + 105 \, A a^{2} d e^{3} + 24 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e - 42 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + 5 \, {\left (B b^{2} d e^{3} + 9 \, {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} d^{2} e^{2} - 3 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} - 21 \, {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{2} + {\left (8 \, B b^{2} d^{3} e + 105 \, A a^{2} e^{4} - 12 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \]
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Time = 1.00 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.01 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (A a^{2} x + \frac {B b^{2} x^{4}}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.24 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{2} - 45 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (112) = 224\).
Time = 0.27 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.80 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} A a^{2} d + 105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} A a^{2} + \frac {105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} B a^{2} d}{e} + \frac {210 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} A a b d}{e} + \frac {42 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} B a b d}{e^{2}} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A b^{2} d}{e^{2}} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} B a^{2}}{e} + \frac {42 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A a b}{e} + \frac {9 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B b^{2} d}{e^{3}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B a b}{e^{2}} + \frac {9 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} A b^{2}}{e^{2}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} B b^{2}}{e^{3}}\right )}}{315 \, e} \]
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Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{7\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{5\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \]
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