Integrand size = 22, antiderivative size = 128 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
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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} \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=-\frac {2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{3/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^3}+\frac {b^2 B (d+e x)^{9/2}}{e^3}\right ) \, dx \\ & = -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 (d+e x)^{5/2} \left (99 a^2 e^2 (-2 B d+7 A e+5 B e x)+22 a b e \left (9 A e (-2 d+5 e x)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]
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Time = 0.94 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {5 x^{2} \left (\frac {9 B x}{11}+A \right ) b^{2}}{9}+\frac {10 x \left (\frac {7 B x}{9}+A \right ) a b}{7}+a^{2} \left (\frac {5 B x}{7}+A \right )\right ) e^{3}-\frac {4 d \left (\frac {5 x \left (\frac {21 B x}{22}+A \right ) b^{2}}{9}+a \left (\frac {10 B x}{9}+A \right ) b +\frac {a^{2} B}{2}\right ) e^{2}}{7}+\frac {8 d^{2} b \left (\left (\frac {15 B x}{11}+A \right ) b +2 B a \right ) e}{63}-\frac {16 b^{2} B \,d^{3}}{231}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{4}}\) | \(117\) |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (315 b^{2} B \,x^{3} e^{3}+385 A \,b^{2} e^{3} x^{2}+770 B a b \,e^{3} x^{2}-210 B \,b^{2} d \,e^{2} x^{2}+990 A a b \,e^{3} x -220 A \,b^{2} d \,e^{2} x +495 B \,a^{2} e^{3} x -440 B a b d \,e^{2} x +120 B \,b^{2} d^{2} e x +693 a^{2} A \,e^{3}-396 A a b d \,e^{2}+88 A \,b^{2} d^{2} e -198 B \,a^{2} d \,e^{2}+176 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{3465 e^{4}}\) | \(169\) |
trager | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
risch | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (112) = 224\).
Time = 0.22 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.26 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 \, {\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \, {\left (12 \, B b^{2} d e^{4} + 11 \, {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (3 \, B b^{2} d^{2} e^{3} + 110 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \]
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Time = 1.07 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.01 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{5 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (A a^{2} x + \frac {B b^{2} x^{4}}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.24 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{2} - 385 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (112) = 224\).
Time = 0.29 (sec) , antiderivative size = 854, normalized size of antiderivative = 6.67 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]
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Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{9\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{7\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \]
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