Integrand size = 22, antiderivative size = 173 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2}}{9 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2}}{11 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2}}{13 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5} \]
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Time = 0.08 (sec) , antiderivative size = 173, 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} \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=-\frac {2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac {6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac {2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^3 (-B d+A e) (d+e x)^{7/2}}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{9/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{13/2}}{e^4}+\frac {b^3 B (d+e x)^{15/2}}{e^4}\right ) \, dx \\ & = \frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2}}{9 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2}}{11 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2}}{13 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.31 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\frac {2 (d+e x)^{9/2} \left (1105 a^3 e^3 (-2 B d+11 A e+9 B e x)+255 a^2 b e^2 \left (13 A e (-2 d+9 e x)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-51 a b^2 e \left (-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )\right )+b^3 \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )\right )}{109395 e^5} \]
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Time = 1.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (a e -b d \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -b d \right )^{2} b B +3 \left (a e -b d \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right )^{3} B +3 \left (a e -b d \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(171\) |
default | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (a e -b d \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -b d \right )^{2} b B +3 \left (a e -b d \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right )^{3} B +3 \left (a e -b d \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(171\) |
pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {9}{17} x^{4} B +\frac {3}{5} A \,x^{3}\right ) b^{3}+\frac {27 x^{2} \left (\frac {13 B x}{15}+A \right ) a \,b^{2}}{13}+\frac {27 x \left (\frac {11 B x}{13}+A \right ) a^{2} b}{11}+a^{3} \left (\frac {9 B x}{11}+A \right )\right ) e^{4}-\frac {6 d \left (\frac {33 x^{2} \left (\frac {52 B x}{51}+A \right ) b^{3}}{65}+\frac {18 \left (\frac {11 B x}{10}+A \right ) x a \,b^{2}}{13}+a^{2} \left (\frac {18 B x}{13}+A \right ) b +\frac {a^{3} B}{3}\right ) e^{3}}{11}+\frac {24 d^{2} b \left (\frac {3 x \left (\frac {22 B x}{17}+A \right ) b^{2}}{5}+a \left (\frac {9 B x}{5}+A \right ) b +a^{2} B \right ) e^{2}}{143}-\frac {16 \left (\left (\frac {36 B x}{17}+A \right ) b +3 B a \right ) d^{3} b^{2} e}{715}+\frac {128 b^{3} B \,d^{4}}{12155}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{5}}\) | \(192\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 B \,x^{4} b^{3} e^{4}+7293 A \,x^{3} b^{3} e^{4}+21879 B \,x^{3} a \,b^{2} e^{4}-3432 B \,x^{3} b^{3} d \,e^{3}+25245 A \,x^{2} a \,b^{2} e^{4}-3366 A \,x^{2} b^{3} d \,e^{3}+25245 B \,x^{2} a^{2} b \,e^{4}-10098 B \,x^{2} a \,b^{2} d \,e^{3}+1584 B \,x^{2} b^{3} d^{2} e^{2}+29835 A x \,a^{2} b \,e^{4}-9180 A x a \,b^{2} d \,e^{3}+1224 A x \,b^{3} d^{2} e^{2}+9945 B x \,a^{3} e^{4}-9180 B x \,a^{2} b d \,e^{3}+3672 B x a \,b^{2} d^{2} e^{2}-576 B x \,b^{3} d^{3} e +12155 a^{3} A \,e^{4}-6630 A \,a^{2} b d \,e^{3}+2040 A a \,b^{2} d^{2} e^{2}-272 A \,b^{3} d^{3} e -2210 B \,a^{3} d \,e^{3}+2040 B \,a^{2} b \,d^{2} e^{2}-816 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{109395 e^{5}}\) | \(301\) |
trager | \(\frac {2 \left (6435 B \,b^{3} e^{8} x^{8}+7293 A \,b^{3} e^{8} x^{7}+21879 B a \,b^{2} e^{8} x^{7}+22308 B \,b^{3} d \,e^{7} x^{7}+25245 A a \,b^{2} e^{8} x^{6}+25806 A \,b^{3} d \,e^{7} x^{6}+25245 B \,a^{2} b \,e^{8} x^{6}+77418 B a \,b^{2} d \,e^{7} x^{6}+26466 B \,b^{3} d^{2} e^{6} x^{6}+29835 A \,a^{2} b \,e^{8} x^{5}+91800 A a \,b^{2} d \,e^{7} x^{5}+31518 A \,b^{3} d^{2} e^{6} x^{5}+9945 B \,a^{3} e^{8} x^{5}+91800 B \,a^{2} b d \,e^{7} x^{5}+94554 B a \,b^{2} d^{2} e^{6} x^{5}+10908 B \,b^{3} d^{3} e^{5} x^{5}+12155 A \,a^{3} e^{8} x^{4}+112710 A \,a^{2} b d \,e^{7} x^{4}+116790 A a \,b^{2} d^{2} e^{6} x^{4}+13600 A \,b^{3} d^{3} e^{5} x^{4}+37570 B \,a^{3} d \,e^{7} x^{4}+116790 B \,a^{2} b \,d^{2} e^{6} x^{4}+40800 B a \,b^{2} d^{3} e^{5} x^{4}+35 B \,b^{3} d^{4} e^{4} x^{4}+48620 A \,a^{3} d \,e^{7} x^{3}+152490 A \,a^{2} b \,d^{2} e^{6} x^{3}+54060 A a \,b^{2} d^{3} e^{5} x^{3}+85 A \,b^{3} d^{4} e^{4} x^{3}+50830 B \,a^{3} d^{2} e^{6} x^{3}+54060 B \,a^{2} b \,d^{3} e^{5} x^{3}+255 B a \,b^{2} d^{4} e^{4} x^{3}-40 B \,b^{3} d^{5} e^{3} x^{3}+72930 A \,a^{3} d^{2} e^{6} x^{2}+79560 A \,a^{2} b \,d^{3} e^{5} x^{2}+765 A a \,b^{2} d^{4} e^{4} x^{2}-102 A \,b^{3} d^{5} e^{3} x^{2}+26520 B \,a^{3} d^{3} e^{5} x^{2}+765 B \,a^{2} b \,d^{4} e^{4} x^{2}-306 B a \,b^{2} d^{5} e^{3} x^{2}+48 B \,b^{3} d^{6} e^{2} x^{2}+48620 A \,a^{3} d^{3} e^{5} x +3315 A \,a^{2} b \,d^{4} e^{4} x -1020 A a \,b^{2} d^{5} e^{3} x +136 A \,b^{3} d^{6} e^{2} x +1105 B \,a^{3} d^{4} e^{4} x -1020 B \,a^{2} b \,d^{5} e^{3} x +408 B a \,b^{2} d^{6} e^{2} x -64 B \,b^{3} d^{7} e x +12155 A \,a^{3} d^{4} e^{4}-6630 A \,a^{2} b \,d^{5} e^{3}+2040 A a \,b^{2} d^{6} e^{2}-272 A \,b^{3} d^{7} e -2210 B \,a^{3} d^{5} e^{3}+2040 B \,a^{2} b \,d^{6} e^{2}-816 B a \,b^{2} d^{7} e +128 B \,b^{3} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(793\) |
risch | \(\frac {2 \left (6435 B \,b^{3} e^{8} x^{8}+7293 A \,b^{3} e^{8} x^{7}+21879 B a \,b^{2} e^{8} x^{7}+22308 B \,b^{3} d \,e^{7} x^{7}+25245 A a \,b^{2} e^{8} x^{6}+25806 A \,b^{3} d \,e^{7} x^{6}+25245 B \,a^{2} b \,e^{8} x^{6}+77418 B a \,b^{2} d \,e^{7} x^{6}+26466 B \,b^{3} d^{2} e^{6} x^{6}+29835 A \,a^{2} b \,e^{8} x^{5}+91800 A a \,b^{2} d \,e^{7} x^{5}+31518 A \,b^{3} d^{2} e^{6} x^{5}+9945 B \,a^{3} e^{8} x^{5}+91800 B \,a^{2} b d \,e^{7} x^{5}+94554 B a \,b^{2} d^{2} e^{6} x^{5}+10908 B \,b^{3} d^{3} e^{5} x^{5}+12155 A \,a^{3} e^{8} x^{4}+112710 A \,a^{2} b d \,e^{7} x^{4}+116790 A a \,b^{2} d^{2} e^{6} x^{4}+13600 A \,b^{3} d^{3} e^{5} x^{4}+37570 B \,a^{3} d \,e^{7} x^{4}+116790 B \,a^{2} b \,d^{2} e^{6} x^{4}+40800 B a \,b^{2} d^{3} e^{5} x^{4}+35 B \,b^{3} d^{4} e^{4} x^{4}+48620 A \,a^{3} d \,e^{7} x^{3}+152490 A \,a^{2} b \,d^{2} e^{6} x^{3}+54060 A a \,b^{2} d^{3} e^{5} x^{3}+85 A \,b^{3} d^{4} e^{4} x^{3}+50830 B \,a^{3} d^{2} e^{6} x^{3}+54060 B \,a^{2} b \,d^{3} e^{5} x^{3}+255 B a \,b^{2} d^{4} e^{4} x^{3}-40 B \,b^{3} d^{5} e^{3} x^{3}+72930 A \,a^{3} d^{2} e^{6} x^{2}+79560 A \,a^{2} b \,d^{3} e^{5} x^{2}+765 A a \,b^{2} d^{4} e^{4} x^{2}-102 A \,b^{3} d^{5} e^{3} x^{2}+26520 B \,a^{3} d^{3} e^{5} x^{2}+765 B \,a^{2} b \,d^{4} e^{4} x^{2}-306 B a \,b^{2} d^{5} e^{3} x^{2}+48 B \,b^{3} d^{6} e^{2} x^{2}+48620 A \,a^{3} d^{3} e^{5} x +3315 A \,a^{2} b \,d^{4} e^{4} x -1020 A a \,b^{2} d^{5} e^{3} x +136 A \,b^{3} d^{6} e^{2} x +1105 B \,a^{3} d^{4} e^{4} x -1020 B \,a^{2} b \,d^{5} e^{3} x +408 B a \,b^{2} d^{6} e^{2} x -64 B \,b^{3} d^{7} e x +12155 A \,a^{3} d^{4} e^{4}-6630 A \,a^{2} b \,d^{5} e^{3}+2040 A a \,b^{2} d^{6} e^{2}-272 A \,b^{3} d^{7} e -2210 B \,a^{3} d^{5} e^{3}+2040 B \,a^{2} b \,d^{6} e^{2}-816 B a \,b^{2} d^{7} e +128 B \,b^{3} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(793\) |
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Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (153) = 306\).
Time = 0.23 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.66 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\frac {2 \, {\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \, {\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \, {\left (52 \, B b^{3} d e^{7} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \, {\left (802 \, B b^{3} d^{2} e^{6} + 782 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \, {\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \, {\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \, {\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1523 vs. \(2 (170) = 340\).
Time = 0.82 (sec) , antiderivative size = 1523, normalized size of antiderivative = 8.80 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\text {Too large to display} \]
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none
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.53 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} B b^{3} - 7293 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 25245 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 9945 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 12155 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{109395 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2684 vs. \(2 (153) = 306\).
Time = 0.31 (sec) , antiderivative size = 2684, normalized size of antiderivative = 15.51 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\text {Too large to display} \]
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Time = 1.40 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89 \[ \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{15\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{11\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{13\,e^5} \]
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