Integrand size = 24, antiderivative size = 126 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{7/2}}{7 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{9/2}}{9 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \]
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Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=-\frac {2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac {2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^{3/2}}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{5/2}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{7/2}}{e^3}+\frac {B c (d+e x)^{9/2}}{e^3}\right ) \, dx \\ & = -\frac {2 d (B d-A e) (c d-b e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{7/2}}{7 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{9/2}}{9 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\frac {2 (d+e x)^{5/2} \left (11 A e \left (9 b e (-2 d+5 e x)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+B \left (11 b e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 c \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]
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Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(-\frac {4 \left (-\frac {5 \left (\frac {7 B c \,x^{2}}{11}+\frac {7 \left (A c +B b \right ) x}{9}+A b \right ) x \,e^{3}}{2}+d \left (\frac {35 B c \,x^{2}}{33}+\frac {10 \left (A c +B b \right ) x}{9}+A b \right ) e^{2}-\frac {4 \left (\frac {15}{11} B c x +A c +B b \right ) d^{2} e}{9}+\frac {8 B c \,d^{3}}{33}\right ) \left (e x +d \right )^{\frac {5}{2}}}{35 e^{4}}\) | \(92\) |
default | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (-A e +B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-A e +B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(112\) |
derivativedivides | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-\left (A e -B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {2 \left (A e -B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(113\) |
gosper | \(-\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (-315 B c \,x^{3} e^{3}-385 A c \,e^{3} x^{2}-385 B \,x^{2} b \,e^{3}+210 B \,x^{2} c d \,e^{2}-495 A b \,e^{3} x +220 A c d \,e^{2} x +220 B x b d \,e^{2}-120 B c \,d^{2} e x +198 A b d \,e^{2}-88 A c \,d^{2} e -88 B b \,d^{2} e +48 B c \,d^{3}\right )}{3465 e^{4}}\) | \(121\) |
trager | \(-\frac {2 \left (-315 B c \,e^{5} x^{5}-385 A c \,e^{5} x^{4}-385 B b \,e^{5} x^{4}-420 B c d \,e^{4} x^{4}-495 A b \,e^{5} x^{3}-550 A c d \,e^{4} x^{3}-550 B b d \,e^{4} x^{3}-15 B c \,d^{2} e^{3} x^{3}-792 A b d \,e^{4} x^{2}-33 A c \,d^{2} e^{3} x^{2}-33 B b \,d^{2} e^{3} x^{2}+18 B c \,d^{3} e^{2} x^{2}-99 A b \,d^{2} e^{3} x +44 A c \,d^{3} e^{2} x +44 B b \,d^{3} e^{2} x -24 B c \,d^{4} e x +198 A b \,d^{3} e^{2}-88 A c \,d^{4} e -88 B b \,d^{4} e +48 B c \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(225\) |
risch | \(-\frac {2 \left (-315 B c \,e^{5} x^{5}-385 A c \,e^{5} x^{4}-385 B b \,e^{5} x^{4}-420 B c d \,e^{4} x^{4}-495 A b \,e^{5} x^{3}-550 A c d \,e^{4} x^{3}-550 B b d \,e^{4} x^{3}-15 B c \,d^{2} e^{3} x^{3}-792 A b d \,e^{4} x^{2}-33 A c \,d^{2} e^{3} x^{2}-33 B b \,d^{2} e^{3} x^{2}+18 B c \,d^{3} e^{2} x^{2}-99 A b \,d^{2} e^{3} x +44 A c \,d^{3} e^{2} x +44 B b \,d^{3} e^{2} x -24 B c \,d^{4} e x +198 A b \,d^{3} e^{2}-88 A c \,d^{4} e -88 B b \,d^{4} e +48 B c \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(225\) |
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Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.51 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} - 198 \, A b d^{3} e^{2} + 88 \, {\left (B b + A c\right )} d^{4} e + 35 \, {\left (12 \, B c d e^{4} + 11 \, {\left (B b + A c\right )} e^{5}\right )} x^{4} + 5 \, {\left (3 \, B c d^{2} e^{3} + 99 \, A b e^{5} + 110 \, {\left (B b + A c\right )} d e^{4}\right )} x^{3} - 3 \, {\left (6 \, B c d^{3} e^{2} - 264 \, A b d e^{4} - 11 \, {\left (B b + A c\right )} d^{2} e^{3}\right )} x^{2} + {\left (24 \, B c d^{4} e + 99 \, A b d^{2} e^{3} - 44 \, {\left (B b + A c\right )} d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \]
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Time = 0.98 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.43 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\begin {cases} \frac {2 \left (\frac {B c \left (d + e x\right )^{\frac {11}{2}}}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A c e + B b e - 3 B c d\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- A b d e^{2} + A c d^{2} e + B b d^{2} e - B c d^{3}\right )}{5 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\frac {A b x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {x^{3} \left (A c + B b\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c - 385 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 632, normalized size of antiderivative = 5.02 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (\frac {1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} A b d^{2}}{e} + \frac {231 \, {\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 b d^{2}}{e^{2}} + \frac {231 \, {\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 c d^{2}}{e^{2}} + \frac {462 \, {\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 d}{e} + \frac {99 \, {\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 c d^{2}}{e^{3}} + \frac {198 \, {\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 d}{e^{2}} + \frac {198 \, {\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 c d}{e^{2}} + \frac {99 \, {\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}{e} + \frac {22 \, {\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 c d}{e^{3}} + \frac {11 \, {\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}{e^{2}} + \frac {11 \, {\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 )} A c}{e^{2}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} B c}{e^{3}}\right )}}{3465 \, e} \]
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Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{7\,e^4}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{9\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \]
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