Integrand size = 26, antiderivative size = 263 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac {2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}-\frac {2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {2 d \sqrt {d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
[In]
[Out]
Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{3/2}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 \sqrt {d+e x}}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{5/2}}{e^5}+\frac {B c^2 (d+e x)^{7/2}}{e^5}\right ) \, dx \\ & = \frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {-6 A e \left (35 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+2 B \left (63 b^2 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+18 b c e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 c^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{315 e^6 \sqrt {d+e x}} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (5 \left (7 e^{5} x^{5}-10 d \,e^{4} x^{4}+16 d^{2} e^{3} x^{3}-32 d^{3} e^{2} x^{2}+128 d^{4} e x +256 d^{5}\right ) B -1152 \left (-\frac {5}{128} e^{4} x^{4}+\frac {1}{16} d \,e^{3} x^{3}-\frac {1}{8} d^{2} e^{2} x^{2}+\frac {1}{2} d^{3} e x +d^{4}\right ) A e \right ) c^{2}}{315}+\frac {64 \left (\frac {\left (\frac {5}{16} e^{4} x^{4}-\frac {1}{2} d \,e^{3} x^{3}+d^{2} e^{2} x^{2}-4 d^{3} e x -8 d^{4}\right ) B}{7}+A e \left (\frac {1}{16} e^{3} x^{3}-\frac {1}{8} d \,e^{2} x^{2}+\frac {1}{2} d^{2} e x +d^{3}\right )\right ) e b c}{5}-\frac {16 \left (\frac {3 \left (-\frac {1}{8} e^{3} x^{3}+\frac {1}{4} d \,e^{2} x^{2}-d^{2} e x -2 d^{3}\right ) B}{5}+A e \left (-\frac {1}{8} e^{2} x^{2}+\frac {1}{2} d e x +d^{2}\right )\right ) e^{2} b^{2}}{3}}{\sqrt {e x +d}\, e^{6}}\) | \(259\) |
risch | \(-\frac {2 \left (-35 B \,c^{2} x^{4} e^{4}-45 A \,c^{2} e^{4} x^{3}-90 B b c \,e^{4} x^{3}+85 B \,c^{2} d \,e^{3} x^{3}-126 A b c \,e^{4} x^{2}+117 A \,c^{2} d \,e^{3} x^{2}-63 B \,b^{2} e^{4} x^{2}+234 B b c d \,e^{3} x^{2}-165 B \,c^{2} d^{2} e^{2} x^{2}-105 A \,b^{2} e^{4} x +378 A b c d \,e^{3} x -261 A \,c^{2} d^{2} e^{2} x +189 B \,b^{2} d \,e^{3} x -522 B b c \,d^{2} e^{2} x +325 B \,c^{2} d^{3} e x +525 A \,b^{2} d \,e^{3}-1386 A b c \,d^{2} e^{2}+837 A \,c^{2} d^{3} e -693 B \,b^{2} d^{2} e^{2}+1674 B b c \,d^{3} e -965 B \,c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{6}}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{e^{6} \sqrt {e x +d}}\) | \(324\) |
gosper | \(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-45 A \,x^{4} c^{2} e^{5}-90 B \,x^{4} b c \,e^{5}+50 B \,x^{4} c^{2} d \,e^{4}-126 A \,x^{3} b c \,e^{5}+72 A \,x^{3} c^{2} d \,e^{4}-63 B \,x^{3} b^{2} e^{5}+144 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}-105 A \,x^{2} b^{2} e^{5}+252 A \,x^{2} b c d \,e^{4}-144 A \,x^{2} c^{2} d^{2} e^{3}+126 B \,x^{2} b^{2} d \,e^{4}-288 B \,x^{2} b c \,d^{2} e^{3}+160 B \,x^{2} c^{2} d^{3} e^{2}+420 A x \,b^{2} d \,e^{4}-1008 A x b c \,d^{2} e^{3}+576 A x \,c^{2} d^{3} e^{2}-504 B x \,b^{2} d^{2} e^{3}+1152 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +840 A \,b^{2} d^{2} e^{3}-2016 A b c \,d^{3} e^{2}+1152 A \,c^{2} d^{4} e -1008 B \,b^{2} d^{3} e^{2}+2304 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\) | \(341\) |
trager | \(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-45 A \,x^{4} c^{2} e^{5}-90 B \,x^{4} b c \,e^{5}+50 B \,x^{4} c^{2} d \,e^{4}-126 A \,x^{3} b c \,e^{5}+72 A \,x^{3} c^{2} d \,e^{4}-63 B \,x^{3} b^{2} e^{5}+144 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}-105 A \,x^{2} b^{2} e^{5}+252 A \,x^{2} b c d \,e^{4}-144 A \,x^{2} c^{2} d^{2} e^{3}+126 B \,x^{2} b^{2} d \,e^{4}-288 B \,x^{2} b c \,d^{2} e^{3}+160 B \,x^{2} c^{2} d^{3} e^{2}+420 A x \,b^{2} d \,e^{4}-1008 A x b c \,d^{2} e^{3}+576 A x \,c^{2} d^{3} e^{2}-504 B x \,b^{2} d^{2} e^{3}+1152 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +840 A \,b^{2} d^{2} e^{3}-2016 A b c \,d^{3} e^{2}+1152 A \,c^{2} d^{4} e -1008 B \,b^{2} d^{3} e^{2}+2304 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\) | \(341\) |
derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 B b c e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {16 B b c d e \left (e x +d \right )^{\frac {5}{2}}}{5}+4 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 A \,b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A b c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+4 A \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-2 B \,b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+8 B b c \,d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 B \,c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A \,b^{2} d \,e^{3} \sqrt {e x +d}+12 A b c \,d^{2} e^{2} \sqrt {e x +d}-8 A \,c^{2} d^{3} e \sqrt {e x +d}+6 B \,b^{2} d^{2} e^{2} \sqrt {e x +d}-16 B b c \,d^{3} e \sqrt {e x +d}+10 B \,c^{2} d^{4} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(405\) |
default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 B b c e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {16 B b c d e \left (e x +d \right )^{\frac {5}{2}}}{5}+4 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 A \,b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A b c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+4 A \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-2 B \,b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+8 B b c \,d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 B \,c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A \,b^{2} d \,e^{3} \sqrt {e x +d}+12 A b c \,d^{2} e^{2} \sqrt {e x +d}-8 A \,c^{2} d^{3} e \sqrt {e x +d}+6 B \,b^{2} d^{2} e^{2} \sqrt {e x +d}-16 B b c \,d^{3} e \sqrt {e x +d}+10 B \,c^{2} d^{4} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(405\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 840 \, A b^{2} d^{2} e^{3} - 1152 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 1008 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \, {\left (10 \, B c^{2} d e^{4} - 9 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + {\left (80 \, B c^{2} d^{2} e^{3} - 72 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 63 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - {\left (160 \, B c^{2} d^{3} e^{2} - 105 \, A b^{2} e^{5} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 105 \, A b^{2} d e^{4} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\right )}} \]
[In]
[Out]
Time = 8.92 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{5}} + \frac {d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{2} - 45 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )}}{\sqrt {e x + d} e^{6}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{2} e^{48} - 225 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} d e^{48} + 630 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} d^{2} e^{48} - 1050 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt {e x + d} B c^{2} d^{4} e^{48} + 90 \, {\left (e x + d\right )}^{\frac {7}{2}} B b c e^{49} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A c^{2} e^{49} - 504 \, {\left (e x + d\right )}^{\frac {5}{2}} B b c d e^{49} - 252 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{2} d e^{49} + 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} B b c d^{2} e^{49} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} d^{2} e^{49} - 2520 \, \sqrt {e x + d} B b c d^{3} e^{49} - 1260 \, \sqrt {e x + d} A c^{2} d^{3} e^{49} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{2} e^{50} + 126 \, {\left (e x + d\right )}^{\frac {5}{2}} A b c e^{50} - 315 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e^{50} - 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c d e^{50} + 945 \, \sqrt {e x + d} B b^{2} d^{2} e^{50} + 1890 \, \sqrt {e x + d} A b c d^{2} e^{50} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{51} - 630 \, \sqrt {e x + d} A b^{2} d e^{51}\right )}}{315 \, e^{54}} \]
[In]
[Out]
Time = 10.40 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{3\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{5\,e^6}+\frac {2\,B\,b^2\,d^3\,e^2-2\,A\,b^2\,d^2\,e^3-4\,B\,b\,c\,d^4\,e+4\,A\,b\,c\,d^3\,e^2+2\,B\,c^2\,d^5-2\,A\,c^2\,d^4\,e}{e^6\,\sqrt {d+e\,x}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{e^6} \]
[In]
[Out]