Integrand size = 24, antiderivative size = 143 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {3 c (2 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {2 c^2 (d+e x)^{4+m}}{e^4 (4+m)} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 143, 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} \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac {(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]
[In]
[Out]
Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^3}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{1+m}}{e^3}-\frac {3 c (2 c d-b e) (d+e x)^{2+m}}{e^3}+\frac {2 c^2 (d+e x)^{3+m}}{e^3}\right ) \, dx \\ & = -\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {3 c (2 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {2 c^2 (d+e x)^{4+m}}{e^4 (4+m)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {6 (-2 c d+b e) \left (c d^2+e (-b d+a e)\right )}{e^2 (1+m)}+\frac {\left (12 c^2 d^2-b^2 e^2 m+4 c e (-3 b d+a e (3+m))\right ) (d+e x)}{e^2 (2+m)}+(-6 c d+b e (6+m)+2 c e (3+m) x) (a+x (b+c x))\right )}{e^2 (3+m) (4+m)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(143)=286\).
Time = 0.22 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.97
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (2 c^{2} e^{3} m^{3} x^{3}+3 b c \,e^{3} m^{3} x^{2}+12 c^{2} e^{3} m^{2} x^{3}+2 a c \,e^{3} m^{3} x +b^{2} e^{3} m^{3} x +21 b c \,e^{3} m^{2} x^{2}-6 c^{2} d \,e^{2} m^{2} x^{2}+22 c^{2} e^{3} m \,x^{3}+a b \,e^{3} m^{3}+16 a c \,e^{3} m^{2} x +8 b^{2} e^{3} m^{2} x -6 b c d \,e^{2} m^{2} x +42 b c \,e^{3} m \,x^{2}-18 c^{2} d \,e^{2} m \,x^{2}+12 c^{2} x^{3} e^{3}+9 a b \,e^{3} m^{2}-2 a c d \,e^{2} m^{2}+38 a c \,e^{3} m x -b^{2} d \,e^{2} m^{2}+19 b^{2} e^{3} m x -30 b c d \,e^{2} m x +24 x^{2} b c \,e^{3}+12 c^{2} d^{2} e m x -12 x^{2} c^{2} d \,e^{2}+26 a b \,e^{3} m -14 a c d \,e^{2} m +24 x a c \,e^{3}-7 b^{2} d \,e^{2} m +12 x \,b^{2} e^{3}+6 b c \,d^{2} e m -24 x b c d \,e^{2}+12 x \,c^{2} d^{2} e +24 a b \,e^{3}-24 a c d \,e^{2}-12 b^{2} d \,e^{2}+24 b c \,d^{2} e -12 c^{2} d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(424\) |
norman | \(\frac {d \left (a b \,e^{3} m^{3}+9 a b \,e^{3} m^{2}-2 a c d \,e^{2} m^{2}-b^{2} d \,e^{2} m^{2}+26 a b \,e^{3} m -14 a c d \,e^{2} m -7 b^{2} d \,e^{2} m +6 b c \,d^{2} e m +24 a b \,e^{3}-24 a c d \,e^{2}-12 b^{2} d \,e^{2}+24 b c \,d^{2} e -12 c^{2} d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (2 a c \,e^{2} m^{2}+b^{2} e^{2} m^{2}+3 b c d e \,m^{2}+14 a c \,e^{2} m +7 b^{2} e^{2} m +12 b c d e m -6 c^{2} d^{2} m +24 a c \,e^{2}+12 b^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (a b \,e^{3} m^{3}+2 a c d \,e^{2} m^{3}+b^{2} d \,e^{2} m^{3}+9 a b \,e^{3} m^{2}+14 a c d \,e^{2} m^{2}+7 b^{2} d \,e^{2} m^{2}-6 b c \,d^{2} e \,m^{2}+26 a b \,e^{3} m +24 a c d \,e^{2} m +12 b^{2} d \,e^{2} m -24 b c \,d^{2} e m +12 c^{2} d^{3} m +24 a b \,e^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 b e m +2 c d m +12 b e \right ) c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {2 c^{2} x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}\) | \(490\) |
risch | \(\frac {\left (2 c^{2} e^{4} m^{3} x^{4}+3 b c \,e^{4} m^{3} x^{3}+2 c^{2} d \,e^{3} m^{3} x^{3}+12 c^{2} e^{4} m^{2} x^{4}+2 a c \,e^{4} m^{3} x^{2}+b^{2} e^{4} m^{3} x^{2}+3 b c d \,e^{3} m^{3} x^{2}+21 b c \,e^{4} m^{2} x^{3}+6 c^{2} d \,e^{3} m^{2} x^{3}+22 c^{2} e^{4} m \,x^{4}+a b \,e^{4} m^{3} x +2 a c d \,e^{3} m^{3} x +16 a c \,e^{4} m^{2} x^{2}+b^{2} d \,e^{3} m^{3} x +8 b^{2} e^{4} m^{2} x^{2}+15 b c d \,e^{3} m^{2} x^{2}+42 b c \,e^{4} m \,x^{3}-6 c^{2} d^{2} e^{2} m^{2} x^{2}+4 c^{2} d \,e^{3} m \,x^{3}+12 c^{2} x^{4} e^{4}+a b d \,e^{3} m^{3}+9 a b \,e^{4} m^{2} x +14 a c d \,e^{3} m^{2} x +38 a c \,e^{4} m \,x^{2}+7 b^{2} d \,e^{3} m^{2} x +19 b^{2} e^{4} m \,x^{2}-6 b c \,d^{2} e^{2} m^{2} x +12 b c d \,e^{3} m \,x^{2}+24 b c \,e^{4} x^{3}-6 c^{2} d^{2} e^{2} m \,x^{2}+9 a b d \,e^{3} m^{2}+26 a b \,e^{4} m x -2 a c \,d^{2} e^{2} m^{2}+24 a c d \,e^{3} m x +24 a c \,e^{4} x^{2}-b^{2} d^{2} e^{2} m^{2}+12 b^{2} d \,e^{3} m x +12 b^{2} e^{4} x^{2}-24 b c \,d^{2} e^{2} m x +12 c^{2} d^{3} e m x +26 a b d \,e^{3} m +24 a b \,e^{4} x -14 a c \,d^{2} e^{2} m -7 b^{2} d^{2} e^{2} m +6 b c \,d^{3} e m +24 a b d \,e^{3}-24 a c \,d^{2} e^{2}-12 b^{2} d^{2} e^{2}+24 b c \,d^{3} e -12 c^{2} d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(622\) |
parallelrisch | \(\frac {9 \left (e x +d \right )^{m} a b d \,e^{3} m^{2}-2 \left (e x +d \right )^{m} a c \,d^{2} e^{2} m^{2}+26 \left (e x +d \right )^{m} a b d \,e^{3} m -14 \left (e x +d \right )^{m} a c \,d^{2} e^{2} m +6 \left (e x +d \right )^{m} b c \,d^{3} e m +2 x^{3} \left (e x +d \right )^{m} c^{2} d \,e^{3} m^{3}+21 x^{3} \left (e x +d \right )^{m} b c \,e^{4} m^{2}+6 x^{3} \left (e x +d \right )^{m} c^{2} d \,e^{3} m^{2}+2 x^{2} \left (e x +d \right )^{m} a c \,e^{4} m^{3}+42 x^{3} \left (e x +d \right )^{m} b c \,e^{4} m +4 x^{3} \left (e x +d \right )^{m} c^{2} d \,e^{3} m +16 x^{2} \left (e x +d \right )^{m} a c \,e^{4} m^{2}-6 x^{2} \left (e x +d \right )^{m} c^{2} d^{2} e^{2} m^{2}+x \left (e x +d \right )^{m} a b \,e^{4} m^{3}+x \left (e x +d \right )^{m} b^{2} d \,e^{3} m^{3}+38 x^{2} \left (e x +d \right )^{m} a c \,e^{4} m -6 x^{2} \left (e x +d \right )^{m} c^{2} d^{2} e^{2} m +9 x \left (e x +d \right )^{m} a b \,e^{4} m^{2}+7 x \left (e x +d \right )^{m} b^{2} d \,e^{3} m^{2}+\left (e x +d \right )^{m} a b d \,e^{3} m^{3}+26 x \left (e x +d \right )^{m} a b \,e^{4} m +12 x \left (e x +d \right )^{m} b^{2} d \,e^{3} m +12 x \left (e x +d \right )^{m} c^{2} d^{3} e m +3 x^{3} \left (e x +d \right )^{m} b c \,e^{4} m^{3}+12 x^{4} \left (e x +d \right )^{m} c^{2} e^{4}+12 x^{2} \left (e x +d \right )^{m} b^{2} e^{4}-12 \left (e x +d \right )^{m} b^{2} d^{2} e^{2}+3 x^{2} \left (e x +d \right )^{m} b c d \,e^{3} m^{3}+15 x^{2} \left (e x +d \right )^{m} b c d \,e^{3} m^{2}+2 x \left (e x +d \right )^{m} a c d \,e^{3} m^{3}+12 x^{2} \left (e x +d \right )^{m} b c d \,e^{3} m +14 x \left (e x +d \right )^{m} a c d \,e^{3} m^{2}-6 x \left (e x +d \right )^{m} b c \,d^{2} e^{2} m^{2}+24 x \left (e x +d \right )^{m} a c d \,e^{3} m -24 x \left (e x +d \right )^{m} b c \,d^{2} e^{2} m +2 x^{4} \left (e x +d \right )^{m} c^{2} e^{4} m^{3}+12 x^{4} \left (e x +d \right )^{m} c^{2} e^{4} m^{2}+22 x^{4} \left (e x +d \right )^{m} c^{2} e^{4} m +x^{2} \left (e x +d \right )^{m} b^{2} e^{4} m^{3}+8 x^{2} \left (e x +d \right )^{m} b^{2} e^{4} m^{2}+24 x^{3} \left (e x +d \right )^{m} b c \,e^{4}+19 x^{2} \left (e x +d \right )^{m} b^{2} e^{4} m +24 x^{2} \left (e x +d \right )^{m} a c \,e^{4}-\left (e x +d \right )^{m} b^{2} d^{2} e^{2} m^{2}+24 x \left (e x +d \right )^{m} a b \,e^{4}-7 \left (e x +d \right )^{m} b^{2} d^{2} e^{2} m +24 \left (e x +d \right )^{m} a b d \,e^{3}-24 \left (e x +d \right )^{m} a c \,d^{2} e^{2}+24 \left (e x +d \right )^{m} b c \,d^{3} e -12 \left (e x +d \right )^{m} c^{2} d^{4}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(965\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (143) = 286\).
Time = 0.47 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.57 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left (a b d e^{3} m^{3} - 12 \, c^{2} d^{4} + 24 \, b c d^{3} e + 24 \, a b d e^{3} - 12 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 2 \, {\left (c^{2} e^{4} m^{3} + 6 \, c^{2} e^{4} m^{2} + 11 \, c^{2} e^{4} m + 6 \, c^{2} e^{4}\right )} x^{4} + {\left (24 \, b c e^{4} + {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} m^{3} + 3 \, {\left (2 \, c^{2} d e^{3} + 7 \, b c e^{4}\right )} m^{2} + 2 \, {\left (2 \, c^{2} d e^{3} + 21 \, b c e^{4}\right )} m\right )} x^{3} + {\left (9 \, a b d e^{3} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m^{2} + {\left (12 \, {\left (b^{2} + 2 \, a c\right )} e^{4} + {\left (3 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{3} - {\left (6 \, c^{2} d^{2} e^{2} - 15 \, b c d e^{3} - 8 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{2} - {\left (6 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} - 19 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m\right )} x^{2} + {\left (6 \, b c d^{3} e + 26 \, a b d e^{3} - 7 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m + {\left (24 \, a b e^{4} + {\left (a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{3} - {\left (6 \, b c d^{2} e^{2} - 9 \, a b e^{4} - 7 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + 13 \, a b e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 4760 vs. \(2 (129) = 258\).
Time = 1.18 (sec) , antiderivative size = 4760, normalized size of antiderivative = 33.29 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (143) = 286\).
Time = 0.22 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.01 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a c}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a b}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (143) = 286\).
Time = 0.29 (sec) , antiderivative size = 978, normalized size of antiderivative = 6.84 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {2 \, {\left (e x + d\right )}^{m} c^{2} e^{4} m^{3} x^{4} + 2 \, {\left (e x + d\right )}^{m} c^{2} d e^{3} m^{3} x^{3} + 3 \, {\left (e x + d\right )}^{m} b c e^{4} m^{3} x^{3} + 12 \, {\left (e x + d\right )}^{m} c^{2} e^{4} m^{2} x^{4} + 3 \, {\left (e x + d\right )}^{m} b c d e^{3} m^{3} x^{2} + {\left (e x + d\right )}^{m} b^{2} e^{4} m^{3} x^{2} + 2 \, {\left (e x + d\right )}^{m} a c e^{4} m^{3} x^{2} + 6 \, {\left (e x + d\right )}^{m} c^{2} d e^{3} m^{2} x^{3} + 21 \, {\left (e x + d\right )}^{m} b c e^{4} m^{2} x^{3} + 22 \, {\left (e x + d\right )}^{m} c^{2} e^{4} m x^{4} + {\left (e x + d\right )}^{m} b^{2} d e^{3} m^{3} x + 2 \, {\left (e x + d\right )}^{m} a c d e^{3} m^{3} x + {\left (e x + d\right )}^{m} a b e^{4} m^{3} x - 6 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{2} m^{2} x^{2} + 15 \, {\left (e x + d\right )}^{m} b c d e^{3} m^{2} x^{2} + 8 \, {\left (e x + d\right )}^{m} b^{2} e^{4} m^{2} x^{2} + 16 \, {\left (e x + d\right )}^{m} a c e^{4} m^{2} x^{2} + 4 \, {\left (e x + d\right )}^{m} c^{2} d e^{3} m x^{3} + 42 \, {\left (e x + d\right )}^{m} b c e^{4} m x^{3} + 12 \, {\left (e x + d\right )}^{m} c^{2} e^{4} x^{4} + {\left (e x + d\right )}^{m} a b d e^{3} m^{3} - 6 \, {\left (e x + d\right )}^{m} b c d^{2} e^{2} m^{2} x + 7 \, {\left (e x + d\right )}^{m} b^{2} d e^{3} m^{2} x + 14 \, {\left (e x + d\right )}^{m} a c d e^{3} m^{2} x + 9 \, {\left (e x + d\right )}^{m} a b e^{4} m^{2} x - 6 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{2} m x^{2} + 12 \, {\left (e x + d\right )}^{m} b c d e^{3} m x^{2} + 19 \, {\left (e x + d\right )}^{m} b^{2} e^{4} m x^{2} + 38 \, {\left (e x + d\right )}^{m} a c e^{4} m x^{2} + 24 \, {\left (e x + d\right )}^{m} b c e^{4} x^{3} - {\left (e x + d\right )}^{m} b^{2} d^{2} e^{2} m^{2} - 2 \, {\left (e x + d\right )}^{m} a c d^{2} e^{2} m^{2} + 9 \, {\left (e x + d\right )}^{m} a b d e^{3} m^{2} + 12 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e m x - 24 \, {\left (e x + d\right )}^{m} b c d^{2} e^{2} m x + 12 \, {\left (e x + d\right )}^{m} b^{2} d e^{3} m x + 24 \, {\left (e x + d\right )}^{m} a c d e^{3} m x + 26 \, {\left (e x + d\right )}^{m} a b e^{4} m x + 12 \, {\left (e x + d\right )}^{m} b^{2} e^{4} x^{2} + 24 \, {\left (e x + d\right )}^{m} a c e^{4} x^{2} + 6 \, {\left (e x + d\right )}^{m} b c d^{3} e m - 7 \, {\left (e x + d\right )}^{m} b^{2} d^{2} e^{2} m - 14 \, {\left (e x + d\right )}^{m} a c d^{2} e^{2} m + 26 \, {\left (e x + d\right )}^{m} a b d e^{3} m + 24 \, {\left (e x + d\right )}^{m} a b e^{4} x - 12 \, {\left (e x + d\right )}^{m} c^{2} d^{4} + 24 \, {\left (e x + d\right )}^{m} b c d^{3} e - 12 \, {\left (e x + d\right )}^{m} b^{2} d^{2} e^{2} - 24 \, {\left (e x + d\right )}^{m} a c d^{2} e^{2} + 24 \, {\left (e x + d\right )}^{m} a b d e^{3}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
[In]
[Out]
Time = 11.06 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.87 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {x\,{\left (d+e\,x\right )}^m\,\left (b^2\,d\,e^3\,m^3+7\,b^2\,d\,e^3\,m^2+12\,b^2\,d\,e^3\,m-6\,b\,c\,d^2\,e^2\,m^2-24\,b\,c\,d^2\,e^2\,m+a\,b\,e^4\,m^3+9\,a\,b\,e^4\,m^2+26\,a\,b\,e^4\,m+24\,a\,b\,e^4+12\,c^2\,d^3\,e\,m+2\,a\,c\,d\,e^3\,m^3+14\,a\,c\,d\,e^3\,m^2+24\,a\,c\,d\,e^3\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {{\left (d+e\,x\right )}^m\,\left (b^2\,d^2\,e^2\,m^2+7\,b^2\,d^2\,e^2\,m+12\,b^2\,d^2\,e^2-6\,b\,c\,d^3\,e\,m-24\,b\,c\,d^3\,e-a\,b\,d\,e^3\,m^3-9\,a\,b\,d\,e^3\,m^2-26\,a\,b\,d\,e^3\,m-24\,a\,b\,d\,e^3+12\,c^2\,d^4+2\,a\,c\,d^2\,e^2\,m^2+14\,a\,c\,d^2\,e^2\,m+24\,a\,c\,d^2\,e^2\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {2\,c^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^2\,e^2\,m^2+7\,b^2\,e^2\,m+12\,b^2\,e^2+3\,b\,c\,d\,e\,m^2+12\,b\,c\,d\,e\,m-6\,c^2\,d^2\,m+2\,a\,c\,e^2\,m^2+14\,a\,c\,e^2\,m+24\,a\,c\,e^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {c\,x^3\,{\left (d+e\,x\right )}^m\,\left (12\,b\,e+3\,b\,e\,m+2\,c\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
[In]
[Out]