3.369 \(\int (d+e x)^m (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx\)

Optimal. Leaf size=292 \[ \frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^{m+1}}{e^7 (m+1)}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^{m+2}}{e^7 (m+2)}-\frac {(120 d+17 e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {20 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

(5*d^2-2*d*e+3*e^2)*(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)*(e*x+d)^(1+m)/e^7/(1+m)-(120*d^5+85*d^4*e+68*d^3*e^2
+12*d^2*e^3+42*d*e^4-7*e^5)*(e*x+d)^(2+m)/e^7/(2+m)+(300*d^4+170*d^3*e+102*d^2*e^2+12*d*e^3+21*e^4)*(e*x+d)^(3
+m)/e^7/(3+m)-2*(200*d^3+85*d^2*e+34*d*e^2+2*e^3)*(e*x+d)^(4+m)/e^7/(4+m)+(300*d^2+85*d*e+17*e^2)*(e*x+d)^(5+m
)/e^7/(5+m)-(120*d+17*e)*(e*x+d)^(6+m)/e^7/(6+m)+20*(e*x+d)^(7+m)/e^7/(7+m)

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Rubi [A]  time = 0.19, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1628} \[ \frac {\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^{m+1}}{e^7 (m+1)}-\frac {\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) (d+e x)^{m+2}}{e^7 (m+2)}+\frac {\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {2 \left (85 d^2 e+200 d^3+34 d e^2+2 e^3\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {(120 d+17 e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {20 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*(d + e*x)^(1 + m))/(e^7*(1 + m)) - ((12
0*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((300*d^4 +
170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - (2*(200*d^3 + 85*d^2*e + 34*d*
e^2 + 2*e^3)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m))
- ((120*d + 17*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (20*(d + e*x)^(7 + m))/(e^7*(7 + m))

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (\frac {\left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) (d+e x)^m}{e^6}+\frac {\left (-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5\right ) (d+e x)^{1+m}}{e^6}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{2+m}}{e^6}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{3+m}}{e^6}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{4+m}}{e^6}+\frac {(-120 d-17 e) (d+e x)^{5+m}}{e^6}+\frac {20 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^{1+m}}{e^7 (1+m)}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {(120 d+17 e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {20 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 261, normalized size = 0.89 \[ \frac {(d+e x)^{m+1} \left (\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^4}{m+5}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^3}{m+4}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^2}{m+3}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{m+1}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)}{m+2}+\frac {20 (d+e x)^6}{m+7}-\frac {(120 d+17 e) (d+e x)^5}{m+6}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

((d + e*x)^(1 + m)*(((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(1 + m) - ((120*d^
5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x))/(2 + m) + ((300*d^4 + 170*d^3*e + 102*d^
2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^2)/(3 + m) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^3)/(4 +
m) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^4)/(5 + m) - ((120*d + 17*e)*(d + e*x)^5)/(6 + m) + (20*(d + e*x)
^6)/(7 + m)))/e^7

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fricas [B]  time = 0.61, size = 1448, normalized size = 4.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

(6*d*e^6*m^6 + 20*(e^7*m^6 + 21*e^7*m^5 + 175*e^7*m^4 + 735*e^7*m^3 + 1624*e^7*m^2 + 1764*e^7*m + 720*e^7)*x^7
+ 14400*d^7 + 14280*d^6*e + 17136*d^5*e^2 + 5040*d^4*e^3 + 35280*d^3*e^4 - 17640*d^2*e^5 + 30240*d*e^6 - (142
80*e^7 - (20*d*e^6 - 17*e^7)*m^6 - 2*(150*d*e^6 - 187*e^7)*m^5 - 170*(10*d*e^6 - 19*e^7)*m^4 - 20*(225*d*e^6 -
697*e^7)*m^3 - (5480*d*e^6 - 31433*e^7)*m^2 - 2*(1200*d*e^6 - 17323*e^7)*m)*x^6 - (7*d^2*e^5 - 162*d*e^6)*m^5
+ (17136*e^7 - 17*(d*e^6 - e^7)*m^6 - (120*d^2*e^5 + 289*d*e^6 - 391*e^7)*m^5 - 3*(400*d^2*e^5 + 595*d*e^6 -
1173*e^7)*m^4 - 5*(840*d^2*e^5 + 1003*d*e^6 - 3145*e^7)*m^3 - 2*(3000*d^2*e^5 + 3179*d*e^6 - 18224*e^7)*m^2 -
12*(240*d^2*e^5 + 238*d*e^6 - 3417*e^7)*m)*x^5 + (42*d^3*e^4 - 175*d^2*e^5 + 1770*d*e^6)*m^4 - (5040*e^7 - (17
*d*e^6 - 4*e^7)*m^6 - (85*d^2*e^5 + 323*d*e^6 - 96*e^7)*m^5 - (600*d^3*e^4 + 1105*d^2*e^5 + 2227*d*e^6 - 904*e
^7)*m^4 - (3600*d^3*e^4 + 4505*d^2*e^5 + 6817*d*e^6 - 4224*e^7)*m^3 - 5*(1320*d^3*e^4 + 1411*d^2*e^5 + 1836*d*
e^6 - 2036*e^7)*m^2 - 6*(600*d^3*e^4 + 595*d^2*e^5 + 714*d*e^6 - 1968*e^7)*m)*x^4 + (24*d^4*e^3 + 924*d^3*e^4
- 1715*d^2*e^5 + 9990*d*e^6)*m^3 + (35280*e^7 - (4*d*e^6 - 21*e^7)*m^6 - (68*d^2*e^5 + 84*d*e^6 - 525*e^7)*m^5
- (340*d^3*e^4 + 1088*d^2*e^5 + 652*d*e^6 - 5187*e^7)*m^4 - (2400*d^4*e^3 + 3400*d^3*e^4 + 5644*d^2*e^5 + 226
8*d*e^6 - 25599*e^7)*m^3 - 4*(1800*d^4*e^3 + 1955*d^3*e^4 + 2584*d^2*e^5 + 844*d*e^6 - 16338*e^7)*m^2 - 4*(120
0*d^4*e^3 + 1190*d^3*e^4 + 1428*d^2*e^5 + 420*d*e^6 - 19929*e^7)*m)*x^3 + (408*d^5*e^2 + 432*d^4*e^3 + 7518*d^
3*e^4 - 8225*d^2*e^5 + 30624*d*e^6)*m^2 + (17640*e^7 + 7*(3*d*e^6 + e^7)*m^6 + (12*d^2*e^5 + 483*d*e^6 + 182*e
^7)*m^5 + 3*(68*d^3*e^4 + 76*d^2*e^5 + 1407*d*e^6 + 630*e^7)*m^4 + (1020*d^4*e^3 + 2856*d^3*e^4 + 1500*d^2*e^5
+ 17157*d*e^6 + 9940*e^7)*m^3 + (7200*d^5*e^2 + 8160*d^4*e^3 + 11220*d^3*e^4 + 3804*d^2*e^5 + 31038*d*e^6 + 2
7503*e^7)*m^2 + 6*(1200*d^5*e^2 + 1190*d^4*e^3 + 1428*d^3*e^4 + 420*d^2*e^5 + 2940*d*e^6 + 6153*e^7)*m)*x^2 +
6*(340*d^6*e + 884*d^5*e^2 + 428*d^4*e^3 + 4466*d^3*e^4 - 3213*d^2*e^5 + 8028*d*e^6)*m + (30240*e^7 + (7*d*e^6
+ 6*e^7)*m^6 - (42*d^2*e^5 - 175*d*e^6 - 162*e^7)*m^5 - (24*d^3*e^4 + 924*d^2*e^5 - 1715*d*e^6 - 1770*e^7)*m^
4 - (408*d^4*e^3 + 432*d^3*e^4 + 7518*d^2*e^5 - 8225*d*e^6 - 9990*e^7)*m^3 - 6*(340*d^5*e^2 + 884*d^4*e^3 + 42
8*d^3*e^4 + 4466*d^2*e^5 - 3213*d*e^6 - 5104*e^7)*m^2 - 24*(600*d^6*e + 595*d^5*e^2 + 714*d^4*e^3 + 210*d^3*e^
4 + 1470*d^2*e^5 - 735*d*e^6 - 2007*e^7)*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4
+ 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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giac [B]  time = 0.27, size = 3098, normalized size = 10.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

(20*(x*e + d)^m*m^6*x^7*e^7 + 20*(x*e + d)^m*d*m^6*x^6*e^6 - 17*(x*e + d)^m*m^6*x^6*e^7 + 420*(x*e + d)^m*m^5*
x^7*e^7 - 17*(x*e + d)^m*d*m^6*x^5*e^6 + 300*(x*e + d)^m*d*m^5*x^6*e^6 - 120*(x*e + d)^m*d^2*m^5*x^5*e^5 + 17*
(x*e + d)^m*m^6*x^5*e^7 - 374*(x*e + d)^m*m^5*x^6*e^7 + 3500*(x*e + d)^m*m^4*x^7*e^7 + 17*(x*e + d)^m*d*m^6*x^
4*e^6 - 289*(x*e + d)^m*d*m^5*x^5*e^6 + 1700*(x*e + d)^m*d*m^4*x^6*e^6 + 85*(x*e + d)^m*d^2*m^5*x^4*e^5 - 1200
*(x*e + d)^m*d^2*m^4*x^5*e^5 + 600*(x*e + d)^m*d^3*m^4*x^4*e^4 - 4*(x*e + d)^m*m^6*x^4*e^7 + 391*(x*e + d)^m*m
^5*x^5*e^7 - 3230*(x*e + d)^m*m^4*x^6*e^7 + 14700*(x*e + d)^m*m^3*x^7*e^7 - 4*(x*e + d)^m*d*m^6*x^3*e^6 + 323*
(x*e + d)^m*d*m^5*x^4*e^6 - 1785*(x*e + d)^m*d*m^4*x^5*e^6 + 4500*(x*e + d)^m*d*m^3*x^6*e^6 - 68*(x*e + d)^m*d
^2*m^5*x^3*e^5 + 1105*(x*e + d)^m*d^2*m^4*x^4*e^5 - 4200*(x*e + d)^m*d^2*m^3*x^5*e^5 - 340*(x*e + d)^m*d^3*m^4
*x^3*e^4 + 3600*(x*e + d)^m*d^3*m^3*x^4*e^4 - 2400*(x*e + d)^m*d^4*m^3*x^3*e^3 + 21*(x*e + d)^m*m^6*x^3*e^7 -
96*(x*e + d)^m*m^5*x^4*e^7 + 3519*(x*e + d)^m*m^4*x^5*e^7 - 13940*(x*e + d)^m*m^3*x^6*e^7 + 32480*(x*e + d)^m*
m^2*x^7*e^7 + 21*(x*e + d)^m*d*m^6*x^2*e^6 - 84*(x*e + d)^m*d*m^5*x^3*e^6 + 2227*(x*e + d)^m*d*m^4*x^4*e^6 - 5
015*(x*e + d)^m*d*m^3*x^5*e^6 + 5480*(x*e + d)^m*d*m^2*x^6*e^6 + 12*(x*e + d)^m*d^2*m^5*x^2*e^5 - 1088*(x*e +
d)^m*d^2*m^4*x^3*e^5 + 4505*(x*e + d)^m*d^2*m^3*x^4*e^5 - 6000*(x*e + d)^m*d^2*m^2*x^5*e^5 + 204*(x*e + d)^m*d
^3*m^4*x^2*e^4 - 3400*(x*e + d)^m*d^3*m^3*x^3*e^4 + 6600*(x*e + d)^m*d^3*m^2*x^4*e^4 + 1020*(x*e + d)^m*d^4*m^
3*x^2*e^3 - 7200*(x*e + d)^m*d^4*m^2*x^3*e^3 + 7200*(x*e + d)^m*d^5*m^2*x^2*e^2 + 7*(x*e + d)^m*m^6*x^2*e^7 +
525*(x*e + d)^m*m^5*x^3*e^7 - 904*(x*e + d)^m*m^4*x^4*e^7 + 15725*(x*e + d)^m*m^3*x^5*e^7 - 31433*(x*e + d)^m*
m^2*x^6*e^7 + 35280*(x*e + d)^m*m*x^7*e^7 + 7*(x*e + d)^m*d*m^6*x*e^6 + 483*(x*e + d)^m*d*m^5*x^2*e^6 - 652*(x
*e + d)^m*d*m^4*x^3*e^6 + 6817*(x*e + d)^m*d*m^3*x^4*e^6 - 6358*(x*e + d)^m*d*m^2*x^5*e^6 + 2400*(x*e + d)^m*d
*m*x^6*e^6 - 42*(x*e + d)^m*d^2*m^5*x*e^5 + 228*(x*e + d)^m*d^2*m^4*x^2*e^5 - 5644*(x*e + d)^m*d^2*m^3*x^3*e^5
+ 7055*(x*e + d)^m*d^2*m^2*x^4*e^5 - 2880*(x*e + d)^m*d^2*m*x^5*e^5 - 24*(x*e + d)^m*d^3*m^4*x*e^4 + 2856*(x*
e + d)^m*d^3*m^3*x^2*e^4 - 7820*(x*e + d)^m*d^3*m^2*x^3*e^4 + 3600*(x*e + d)^m*d^3*m*x^4*e^4 - 408*(x*e + d)^m
*d^4*m^3*x*e^3 + 8160*(x*e + d)^m*d^4*m^2*x^2*e^3 - 4800*(x*e + d)^m*d^4*m*x^3*e^3 - 2040*(x*e + d)^m*d^5*m^2*
x*e^2 + 7200*(x*e + d)^m*d^5*m*x^2*e^2 - 14400*(x*e + d)^m*d^6*m*x*e + 6*(x*e + d)^m*m^6*x*e^7 + 182*(x*e + d)
^m*m^5*x^2*e^7 + 5187*(x*e + d)^m*m^4*x^3*e^7 - 4224*(x*e + d)^m*m^3*x^4*e^7 + 36448*(x*e + d)^m*m^2*x^5*e^7 -
34646*(x*e + d)^m*m*x^6*e^7 + 14400*(x*e + d)^m*x^7*e^7 + 6*(x*e + d)^m*d*m^6*e^6 + 175*(x*e + d)^m*d*m^5*x*e
^6 + 4221*(x*e + d)^m*d*m^4*x^2*e^6 - 2268*(x*e + d)^m*d*m^3*x^3*e^6 + 9180*(x*e + d)^m*d*m^2*x^4*e^6 - 2856*(
x*e + d)^m*d*m*x^5*e^6 - 7*(x*e + d)^m*d^2*m^5*e^5 - 924*(x*e + d)^m*d^2*m^4*x*e^5 + 1500*(x*e + d)^m*d^2*m^3*
x^2*e^5 - 10336*(x*e + d)^m*d^2*m^2*x^3*e^5 + 3570*(x*e + d)^m*d^2*m*x^4*e^5 + 42*(x*e + d)^m*d^3*m^4*e^4 - 43
2*(x*e + d)^m*d^3*m^3*x*e^4 + 11220*(x*e + d)^m*d^3*m^2*x^2*e^4 - 4760*(x*e + d)^m*d^3*m*x^3*e^4 + 24*(x*e + d
)^m*d^4*m^3*e^3 - 5304*(x*e + d)^m*d^4*m^2*x*e^3 + 7140*(x*e + d)^m*d^4*m*x^2*e^3 + 408*(x*e + d)^m*d^5*m^2*e^
2 - 14280*(x*e + d)^m*d^5*m*x*e^2 + 2040*(x*e + d)^m*d^6*m*e + 14400*(x*e + d)^m*d^7 + 162*(x*e + d)^m*m^5*x*e
^7 + 1890*(x*e + d)^m*m^4*x^2*e^7 + 25599*(x*e + d)^m*m^3*x^3*e^7 - 10180*(x*e + d)^m*m^2*x^4*e^7 + 41004*(x*e
+ d)^m*m*x^5*e^7 - 14280*(x*e + d)^m*x^6*e^7 + 162*(x*e + d)^m*d*m^5*e^6 + 1715*(x*e + d)^m*d*m^4*x*e^6 + 171
57*(x*e + d)^m*d*m^3*x^2*e^6 - 3376*(x*e + d)^m*d*m^2*x^3*e^6 + 4284*(x*e + d)^m*d*m*x^4*e^6 - 175*(x*e + d)^m
*d^2*m^4*e^5 - 7518*(x*e + d)^m*d^2*m^3*x*e^5 + 3804*(x*e + d)^m*d^2*m^2*x^2*e^5 - 5712*(x*e + d)^m*d^2*m*x^3*
e^5 + 924*(x*e + d)^m*d^3*m^3*e^4 - 2568*(x*e + d)^m*d^3*m^2*x*e^4 + 8568*(x*e + d)^m*d^3*m*x^2*e^4 + 432*(x*e
+ d)^m*d^4*m^2*e^3 - 17136*(x*e + d)^m*d^4*m*x*e^3 + 5304*(x*e + d)^m*d^5*m*e^2 + 14280*(x*e + d)^m*d^6*e + 1
770*(x*e + d)^m*m^4*x*e^7 + 9940*(x*e + d)^m*m^3*x^2*e^7 + 65352*(x*e + d)^m*m^2*x^3*e^7 - 11808*(x*e + d)^m*m
*x^4*e^7 + 17136*(x*e + d)^m*x^5*e^7 + 1770*(x*e + d)^m*d*m^4*e^6 + 8225*(x*e + d)^m*d*m^3*x*e^6 + 31038*(x*e
+ d)^m*d*m^2*x^2*e^6 - 1680*(x*e + d)^m*d*m*x^3*e^6 - 1715*(x*e + d)^m*d^2*m^3*e^5 - 26796*(x*e + d)^m*d^2*m^2
*x*e^5 + 2520*(x*e + d)^m*d^2*m*x^2*e^5 + 7518*(x*e + d)^m*d^3*m^2*e^4 - 5040*(x*e + d)^m*d^3*m*x*e^4 + 2568*(
x*e + d)^m*d^4*m*e^3 + 17136*(x*e + d)^m*d^5*e^2 + 9990*(x*e + d)^m*m^3*x*e^7 + 27503*(x*e + d)^m*m^2*x^2*e^7
+ 79716*(x*e + d)^m*m*x^3*e^7 - 5040*(x*e + d)^m*x^4*e^7 + 9990*(x*e + d)^m*d*m^3*e^6 + 19278*(x*e + d)^m*d*m^
2*x*e^6 + 17640*(x*e + d)^m*d*m*x^2*e^6 - 8225*(x*e + d)^m*d^2*m^2*e^5 - 35280*(x*e + d)^m*d^2*m*x*e^5 + 26796
*(x*e + d)^m*d^3*m*e^4 + 5040*(x*e + d)^m*d^4*e^3 + 30624*(x*e + d)^m*m^2*x*e^7 + 36918*(x*e + d)^m*m*x^2*e^7
+ 35280*(x*e + d)^m*x^3*e^7 + 30624*(x*e + d)^m*d*m^2*e^6 + 17640*(x*e + d)^m*d*m*x*e^6 - 19278*(x*e + d)^m*d^
2*m*e^5 + 35280*(x*e + d)^m*d^3*e^4 + 48168*(x*e + d)^m*m*x*e^7 + 17640*(x*e + d)^m*x^2*e^7 + 48168*(x*e + d)^
m*d*m*e^6 - 17640*(x*e + d)^m*d^2*e^5 + 30240*(x*e + d)^m*x*e^7 + 30240*(x*e + d)^m*d*e^6)/(m^7*e^7 + 28*m^6*e
^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e^7 + 13068*m*e^7 + 5040*e^7)

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maple [B]  time = 0.02, size = 1504, normalized size = 5.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x)

[Out]

(e*x+d)^(1+m)*(20*e^6*m^6*x^6-17*e^6*m^6*x^5+420*e^6*m^5*x^6-120*d*e^5*m^5*x^5+17*e^6*m^6*x^4-374*e^6*m^5*x^5+
3500*e^6*m^4*x^6+85*d*e^5*m^5*x^4-1800*d*e^5*m^4*x^5-4*e^6*m^6*x^3+391*e^6*m^5*x^4-3230*e^6*m^4*x^5+14700*e^6*
m^3*x^6+600*d^2*e^4*m^4*x^4-68*d*e^5*m^5*x^3+1445*d*e^5*m^4*x^4-10200*d*e^5*m^3*x^5+21*e^6*m^6*x^2-96*e^6*m^5*
x^3+3519*e^6*m^4*x^4-13940*e^6*m^3*x^5+32480*e^6*m^2*x^6-340*d^2*e^4*m^4*x^3+6000*d^2*e^4*m^3*x^4+12*d*e^5*m^5
*x^2-1292*d*e^5*m^4*x^3+8925*d*e^5*m^3*x^4-27000*d*e^5*m^2*x^5+7*e^6*m^6*x+525*e^6*m^5*x^2-904*e^6*m^4*x^3+157
25*e^6*m^3*x^4-31433*e^6*m^2*x^5+35280*e^6*m*x^6-2400*d^3*e^3*m^3*x^3+204*d^2*e^4*m^4*x^2-4420*d^2*e^4*m^3*x^3
+21000*d^2*e^4*m^2*x^4-42*d*e^5*m^5*x+252*d*e^5*m^4*x^2-8908*d*e^5*m^3*x^3+25075*d*e^5*m^2*x^4-32880*d*e^5*m*x
^5+6*e^6*m^6+182*e^6*m^5*x+5187*e^6*m^4*x^2-4224*e^6*m^3*x^3+36448*e^6*m^2*x^4-34646*e^6*m*x^5+14400*e^6*x^6+1
020*d^3*e^3*m^3*x^2-14400*d^3*e^3*m^2*x^3-24*d^2*e^4*m^4*x+3264*d^2*e^4*m^3*x^2-18020*d^2*e^4*m^2*x^3+30000*d^
2*e^4*m*x^4-7*d*e^5*m^5-966*d*e^5*m^4*x+1956*d*e^5*m^3*x^2-27268*d*e^5*m^2*x^3+31790*d*e^5*m*x^4-14400*d*e^5*x
^5+162*e^6*m^5+1890*e^6*m^4*x+25599*e^6*m^3*x^2-10180*e^6*m^2*x^3+41004*e^6*m*x^4-14280*e^6*x^5+7200*d^4*e^2*m
^2*x^2-408*d^3*e^3*m^3*x+10200*d^3*e^3*m^2*x^2-26400*d^3*e^3*m*x^3+42*d^2*e^4*m^4-456*d^2*e^4*m^3*x+16932*d^2*
e^4*m^2*x^2-28220*d^2*e^4*m*x^3+14400*d^2*e^4*x^4-175*d*e^5*m^4-8442*d*e^5*m^3*x+6804*d*e^5*m^2*x^2-36720*d*e^
5*m*x^3+14280*d*e^5*x^4+1770*e^6*m^4+9940*e^6*m^3*x+65352*e^6*m^2*x^2-11808*e^6*m*x^3+17136*e^6*x^4-2040*d^4*e
^2*m^2*x+21600*d^4*e^2*m*x^2+24*d^3*e^3*m^3-5712*d^3*e^3*m^2*x+23460*d^3*e^3*m*x^2-14400*d^3*e^3*x^3+924*d^2*e
^4*m^3-3000*d^2*e^4*m^2*x+31008*d^2*e^4*m*x^2-14280*d^2*e^4*x^3-1715*d*e^5*m^3-34314*d*e^5*m^2*x+10128*d*e^5*m
*x^2-17136*d*e^5*x^3+9990*e^6*m^3+27503*e^6*m^2*x+79716*e^6*m*x^2-5040*e^6*x^3-14400*d^5*e*m*x+408*d^4*e^2*m^2
-16320*d^4*e^2*m*x+14400*d^4*e^2*x^2+432*d^3*e^3*m^2-22440*d^3*e^3*m*x+14280*d^3*e^3*x^2+7518*d^2*e^4*m^2-7608
*d^2*e^4*m*x+17136*d^2*e^4*x^2-8225*d*e^5*m^2-62076*d*e^5*m*x+5040*d*e^5*x^2+30624*e^6*m^2+36918*e^6*m*x+35280
*e^6*x^2+2040*d^5*e*m-14400*d^5*e*x+5304*d^4*e^2*m-14280*d^4*e^2*x+2568*d^3*e^3*m-17136*d^3*e^3*x+26796*d^2*e^
4*m-5040*d^2*e^4*x-19278*d*e^5*m-35280*d*e^5*x+48168*e^6*m+17640*e^6*x+14400*d^6+14280*d^5*e+17136*d^4*e^2+504
0*d^3*e^3+35280*d^2*e^4-17640*d*e^5+30240*e^6)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+504
0)

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maxima [B]  time = 0.55, size = 788, normalized size = 2.70 \[ \frac {7 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {6 \, {\left (e x + d\right )}^{m + 1}}{e {\left (m + 1\right )}} + \frac {21 \, {\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}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} - \frac {4 \, {\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}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {17 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} - \frac {17 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} + \frac {20 \, {\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="maxima")

[Out]

7*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m/((m^2 + 3*m + 2)*e^2) + 6*(e*x + d)^(m + 1)/(e*(m + 1)) + 21*(
(m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m/((m^3 + 6*m^2 + 11*m + 6)*e^3
) - 4*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*
x - 6*d^4)*(e*x + d)^m/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 17*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5
*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 -
24*d^4*e*m*x + 24*d^5)*(e*x + d)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) - 17*((m^5 + 15*m^4
+ 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3
+ 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x -
120*d^6)*(e*x + d)^m/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + 20*((m^6 + 21*m^5 +
175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*
e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4
- 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m/((m^7
+ 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)

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mupad [B]  time = 5.09, size = 1425, normalized size = 4.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m*(2*x + 5*x^2 + 3)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2),x)

[Out]

((d + e*x)^m*(30240*d*e^6 + 14280*d^6*e + 14400*d^7 - 17640*d^2*e^5 + 35280*d^3*e^4 + 5040*d^4*e^3 + 17136*d^5
*e^2 - 19278*d^2*e^5*m + 26796*d^3*e^4*m + 2568*d^4*e^3*m + 5304*d^5*e^2*m + 30624*d*e^6*m^2 + 9990*d*e^6*m^3
+ 1770*d*e^6*m^4 + 162*d*e^6*m^5 + 6*d*e^6*m^6 - 8225*d^2*e^5*m^2 + 7518*d^3*e^4*m^2 + 432*d^4*e^3*m^2 + 408*d
^5*e^2*m^2 - 1715*d^2*e^5*m^3 + 924*d^3*e^4*m^3 + 24*d^4*e^3*m^3 - 175*d^2*e^5*m^4 + 42*d^3*e^4*m^4 - 7*d^2*e^
5*m^5 + 48168*d*e^6*m + 2040*d^6*e*m))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^
7 + 5040)) + (20*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 1313
2*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (x*(d + e*x)^m*(35280*d^2*e^5*m - 30240*e^7 - 3
0624*e^7*m^2 - 9990*e^7*m^3 - 1770*e^7*m^4 - 162*e^7*m^5 - 6*e^7*m^6 - 48168*e^7*m + 5040*d^3*e^4*m + 17136*d^
4*e^3*m + 14280*d^5*e^2*m - 19278*d*e^6*m^2 - 8225*d*e^6*m^3 - 1715*d*e^6*m^4 - 175*d*e^6*m^5 - 7*d*e^6*m^6 +
26796*d^2*e^5*m^2 + 2568*d^3*e^4*m^2 + 5304*d^4*e^3*m^2 + 2040*d^5*e^2*m^2 + 7518*d^2*e^5*m^3 + 432*d^3*e^4*m^
3 + 408*d^4*e^3*m^3 + 924*d^2*e^5*m^4 + 24*d^3*e^4*m^4 + 42*d^2*e^5*m^5 - 17640*d*e^6*m + 14400*d^6*e*m))/(e^7
*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (x^3*(d + e*x)^m*(3*m + m^2 +
2)*(2400*d^4*m - 13398*e^4*m - 17640*e^4 - 3759*e^4*m^2 - 462*e^4*m^3 - 21*e^4*m^4 + 2856*d^2*e^2*m + 428*d*e^
3*m^2 + 340*d^3*e*m^2 + 72*d*e^3*m^3 + 4*d*e^3*m^4 + 884*d^2*e^2*m^2 + 68*d^2*e^2*m^3 + 840*d*e^3*m + 2380*d^3
*e*m))/(e^4*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (x^5*(d + e*x)^m*(5
0*m + 35*m^2 + 10*m^3 + m^4 + 24)*(120*d^2*m - 221*e^2*m - 714*e^2 - 17*e^2*m^2 + 119*d*e*m + 17*d*e*m^2))/(e^
2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x^4*(d + e*x)^m*(11*m + 6*m^
2 + m^3 + 6)*(600*d^3*m - 428*e^3*m - 840*e^3 - 72*e^3*m^2 - 4*e^3*m^3 + 221*d*e^2*m^2 + 85*d^2*e*m^2 + 17*d*e
^2*m^3 + 714*d*e^2*m + 595*d^2*e*m))/(e^3*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7
+ 5040)) + (x^2*(m + 1)*(d + e*x)^m*(7200*d^5*m + 19278*e^5*m + 17640*e^5 + 8225*e^5*m^2 + 1715*e^5*m^3 + 175*
e^5*m^4 + 7*e^5*m^5 + 2520*d^2*e^3*m + 8568*d^3*e^2*m + 13398*d*e^4*m^2 + 1020*d^4*e*m^2 + 3759*d*e^4*m^3 + 46
2*d*e^4*m^4 + 21*d*e^4*m^5 + 1284*d^2*e^3*m^2 + 2652*d^3*e^2*m^2 + 216*d^2*e^3*m^3 + 204*d^3*e^2*m^3 + 12*d^2*
e^3*m^4 + 17640*d*e^4*m + 7140*d^4*e*m))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 +
m^7 + 5040)) - (x^6*(d + e*x)^m*(119*e - 20*d*m + 17*e*m)*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(e*
(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(5*x**2+2*x+3)*(4*x**4-5*x**3+3*x**2+x+2),x)

[Out]

Timed out

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