3.3.89 \(\int (d+e x)^3 (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx\) [289]

Optimal. Leaf size=258 \[ \frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac {(120 d+17 e) (d+e x)^9}{9 e^7}+\frac {2 (d+e x)^{10}}{e^7} \]

[Out]

1/4*(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)^4/e^7-1/5*(120*d^5+85*d^4*e+68*d^3*e^2+1
2*d^2*e^3+42*d*e^4-7*e^5)*(e*x+d)^5/e^7+1/6*(300*d^4+170*d^3*e+102*d^2*e^2+12*d*e^3+21*e^4)*(e*x+d)^6/e^7-2/7*
(200*d^3+85*d^2*e+34*d*e^2+2*e^3)*(e*x+d)^7/e^7+1/8*(300*d^2+85*d*e+17*e^2)*(e*x+d)^8/e^7-1/9*(120*d+17*e)*(e*
x+d)^9/e^7+2*(e*x+d)^10/e^7

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Rubi [A]
time = 0.16, antiderivative size = 258, 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 = {1642} \begin {gather*} \frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\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)^6}{6 e^7}+\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)^4}{4 e^7}-\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)^5}{5 e^7}+\frac {2 (d+e x)^{10}}{e^7}-\frac {(120 d+17 e) (d+e x)^9}{9 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(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)^4)/(4*e^7) - ((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)^5)/(5*e^7) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2
+ 12*d*e^3 + 21*e^4)*(d + e*x)^6)/(6*e^7) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^7)/(7*e^7) +
((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^8)/(8*e^7) - ((120*d + 17*e)*(d + e*x)^9)/(9*e^7) + (2*(d + e*x)^10)/e^
7

Rule 1642

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)^3 \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)^3}{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)^4}{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)^5}{e^6}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^6}{e^6}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^7}{e^6}+\frac {(-120 d-17 e) (d+e x)^8}{e^6}+\frac {20 (d+e x)^9}{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)^4}{4 e^7}-\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)^5}{5 e^7}+\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)^6}{6 e^7}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac {(120 d+17 e) (d+e x)^9}{9 e^7}+\frac {2 (d+e x)^{10}}{e^7}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 212, normalized size = 0.82 \begin {gather*} 6 d^3 x+\frac {1}{2} d^2 (7 d+18 e) x^2+d \left (7 d^2+7 d e+6 e^2\right ) x^3+\frac {1}{4} \left (-4 d^3+63 d^2 e+21 d e^2+6 e^3\right ) x^4+\frac {1}{5} \left (17 d^3-12 d^2 e+63 d e^2+7 e^3\right ) x^5+\frac {1}{6} \left (-17 d^3+51 d^2 e-12 d e^2+21 e^3\right ) x^6+\frac {1}{7} \left (20 d^3-51 d^2 e+51 d e^2-4 e^3\right ) x^7+\frac {1}{8} e \left (60 d^2-51 d e+17 e^2\right ) x^8+\frac {1}{9} (60 d-17 e) e^2 x^9+2 e^3 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

6*d^3*x + (d^2*(7*d + 18*e)*x^2)/2 + d*(7*d^2 + 7*d*e + 6*e^2)*x^3 + ((-4*d^3 + 63*d^2*e + 21*d*e^2 + 6*e^3)*x
^4)/4 + ((17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e^3)*x^5)/5 + ((-17*d^3 + 51*d^2*e - 12*d*e^2 + 21*e^3)*x^6)/6 + ((
20*d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7)/7 + (e*(60*d^2 - 51*d*e + 17*e^2)*x^8)/8 + ((60*d - 17*e)*e^2*x^9)/
9 + 2*e^3*x^10

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Maple [A]
time = 0.12, size = 208, normalized size = 0.81

method	result	size
norman	\(2 e^{3} x^{10}+\left (\frac {20}{3} d \,e^{2}-\frac {17}{9} e^{3}\right ) x^{9}+\left (\frac {15}{2} d^{2} e -\frac {51}{8} d \,e^{2}+\frac {17}{8} e^{3}\right ) x^{8}+\left (\frac {20}{7} d^{3}-\frac {51}{7} d^{2} e +\frac {51}{7} d \,e^{2}-\frac {4}{7} e^{3}\right ) x^{7}+\left (-\frac {17}{6} d^{3}+\frac {17}{2} d^{2} e -2 d \,e^{2}+\frac {7}{2} e^{3}\right ) x^{6}+\left (\frac {17}{5} d^{3}-\frac {12}{5} d^{2} e +\frac {63}{5} d \,e^{2}+\frac {7}{5} e^{3}\right ) x^{5}+\left (-d^{3}+\frac {63}{4} d^{2} e +\frac {21}{4} d \,e^{2}+\frac {3}{2} e^{3}\right ) x^{4}+\left (7 d^{3}+7 d^{2} e +6 d \,e^{2}\right ) x^{3}+\left (\frac {7}{2} d^{3}+9 d^{2} e \right ) x^{2}+6 d^{3} x\)	\(200\)
default	\(2 e^{3} x^{10}+\frac {\left (60 d \,e^{2}-17 e^{3}\right ) x^{9}}{9}+\frac {\left (60 d^{2} e -51 d \,e^{2}+17 e^{3}\right ) x^{8}}{8}+\frac {\left (20 d^{3}-51 d^{2} e +51 d \,e^{2}-4 e^{3}\right ) x^{7}}{7}+\frac {\left (-17 d^{3}+51 d^{2} e -12 d \,e^{2}+21 e^{3}\right ) x^{6}}{6}+\frac {\left (17 d^{3}-12 d^{2} e +63 d \,e^{2}+7 e^{3}\right ) x^{5}}{5}+\frac {\left (-4 d^{3}+63 d^{2} e +21 d \,e^{2}+6 e^{3}\right ) x^{4}}{4}+\frac {\left (21 d^{3}+21 d^{2} e +18 d \,e^{2}\right ) x^{3}}{3}+\frac {\left (7 d^{3}+18 d^{2} e \right ) x^{2}}{2}+6 d^{3} x\)	\(208\)
gosper	\(-\frac {51}{7} x^{7} d^{2} e +\frac {51}{7} x^{7} d \,e^{2}+\frac {17}{2} x^{6} d^{2} e -2 x^{6} d \,e^{2}-\frac {12}{5} x^{5} d^{2} e +\frac {63}{5} x^{5} d \,e^{2}+\frac {63}{4} x^{4} d^{2} e +\frac {21}{4} x^{4} d \,e^{2}+7 d^{2} e \,x^{3}+\frac {15}{2} x^{8} d^{2} e -\frac {51}{8} x^{8} d \,e^{2}+\frac {20}{3} x^{9} d \,e^{2}+6 d^{3} x +\frac {3}{2} e^{3} x^{4}-\frac {17}{6} x^{6} d^{3}+\frac {17}{5} x^{5} d^{3}+\frac {7}{5} x^{5} e^{3}-x^{4} d^{3}+\frac {7}{2} x^{2} d^{3}+7 d^{3} x^{3}+2 e^{3} x^{10}-\frac {17}{9} x^{9} e^{3}+\frac {17}{8} x^{8} e^{3}+\frac {20}{7} x^{7} d^{3}-\frac {4}{7} x^{7} e^{3}+6 d \,e^{2} x^{3}+9 d^{2} e \,x^{2}+\frac {7}{2} e^{3} x^{6}\)	\(238\)
risch	\(-\frac {51}{7} x^{7} d^{2} e +\frac {51}{7} x^{7} d \,e^{2}+\frac {17}{2} x^{6} d^{2} e -2 x^{6} d \,e^{2}-\frac {12}{5} x^{5} d^{2} e +\frac {63}{5} x^{5} d \,e^{2}+\frac {63}{4} x^{4} d^{2} e +\frac {21}{4} x^{4} d \,e^{2}+7 d^{2} e \,x^{3}+\frac {15}{2} x^{8} d^{2} e -\frac {51}{8} x^{8} d \,e^{2}+\frac {20}{3} x^{9} d \,e^{2}+6 d^{3} x +\frac {3}{2} e^{3} x^{4}-\frac {17}{6} x^{6} d^{3}+\frac {17}{5} x^{5} d^{3}+\frac {7}{5} x^{5} e^{3}-x^{4} d^{3}+\frac {7}{2} x^{2} d^{3}+7 d^{3} x^{3}+2 e^{3} x^{10}-\frac {17}{9} x^{9} e^{3}+\frac {17}{8} x^{8} e^{3}+\frac {20}{7} x^{7} d^{3}-\frac {4}{7} x^{7} e^{3}+6 d \,e^{2} x^{3}+9 d^{2} e \,x^{2}+\frac {7}{2} e^{3} x^{6}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x,method=_RETURNVERBOSE)

[Out]

2*e^3*x^10+1/9*(60*d*e^2-17*e^3)*x^9+1/8*(60*d^2*e-51*d*e^2+17*e^3)*x^8+1/7*(20*d^3-51*d^2*e+51*d*e^2-4*e^3)*x
^7+1/6*(-17*d^3+51*d^2*e-12*d*e^2+21*e^3)*x^6+1/5*(17*d^3-12*d^2*e+63*d*e^2+7*e^3)*x^5+1/4*(-4*d^3+63*d^2*e+21
*d*e^2+6*e^3)*x^4+1/3*(21*d^3+21*d^2*e+18*d*e^2)*x^3+1/2*(7*d^3+18*d^2*e)*x^2+6*d^3*x

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Maxima [A]
time = 0.34, size = 199, normalized size = 0.77 \begin {gather*} 2 \, x^{10} e^{3} + \frac {1}{9} \, {\left (60 \, d e^{2} - 17 \, e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (60 \, d^{2} e - 51 \, d e^{2} + 17 \, e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{3} - 51 \, d^{2} e + 51 \, d e^{2} - 4 \, e^{3}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{3} - 51 \, d^{2} e + 12 \, d e^{2} - 21 \, e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{3} - 12 \, d^{2} e + 63 \, d e^{2} + 7 \, e^{3}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{3} - 63 \, d^{2} e - 21 \, d e^{2} - 6 \, e^{3}\right )} x^{4} + 6 \, d^{3} x + {\left (7 \, d^{3} + 7 \, d^{2} e + 6 \, d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^10*e^3 + 1/9*(60*d*e^2 - 17*e^3)*x^9 + 1/8*(60*d^2*e - 51*d*e^2 + 17*e^3)*x^8 + 1/7*(20*d^3 - 51*d^2*e + 5
1*d*e^2 - 4*e^3)*x^7 - 1/6*(17*d^3 - 51*d^2*e + 12*d*e^2 - 21*e^3)*x^6 + 1/5*(17*d^3 - 12*d^2*e + 63*d*e^2 + 7
*e^3)*x^5 - 1/4*(4*d^3 - 63*d^2*e - 21*d*e^2 - 6*e^3)*x^4 + 6*d^3*x + (7*d^3 + 7*d^2*e + 6*d*e^2)*x^3 + 1/2*(7
*d^3 + 18*d^2*e)*x^2

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Fricas [A]
time = 0.33, size = 203, normalized size = 0.79 \begin {gather*} \frac {20}{7} \, d^{3} x^{7} - \frac {17}{6} \, d^{3} x^{6} + \frac {17}{5} \, d^{3} x^{5} - d^{3} x^{4} + 7 \, d^{3} x^{3} + \frac {7}{2} \, d^{3} x^{2} + 6 \, d^{3} x + \frac {1}{2520} \, {\left (5040 \, x^{10} - 4760 \, x^{9} + 5355 \, x^{8} - 1440 \, x^{7} + 8820 \, x^{6} + 3528 \, x^{5} + 3780 \, x^{4}\right )} e^{3} + \frac {1}{840} \, {\left (5600 \, d x^{9} - 5355 \, d x^{8} + 6120 \, d x^{7} - 1680 \, d x^{6} + 10584 \, d x^{5} + 4410 \, d x^{4} + 5040 \, d x^{3}\right )} e^{2} + \frac {1}{140} \, {\left (1050 \, d^{2} x^{8} - 1020 \, d^{2} x^{7} + 1190 \, d^{2} x^{6} - 336 \, d^{2} x^{5} + 2205 \, d^{2} x^{4} + 980 \, d^{2} x^{3} + 1260 \, d^{2} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*d^3*x^7 - 17/6*d^3*x^6 + 17/5*d^3*x^5 - d^3*x^4 + 7*d^3*x^3 + 7/2*d^3*x^2 + 6*d^3*x + 1/2520*(5040*x^10 -
4760*x^9 + 5355*x^8 - 1440*x^7 + 8820*x^6 + 3528*x^5 + 3780*x^4)*e^3 + 1/840*(5600*d*x^9 - 5355*d*x^8 + 6120*
d*x^7 - 1680*d*x^6 + 10584*d*x^5 + 4410*d*x^4 + 5040*d*x^3)*e^2 + 1/140*(1050*d^2*x^8 - 1020*d^2*x^7 + 1190*d^
2*x^6 - 336*d^2*x^5 + 2205*d^2*x^4 + 980*d^2*x^3 + 1260*d^2*x^2)*e

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Sympy [A]
time = 0.03, size = 230, normalized size = 0.89 \begin {gather*} 6 d^{3} x + 2 e^{3} x^{10} + x^{9} \cdot \left (\frac {20 d e^{2}}{3} - \frac {17 e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {15 d^{2} e}{2} - \frac {51 d e^{2}}{8} + \frac {17 e^{3}}{8}\right ) + x^{7} \cdot \left (\frac {20 d^{3}}{7} - \frac {51 d^{2} e}{7} + \frac {51 d e^{2}}{7} - \frac {4 e^{3}}{7}\right ) + x^{6} \left (- \frac {17 d^{3}}{6} + \frac {17 d^{2} e}{2} - 2 d e^{2} + \frac {7 e^{3}}{2}\right ) + x^{5} \cdot \left (\frac {17 d^{3}}{5} - \frac {12 d^{2} e}{5} + \frac {63 d e^{2}}{5} + \frac {7 e^{3}}{5}\right ) + x^{4} \left (- d^{3} + \frac {63 d^{2} e}{4} + \frac {21 d e^{2}}{4} + \frac {3 e^{3}}{2}\right ) + x^{3} \cdot \left (7 d^{3} + 7 d^{2} e + 6 d e^{2}\right ) + x^{2} \cdot \left (\frac {7 d^{3}}{2} + 9 d^{2} e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*d**3*x + 2*e**3*x**10 + x**9*(20*d*e**2/3 - 17*e**3/9) + x**8*(15*d**2*e/2 - 51*d*e**2/8 + 17*e**3/8) + x**7
*(20*d**3/7 - 51*d**2*e/7 + 51*d*e**2/7 - 4*e**3/7) + x**6*(-17*d**3/6 + 17*d**2*e/2 - 2*d*e**2 + 7*e**3/2) +
x**5*(17*d**3/5 - 12*d**2*e/5 + 63*d*e**2/5 + 7*e**3/5) + x**4*(-d**3 + 63*d**2*e/4 + 21*d*e**2/4 + 3*e**3/2)
+ x**3*(7*d**3 + 7*d**2*e + 6*d*e**2) + x**2*(7*d**3/2 + 9*d**2*e)

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Giac [A]
time = 4.69, size = 230, normalized size = 0.89 \begin {gather*} 2 \, x^{10} e^{3} + \frac {20}{3} \, d x^{9} e^{2} + \frac {15}{2} \, d^{2} x^{8} e + \frac {20}{7} \, d^{3} x^{7} - \frac {17}{9} \, x^{9} e^{3} - \frac {51}{8} \, d x^{8} e^{2} - \frac {51}{7} \, d^{2} x^{7} e - \frac {17}{6} \, d^{3} x^{6} + \frac {17}{8} \, x^{8} e^{3} + \frac {51}{7} \, d x^{7} e^{2} + \frac {17}{2} \, d^{2} x^{6} e + \frac {17}{5} \, d^{3} x^{5} - \frac {4}{7} \, x^{7} e^{3} - 2 \, d x^{6} e^{2} - \frac {12}{5} \, d^{2} x^{5} e - d^{3} x^{4} + \frac {7}{2} \, x^{6} e^{3} + \frac {63}{5} \, d x^{5} e^{2} + \frac {63}{4} \, d^{2} x^{4} e + 7 \, d^{3} x^{3} + \frac {7}{5} \, x^{5} e^{3} + \frac {21}{4} \, d x^{4} e^{2} + 7 \, d^{2} x^{3} e + \frac {7}{2} \, d^{3} x^{2} + \frac {3}{2} \, x^{4} e^{3} + 6 \, d x^{3} e^{2} + 9 \, d^{2} x^{2} e + 6 \, d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^10*e^3 + 20/3*d*x^9*e^2 + 15/2*d^2*x^8*e + 20/7*d^3*x^7 - 17/9*x^9*e^3 - 51/8*d*x^8*e^2 - 51/7*d^2*x^7*e -
17/6*d^3*x^6 + 17/8*x^8*e^3 + 51/7*d*x^7*e^2 + 17/2*d^2*x^6*e + 17/5*d^3*x^5 - 4/7*x^7*e^3 - 2*d*x^6*e^2 - 12
/5*d^2*x^5*e - d^3*x^4 + 7/2*x^6*e^3 + 63/5*d*x^5*e^2 + 63/4*d^2*x^4*e + 7*d^3*x^3 + 7/5*x^5*e^3 + 21/4*d*x^4*
e^2 + 7*d^2*x^3*e + 7/2*d^3*x^2 + 3/2*x^4*e^3 + 6*d*x^3*e^2 + 9*d^2*x^2*e + 6*d^3*x

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Mupad [B]
time = 4.20, size = 196, normalized size = 0.76 \begin {gather*} 6\,d^3\,x+x^8\,\left (\frac {15\,d^2\,e}{2}-\frac {51\,d\,e^2}{8}+\frac {17\,e^3}{8}\right )-x^6\,\left (\frac {17\,d^3}{6}-\frac {17\,d^2\,e}{2}+2\,d\,e^2-\frac {7\,e^3}{2}\right )+x^4\,\left (-d^3+\frac {63\,d^2\,e}{4}+\frac {21\,d\,e^2}{4}+\frac {3\,e^3}{2}\right )+x^5\,\left (\frac {17\,d^3}{5}-\frac {12\,d^2\,e}{5}+\frac {63\,d\,e^2}{5}+\frac {7\,e^3}{5}\right )+x^7\,\left (\frac {20\,d^3}{7}-\frac {51\,d^2\,e}{7}+\frac {51\,d\,e^2}{7}-\frac {4\,e^3}{7}\right )+2\,e^3\,x^{10}+d\,x^3\,\left (7\,d^2+7\,d\,e+6\,e^2\right )+\frac {d^2\,x^2\,\left (7\,d+18\,e\right )}{2}+\frac {e^2\,x^9\,\left (60\,d-17\,e\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*d^3*x + x^8*((15*d^2*e)/2 - (51*d*e^2)/8 + (17*e^3)/8) - x^6*(2*d*e^2 - (17*d^2*e)/2 + (17*d^3)/6 - (7*e^3)/
2) + x^4*((21*d*e^2)/4 + (63*d^2*e)/4 - d^3 + (3*e^3)/2) + x^5*((63*d*e^2)/5 - (12*d^2*e)/5 + (17*d^3)/5 + (7*
e^3)/5) + x^7*((51*d*e^2)/7 - (51*d^2*e)/7 + (20*d^3)/7 - (4*e^3)/7) + 2*e^3*x^10 + d*x^3*(7*d*e + 7*d^2 + 6*e
^2) + (d^2*x^2*(7*d + 18*e))/2 + (e^2*x^9*(60*d - 17*e))/9

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