3.15.38 \(\int (b+2 c x) (d+e x)^m (a+b x+c x^2) \, dx\)

Optimal. Leaf size=143 \[ \frac {(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac {(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]

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Rubi [A]  time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*c*(2*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (2*c^2*(d
+ e*x)^(4 + m))/(e^4*(4 + m))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx &=\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*}

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Mathematica [A]  time = 0.20, size = 134, normalized size = 0.94 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {(d+e x) \left (4 c e (a e (m+3)-3 b d)-b^2 e^2 m+12 c^2 d^2\right )}{e^2 (m+2)}+\frac {6 (b e-2 c d) \left (e (a e-b d)+c d^2\right )}{e^2 (m+1)}+(a+x (b+c x)) (b e (m+6)-6 c d+2 c e (m+3) x)\right )}{e^2 (m+3) (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

((d + e*x)^(1 + m)*((6*(-2*c*d + b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(e^2*(1 + m)) + ((12*c^2*d^2 - b^2*e^2*m + 4
*c*e*(-3*b*d + a*e*(3 + m)))*(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))))/(e^2*(3 + m)*(4 + m))

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IntegrateAlgebraic [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

Defer[IntegrateAlgebraic][(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2), x]

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fricas [B]  time = 0.44, size = 511, normalized size = 3.57 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(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*(b^2 + 2*a*c)*d^2*e^2 + 2*(c^2*e^4*m^3 + 6*c^2*
e^4*m^2 + 11*c^2*e^4*m + 6*c^2*e^4)*x^4 + (24*b*c*e^4 + (2*c^2*d*e^3 + 3*b*c*e^4)*m^3 + 3*(2*c^2*d*e^3 + 7*b*c
*e^4)*m^2 + 2*(2*c^2*d*e^3 + 21*b*c*e^4)*m)*x^3 + (9*a*b*d*e^3 - (b^2 + 2*a*c)*d^2*e^2)*m^2 + (12*(b^2 + 2*a*c
)*e^4 + (3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*m^3 - (6*c^2*d^2*e^2 - 15*b*c*d*e^3 - 8*(b^2 + 2*a*c)*e^4)*m^2 - (6*
c^2*d^2*e^2 - 12*b*c*d*e^3 - 19*(b^2 + 2*a*c)*e^4)*m)*x^2 + (6*b*c*d^3*e + 26*a*b*d*e^3 - 7*(b^2 + 2*a*c)*d^2*
e^2)*m + (24*a*b*e^4 + (a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*m^3 - (6*b*c*d^2*e^2 - 9*a*b*e^4 - 7*(b^2 + 2*a*c)*d*e^
3)*m^2 + 2*(6*c^2*d^3*e - 12*b*c*d^2*e^2 + 13*a*b*e^4 + 6*(b^2 + 2*a*c)*d*e^3)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10
*e^4*m^3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)

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giac [B]  time = 0.19, size = 980, normalized size = 6.85 \begin {gather*} \frac {2 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{3} e^{3} + 3 \, {\left (x e + d\right )}^{m} b c m^{3} x^{3} e^{4} + 12 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} b c d m^{3} x^{2} e^{3} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{3} e^{3} - 6 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} b^{2} m^{3} x^{2} e^{4} + 2 \, {\left (x e + d\right )}^{m} a c m^{3} x^{2} e^{4} + 21 \, {\left (x e + d\right )}^{m} b c m^{2} x^{3} e^{4} + 22 \, {\left (x e + d\right )}^{m} c^{2} m x^{4} e^{4} + {\left (x e + d\right )}^{m} b^{2} d m^{3} x e^{3} + 2 \, {\left (x e + d\right )}^{m} a c d m^{3} x e^{3} + 15 \, {\left (x e + d\right )}^{m} b c d m^{2} x^{2} e^{3} + 4 \, {\left (x e + d\right )}^{m} c^{2} d m x^{3} e^{3} - 6 \, {\left (x e + d\right )}^{m} b c d^{2} m^{2} x e^{2} - 6 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{2} e^{2} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x e + {\left (x e + d\right )}^{m} a b m^{3} x e^{4} + 8 \, {\left (x e + d\right )}^{m} b^{2} m^{2} x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} a c m^{2} x^{2} e^{4} + 42 \, {\left (x e + d\right )}^{m} b c m x^{3} e^{4} + 12 \, {\left (x e + d\right )}^{m} c^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} a b d m^{3} e^{3} + 7 \, {\left (x e + d\right )}^{m} b^{2} d m^{2} x e^{3} + 14 \, {\left (x e + d\right )}^{m} a c d m^{2} x e^{3} + 12 \, {\left (x e + d\right )}^{m} b c d m x^{2} e^{3} - {\left (x e + d\right )}^{m} b^{2} d^{2} m^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} b c d^{2} m x e^{2} + 6 \, {\left (x e + d\right )}^{m} b c d^{3} m e - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{4} + 9 \, {\left (x e + d\right )}^{m} a b m^{2} x e^{4} + 19 \, {\left (x e + d\right )}^{m} b^{2} m x^{2} e^{4} + 38 \, {\left (x e + d\right )}^{m} a c m x^{2} e^{4} + 24 \, {\left (x e + d\right )}^{m} b c x^{3} e^{4} + 9 \, {\left (x e + d\right )}^{m} a b d m^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} b^{2} d m x e^{3} + 24 \, {\left (x e + d\right )}^{m} a c d m x e^{3} - 7 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m e^{2} - 14 \, {\left (x e + d\right )}^{m} a c d^{2} m e^{2} + 24 \, {\left (x e + d\right )}^{m} b c d^{3} e + 26 \, {\left (x e + d\right )}^{m} a b m x e^{4} + 12 \, {\left (x e + d\right )}^{m} b^{2} x^{2} e^{4} + 24 \, {\left (x e + d\right )}^{m} a c x^{2} e^{4} + 26 \, {\left (x e + d\right )}^{m} a b d m e^{3} - 12 \, {\left (x e + d\right )}^{m} b^{2} d^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} a c d^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} a b x e^{4} + 24 \, {\left (x e + d\right )}^{m} a b d e^{3}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

(2*(x*e + d)^m*c^2*m^3*x^4*e^4 + 2*(x*e + d)^m*c^2*d*m^3*x^3*e^3 + 3*(x*e + d)^m*b*c*m^3*x^3*e^4 + 12*(x*e + d
)^m*c^2*m^2*x^4*e^4 + 3*(x*e + d)^m*b*c*d*m^3*x^2*e^3 + 6*(x*e + d)^m*c^2*d*m^2*x^3*e^3 - 6*(x*e + d)^m*c^2*d^
2*m^2*x^2*e^2 + (x*e + d)^m*b^2*m^3*x^2*e^4 + 2*(x*e + d)^m*a*c*m^3*x^2*e^4 + 21*(x*e + d)^m*b*c*m^2*x^3*e^4 +
22*(x*e + d)^m*c^2*m*x^4*e^4 + (x*e + d)^m*b^2*d*m^3*x*e^3 + 2*(x*e + d)^m*a*c*d*m^3*x*e^3 + 15*(x*e + d)^m*b
*c*d*m^2*x^2*e^3 + 4*(x*e + d)^m*c^2*d*m*x^3*e^3 - 6*(x*e + d)^m*b*c*d^2*m^2*x*e^2 - 6*(x*e + d)^m*c^2*d^2*m*x
^2*e^2 + 12*(x*e + d)^m*c^2*d^3*m*x*e + (x*e + d)^m*a*b*m^3*x*e^4 + 8*(x*e + d)^m*b^2*m^2*x^2*e^4 + 16*(x*e +
d)^m*a*c*m^2*x^2*e^4 + 42*(x*e + d)^m*b*c*m*x^3*e^4 + 12*(x*e + d)^m*c^2*x^4*e^4 + (x*e + d)^m*a*b*d*m^3*e^3 +
7*(x*e + d)^m*b^2*d*m^2*x*e^3 + 14*(x*e + d)^m*a*c*d*m^2*x*e^3 + 12*(x*e + d)^m*b*c*d*m*x^2*e^3 - (x*e + d)^m
*b^2*d^2*m^2*e^2 - 2*(x*e + d)^m*a*c*d^2*m^2*e^2 - 24*(x*e + d)^m*b*c*d^2*m*x*e^2 + 6*(x*e + d)^m*b*c*d^3*m*e
- 12*(x*e + d)^m*c^2*d^4 + 9*(x*e + d)^m*a*b*m^2*x*e^4 + 19*(x*e + d)^m*b^2*m*x^2*e^4 + 38*(x*e + d)^m*a*c*m*x
^2*e^4 + 24*(x*e + d)^m*b*c*x^3*e^4 + 9*(x*e + d)^m*a*b*d*m^2*e^3 + 12*(x*e + d)^m*b^2*d*m*x*e^3 + 24*(x*e + d
)^m*a*c*d*m*x*e^3 - 7*(x*e + d)^m*b^2*d^2*m*e^2 - 14*(x*e + d)^m*a*c*d^2*m*e^2 + 24*(x*e + d)^m*b*c*d^3*e + 26
*(x*e + d)^m*a*b*m*x*e^4 + 12*(x*e + d)^m*b^2*x^2*e^4 + 24*(x*e + d)^m*a*c*x^2*e^4 + 26*(x*e + d)^m*a*b*d*m*e^
3 - 12*(x*e + d)^m*b^2*d^2*e^2 - 24*(x*e + d)^m*a*c*d^2*e^2 + 24*(x*e + d)^m*a*b*x*e^4 + 24*(x*e + d)^m*a*b*d*
e^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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maple [B]  time = 0.05, size = 424, normalized size = 2.97 \begin {gather*} \frac {\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} e^{3} x^{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 b c \,e^{3} x^{2}+12 c^{2} d^{2} e m x -12 c^{2} d \,e^{2} x^{2}+26 a b \,e^{3} m -14 a c d \,e^{2} m +24 a c \,e^{3} x -7 b^{2} d \,e^{2} m +12 b^{2} e^{3} x +6 b c \,d^{2} e m -24 b c d \,e^{2} x +12 c^{2} d^{2} e x +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 ) \left (e x +d \right )^{m +1}}{\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x)

[Out]

(e*x+d)^(m+1)*(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+4
2*b*c*e^3*m*x^2-18*c^2*d*e^2*m*x^2+12*c^2*e^3*x^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+1
9*b^2*e^3*m*x-30*b*c*d*e^2*m*x+24*b*c*e^3*x^2+12*c^2*d^2*e*m*x-12*c^2*d*e^2*x^2+26*a*b*e^3*m-14*a*c*d*e^2*m+24
*a*c*e^3*x-7*b^2*d*e^2*m+12*b^2*e^3*x+6*b*c*d^2*e*m-24*b*c*d*e^2*x+12*c^2*d^2*e*x+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)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [B]  time = 0.55, size = 287, normalized size = 2.01 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*b^2/((m^2 + 3*m + 2)*e^2) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*
(e*x + d)^m*a*c/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a*b/(e*(m + 1)) + 3*((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*b*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((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*c^2/
((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4)

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mupad [B]  time = 2.25, size = 554, normalized size = 3.87 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x)

[Out]

(x*(d + e*x)^m*(24*a*b*e^4 + 7*b^2*d*e^3*m^2 + b^2*d*e^3*m^3 + 26*a*b*e^4*m + 9*a*b*e^4*m^2 + a*b*e^4*m^3 + 12
*b^2*d*e^3*m + 12*c^2*d^3*e*m + 14*a*c*d*e^3*m^2 + 2*a*c*d*e^3*m^3 - 24*b*c*d^2*e^2*m - 6*b*c*d^2*e^2*m^2 + 24
*a*c*d*e^3*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - ((d + e*x)^m*(12*c^2*d^4 + 12*b^2*d^2*e^2 + 7*b^2*d
^2*e^2*m - 24*a*b*d*e^3 - 24*b*c*d^3*e + b^2*d^2*e^2*m^2 + 24*a*c*d^2*e^2 - 9*a*b*d*e^3*m^2 - a*b*d*e^3*m^3 +
14*a*c*d^2*e^2*m + 2*a*c*d^2*e^2*m^2 - 26*a*b*d*e^3*m - 6*b*c*d^3*e*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 2
4)) + (2*c^2*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (x^2*(m + 1)*(d +
e*x)^m*(12*b^2*e^2 + 7*b^2*e^2*m - 6*c^2*d^2*m + b^2*e^2*m^2 + 24*a*c*e^2 + 14*a*c*e^2*m + 2*a*c*e^2*m^2 + 12
*b*c*d*e*m + 3*b*c*d*e*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (c*x^3*(d + e*x)^m*(12*b*e + 3*b*e*m
+ 2*c*d*m)*(3*m + m^2 + 2))/(e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 4.71, size = 4760, normalized size = 33.29

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**m*(c*x**2+b*x+a),x)

[Out]

Piecewise((d**m*(a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2), Eq(e, 0)), (-2*a*b*e**3/(6*d**3*e**
4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*a*c*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x*
*2 + 6*e**7*x**3) - 6*a*c*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - b**2*d*e**2/(
6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 3*b**2*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 1
8*d*e**6*x**2 + 6*e**7*x**3) - 6*b*c*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18
*b*c*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*b*c*e**3*x**2/(6*d**3*e**4 +
18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 12*c**2*d**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*
d*e**6*x**2 + 6*e**7*x**3) + 22*c**2*d**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 36*c
**2*d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 54*c**2*d**2*e*x/(6*
d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 36*c**2*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 1
8*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 36*c**2*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*
x**2 + 6*e**7*x**3) + 12*c**2*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x
**3), Eq(m, -4)), (-a*b*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*a*c*d*e**2/(2*d**2*e**4 + 4*d*e**5*x
+ 2*e**6*x**2) - 4*a*c*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - b**2*d*e**2/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) - 2*b**2*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*b*c*d**2*e*log(d/e + x)/(2*d**2*
e**4 + 4*d*e**5*x + 2*e**6*x**2) + 9*b*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*b*c*d*e**2*x*log
(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*b*c*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)
+ 6*b*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c**2*d**3*log(d/e + x)/(2*d**2*e*
*4 + 4*d*e**5*x + 2*e**6*x**2) - 18*c**2*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**2*d**2*e*x*log(
d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**2*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)
- 12*c**2*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*c**2*e**3*x**3/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-a*b*e**3/(d*e**4 + e**5*x) + 2*a*c*d*e**2*log(d/e + x)/(d*e**4 + e**5
*x) + 2*a*c*d*e**2/(d*e**4 + e**5*x) + 2*a*c*e**3*x*log(d/e + x)/(d*e**4 + e**5*x) + b**2*d*e**2*log(d/e + x)/
(d*e**4 + e**5*x) + b**2*d*e**2/(d*e**4 + e**5*x) + b**2*e**3*x*log(d/e + x)/(d*e**4 + e**5*x) - 6*b*c*d**2*e*
log(d/e + x)/(d*e**4 + e**5*x) - 6*b*c*d**2*e/(d*e**4 + e**5*x) - 6*b*c*d*e**2*x*log(d/e + x)/(d*e**4 + e**5*x
) + 3*b*c*e**3*x**2/(d*e**4 + e**5*x) + 6*c**2*d**3*log(d/e + x)/(d*e**4 + e**5*x) + 6*c**2*d**3/(d*e**4 + e**
5*x) + 6*c**2*d**2*e*x*log(d/e + x)/(d*e**4 + e**5*x) - 3*c**2*d*e**2*x**2/(d*e**4 + e**5*x) + c**2*e**3*x**3/
(d*e**4 + e**5*x), Eq(m, -2)), (a*b*log(d/e + x)/e - 2*a*c*d*log(d/e + x)/e**2 + 2*a*c*x/e - b**2*d*log(d/e +
x)/e**2 + b**2*x/e + 3*b*c*d**2*log(d/e + x)/e**3 - 3*b*c*d*x/e**2 + 3*b*c*x**2/(2*e) - 2*c**2*d**3*log(d/e +
x)/e**4 + 2*c**2*d**2*x/e**3 - c**2*d*x**2/e**2 + 2*c**2*x**3/(3*e), Eq(m, -1)), (a*b*d*e**3*m**3*(d + e*x)**m
/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*a*b*d*e**3*m**2*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*a*b*d*e**3*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*a*b*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + a*b*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e
**4) + 9*a*b*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*a*b
*e**4*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*a*b*e**4*x*(d + e*
x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*a*c*d**2*e**2*m**2*(d + e*x)**m/(e**
4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 14*a*c*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*
e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 24*a*c*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*a*c*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2
+ 50*e**4*m + 24*e**4) + 14*a*c*d*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*
m + 24*e**4) + 24*a*c*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
+ 2*a*c*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 16*a*c*e
**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 38*a*c*e**4*m*x**
2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*a*c*e**4*x**2*(d + e*x)**m
/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - b**2*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4
+ 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 7*b**2*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m
**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*b**2*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4
*m**2 + 50*e**4*m + 24*e**4) + b**2*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + 7*b**2*d*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24
*e**4) + 12*b**2*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + b**
2*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*b**2*e**4*m*
*2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 19*b**2*e**4*m*x**2*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*b**2*e**4*x**2*(d + e*x)**m/(e*
*4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*b*c*d**3*e*m*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*b*c*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*
m**2 + 50*e**4*m + 24*e**4) - 6*b*c*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) - 24*b*c*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 3*b*c*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 15*b*c*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*
b*c*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*b*c*e**4*m*
*3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 21*b*c*e**4*m**2*x**3*(
d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 42*b*c*e**4*m*x**3*(d + e*x)**m/
(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*b*c*e**4*x**3*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*c**2*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e
**4*m**2 + 50*e**4*m + 24*e**4) + 12*c**2*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) - 6*c**2*d**2*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e*
*4*m + 24*e**4) - 6*c**2*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 2*c**2*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 6*c**2*d*e**3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 4*c
**2*d*e**3*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*c**2*e**4*m
**3*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*c**2*e**4*m**2*x**4
*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 22*c**2*e**4*m*x**4*(d + e*x)*
*m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*c**2*e**4*x**4*(d + e*x)**m/(e**4*m**4
+ 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4), True))

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