3.1102 \(\int (A+B x) (d+e x)^m (b x+c x^2) \, dx\)

Optimal. Leaf size=136 \[ -\frac {d (B d-A e) (c d-b e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(d+e x)^{m+2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 (m+2)}-\frac {(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \[ -\frac {d (B d-A e) (c d-b e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(d+e x)^{m+2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 (m+2)}-\frac {(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*(d
+ e*x)^(2 + m))/(e^4*(2 + m)) - ((3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (B*c*(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 (A+B x) (d+e x)^m \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^m}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{1+m}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{2+m}}{e^3}+\frac {B c (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac {d (B d-A e) (c d-b e) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {(3 B c d-b B e-A c e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {B c (d+e x)^{4+m}}{e^4 (4+m)}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 118, normalized size = 0.87 \[ \frac {(d+e x)^{m+1} \left (-\frac {(d+e x)^2 (-A c e-b B e+3 B c d)}{m+3}+\frac {(d+e x) (A e (b e-2 c d)+B d (3 c d-2 b e))}{m+2}-\frac {d (B d-A e) (c d-b e)}{m+1}+\frac {B c (d+e x)^3}{m+4}\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.89, size = 427, normalized size = 3.14 \[ -\frac {{\left (A b d^{2} e^{2} m^{2} + 6 \, B c d^{4} + 12 \, A b d^{2} e^{2} - 8 \, {\left (B b + A c\right )} d^{3} e - {\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} - {\left (8 \, {\left (B b + A c\right )} e^{4} + {\left (B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} m^{3} + {\left (3 \, B c d e^{3} + 7 \, {\left (B b + A c\right )} e^{4}\right )} m^{2} + 2 \, {\left (B c d e^{3} + 7 \, {\left (B b + A c\right )} e^{4}\right )} m\right )} x^{3} - {\left (12 \, A b e^{4} + {\left (A b e^{4} + {\left (B b + A c\right )} d e^{3}\right )} m^{3} - {\left (3 \, B c d^{2} e^{2} - 8 \, A b e^{4} - 5 \, {\left (B b + A c\right )} d e^{3}\right )} m^{2} - {\left (3 \, B c d^{2} e^{2} - 19 \, A b e^{4} - 4 \, {\left (B b + A c\right )} d e^{3}\right )} m\right )} x^{2} + {\left (7 \, A b d^{2} e^{2} - 2 \, {\left (B b + A c\right )} d^{3} e\right )} m - {\left (A b d e^{3} m^{3} + {\left (7 \, A b d e^{3} - 2 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} m^{2} + 2 \, {\left (3 \, B c d^{3} e + 6 \, A b d e^{3} - 4 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A*b*d^2*e^2*m^2 + 6*B*c*d^4 + 12*A*b*d^2*e^2 - 8*(B*b + A*c)*d^3*e - (B*c*e^4*m^3 + 6*B*c*e^4*m^2 + 11*B*c*e
^4*m + 6*B*c*e^4)*x^4 - (8*(B*b + A*c)*e^4 + (B*c*d*e^3 + (B*b + A*c)*e^4)*m^3 + (3*B*c*d*e^3 + 7*(B*b + A*c)*
e^4)*m^2 + 2*(B*c*d*e^3 + 7*(B*b + A*c)*e^4)*m)*x^3 - (12*A*b*e^4 + (A*b*e^4 + (B*b + A*c)*d*e^3)*m^3 - (3*B*c
*d^2*e^2 - 8*A*b*e^4 - 5*(B*b + A*c)*d*e^3)*m^2 - (3*B*c*d^2*e^2 - 19*A*b*e^4 - 4*(B*b + A*c)*d*e^3)*m)*x^2 +
(7*A*b*d^2*e^2 - 2*(B*b + A*c)*d^3*e)*m - (A*b*d*e^3*m^3 + (7*A*b*d*e^3 - 2*(B*b + A*c)*d^2*e^2)*m^2 + 2*(3*B*
c*d^3*e + 6*A*b*d*e^3 - 4*(B*b + A*c)*d^2*e^2)*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 = 0.18, size = 847, normalized size = 6.23 \[ \frac {{\left (x e + d\right )}^{m} B c m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} B c d m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} B b m^{3} x^{3} e^{4} + {\left (x e + d\right )}^{m} A c m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B c m^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} B b d m^{3} x^{2} e^{3} + {\left (x e + d\right )}^{m} A c d m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} B c d m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} B c d^{2} m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} A b m^{3} x^{2} e^{4} + 7 \, {\left (x e + d\right )}^{m} B b m^{2} x^{3} e^{4} + 7 \, {\left (x e + d\right )}^{m} A c m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} B c m x^{4} e^{4} + {\left (x e + d\right )}^{m} A b d m^{3} x e^{3} + 5 \, {\left (x e + d\right )}^{m} B b d m^{2} x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} A c d m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} B c d m x^{3} e^{3} - 2 \, {\left (x e + d\right )}^{m} B b d^{2} m^{2} x e^{2} - 2 \, {\left (x e + d\right )}^{m} A c d^{2} m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} B c d^{2} m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} B c d^{3} m x e + 8 \, {\left (x e + d\right )}^{m} A b m^{2} x^{2} e^{4} + 14 \, {\left (x e + d\right )}^{m} B b m x^{3} e^{4} + 14 \, {\left (x e + d\right )}^{m} A c m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B c x^{4} e^{4} + 7 \, {\left (x e + d\right )}^{m} A b d m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} B b d m x^{2} e^{3} + 4 \, {\left (x e + d\right )}^{m} A c d m x^{2} e^{3} - {\left (x e + d\right )}^{m} A b d^{2} m^{2} e^{2} - 8 \, {\left (x e + d\right )}^{m} B b d^{2} m x e^{2} - 8 \, {\left (x e + d\right )}^{m} A c d^{2} m x e^{2} + 2 \, {\left (x e + d\right )}^{m} B b d^{3} m e + 2 \, {\left (x e + d\right )}^{m} A c d^{3} m e - 6 \, {\left (x e + d\right )}^{m} B c d^{4} + 19 \, {\left (x e + d\right )}^{m} A b m x^{2} e^{4} + 8 \, {\left (x e + d\right )}^{m} B b x^{3} e^{4} + 8 \, {\left (x e + d\right )}^{m} A c x^{3} e^{4} + 12 \, {\left (x e + d\right )}^{m} A b d m x e^{3} - 7 \, {\left (x e + d\right )}^{m} A b d^{2} m e^{2} + 8 \, {\left (x e + d\right )}^{m} B b d^{3} e + 8 \, {\left (x e + d\right )}^{m} A c d^{3} e + 12 \, {\left (x e + d\right )}^{m} A b x^{2} e^{4} - 12 \, {\left (x e + d\right )}^{m} A b d^{2} e^{2}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*B*c*m^3*x^4*e^4 + (x*e + d)^m*B*c*d*m^3*x^3*e^3 + (x*e + d)^m*B*b*m^3*x^3*e^4 + (x*e + d)^m*A*c*m
^3*x^3*e^4 + 6*(x*e + d)^m*B*c*m^2*x^4*e^4 + (x*e + d)^m*B*b*d*m^3*x^2*e^3 + (x*e + d)^m*A*c*d*m^3*x^2*e^3 + 3
*(x*e + d)^m*B*c*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*c*d^2*m^2*x^2*e^2 + (x*e + d)^m*A*b*m^3*x^2*e^4 + 7*(x*e + d)
^m*B*b*m^2*x^3*e^4 + 7*(x*e + d)^m*A*c*m^2*x^3*e^4 + 11*(x*e + d)^m*B*c*m*x^4*e^4 + (x*e + d)^m*A*b*d*m^3*x*e^
3 + 5*(x*e + d)^m*B*b*d*m^2*x^2*e^3 + 5*(x*e + d)^m*A*c*d*m^2*x^2*e^3 + 2*(x*e + d)^m*B*c*d*m*x^3*e^3 - 2*(x*e
+ d)^m*B*b*d^2*m^2*x*e^2 - 2*(x*e + d)^m*A*c*d^2*m^2*x*e^2 - 3*(x*e + d)^m*B*c*d^2*m*x^2*e^2 + 6*(x*e + d)^m*
B*c*d^3*m*x*e + 8*(x*e + d)^m*A*b*m^2*x^2*e^4 + 14*(x*e + d)^m*B*b*m*x^3*e^4 + 14*(x*e + d)^m*A*c*m*x^3*e^4 +
6*(x*e + d)^m*B*c*x^4*e^4 + 7*(x*e + d)^m*A*b*d*m^2*x*e^3 + 4*(x*e + d)^m*B*b*d*m*x^2*e^3 + 4*(x*e + d)^m*A*c*
d*m*x^2*e^3 - (x*e + d)^m*A*b*d^2*m^2*e^2 - 8*(x*e + d)^m*B*b*d^2*m*x*e^2 - 8*(x*e + d)^m*A*c*d^2*m*x*e^2 + 2*
(x*e + d)^m*B*b*d^3*m*e + 2*(x*e + d)^m*A*c*d^3*m*e - 6*(x*e + d)^m*B*c*d^4 + 19*(x*e + d)^m*A*b*m*x^2*e^4 + 8
*(x*e + d)^m*B*b*x^3*e^4 + 8*(x*e + d)^m*A*c*x^3*e^4 + 12*(x*e + d)^m*A*b*d*m*x*e^3 - 7*(x*e + d)^m*A*b*d^2*m*
e^2 + 8*(x*e + d)^m*B*b*d^3*e + 8*(x*e + d)^m*A*c*d^3*e + 12*(x*e + d)^m*A*b*x^2*e^4 - 12*(x*e + d)^m*A*b*d^2*
e^2)/(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 = 402, normalized size = 2.96 \[ -\frac {\left (-B c \,e^{3} m^{3} x^{3}-A c \,e^{3} m^{3} x^{2}-B b \,e^{3} m^{3} x^{2}-6 B c \,e^{3} m^{2} x^{3}-A b \,e^{3} m^{3} x -7 A c \,e^{3} m^{2} x^{2}-7 B b \,e^{3} m^{2} x^{2}+3 B c d \,e^{2} m^{2} x^{2}-11 B c \,e^{3} m \,x^{3}-8 A b \,e^{3} m^{2} x +2 A c d \,e^{2} m^{2} x -14 A c \,e^{3} m \,x^{2}+2 B b d \,e^{2} m^{2} x -14 B b \,e^{3} m \,x^{2}+9 B c d \,e^{2} m \,x^{2}-6 B c \,x^{3} e^{3}+A b d \,e^{2} m^{2}-19 A b \,e^{3} m x +10 A c d \,e^{2} m x -8 A c \,e^{3} x^{2}+10 B b d \,e^{2} m x -8 B b \,e^{3} x^{2}-6 B c \,d^{2} e m x +6 B c d \,e^{2} x^{2}+7 A b d \,e^{2} m -12 A b \,e^{3} x -2 A c \,d^{2} e m +8 A c d \,e^{2} x -2 B b \,d^{2} e m +8 B b d \,e^{2} x -6 B c \,d^{2} e x +12 A b d \,e^{2}-8 A c \,d^{2} e -8 B b \,d^{2} e +6 B c \,d^{3}\right ) \left (e x +d \right )^{m +1}}{\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x+d)^(m+1)*(-B*c*e^3*m^3*x^3-A*c*e^3*m^3*x^2-B*b*e^3*m^3*x^2-6*B*c*e^3*m^2*x^3-A*b*e^3*m^3*x-7*A*c*e^3*m^2
*x^2-7*B*b*e^3*m^2*x^2+3*B*c*d*e^2*m^2*x^2-11*B*c*e^3*m*x^3-8*A*b*e^3*m^2*x+2*A*c*d*e^2*m^2*x-14*A*c*e^3*m*x^2
+2*B*b*d*e^2*m^2*x-14*B*b*e^3*m*x^2+9*B*c*d*e^2*m*x^2-6*B*c*e^3*x^3+A*b*d*e^2*m^2-19*A*b*e^3*m*x+10*A*c*d*e^2*
m*x-8*A*c*e^3*x^2+10*B*b*d*e^2*m*x-8*B*b*e^3*x^2-6*B*c*d^2*e*m*x+6*B*c*d*e^2*x^2+7*A*b*d*e^2*m-12*A*b*e^3*x-2*
A*c*d^2*e*m+8*A*c*d*e^2*x-2*B*b*d^2*e*m+8*B*b*d*e^2*x-6*B*c*d^2*e*x+12*A*b*d*e^2-8*A*c*d^2*e-8*B*b*d^2*e+6*B*c
*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [B]  time = 0.76, size = 288, normalized size = 2.12 \[ \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.64, size = 400, normalized size = 2.94 \[ \frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,A\,b\,e^2+7\,A\,b\,e^2\,m-3\,B\,c\,d^2\,m+A\,b\,e^2\,m^2+4\,A\,c\,d\,e\,m+4\,B\,b\,d\,e\,m+A\,c\,d\,e\,m^2+B\,b\,d\,e\,m^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {d^2\,{\left (d+e\,x\right )}^m\,\left (12\,A\,b\,e^2+6\,B\,c\,d^2+7\,A\,b\,e^2\,m+A\,b\,e^2\,m^2-8\,A\,c\,d\,e-8\,B\,b\,d\,e-2\,A\,c\,d\,e\,m-2\,B\,b\,d\,e\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,A\,c\,e+4\,B\,b\,e+A\,c\,e\,m+B\,b\,e\,m+B\,c\,d\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {B\,c\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {d\,m\,x\,{\left (d+e\,x\right )}^m\,\left (12\,A\,b\,e^2+6\,B\,c\,d^2+7\,A\,b\,e^2\,m+A\,b\,e^2\,m^2-8\,A\,c\,d\,e-8\,B\,b\,d\,e-2\,A\,c\,d\,e\,m-2\,B\,b\,d\,e\,m\right )}{e^3\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(m + 1)*(d + e*x)^m*(12*A*b*e^2 + 7*A*b*e^2*m - 3*B*c*d^2*m + A*b*e^2*m^2 + 4*A*c*d*e*m + 4*B*b*d*e*m + A
*c*d*e*m^2 + B*b*d*e*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (d^2*(d + e*x)^m*(12*A*b*e^2 + 6*B*c*d^
2 + 7*A*b*e^2*m + A*b*e^2*m^2 - 8*A*c*d*e - 8*B*b*d*e - 2*A*c*d*e*m - 2*B*b*d*e*m))/(e^4*(50*m + 35*m^2 + 10*m
^3 + m^4 + 24)) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(4*A*c*e + 4*B*b*e + A*c*e*m + B*b*e*m + B*c*d*m))/(e*(50*m
+ 35*m^2 + 10*m^3 + m^4 + 24)) + (B*c*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) + (d*m*x*(d + e*x)^m*(12*A*b*e^2 + 6*B*c*d^2 + 7*A*b*e^2*m + A*b*e^2*m^2 - 8*A*c*d*e - 8*B*b*d*e - 2*A*
c*d*e*m - 2*B*b*d*e*m))/(e^3*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 4.75, size = 4537, normalized size = 33.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4), Eq(e, 0)), (-A*b*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*A*b*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) - 2*A*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) - 6*A*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) - 6*A*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) - 2*B*b*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) - 6*B
*b*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) - 6*B*b*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) + 6*B*c*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) + 11*B*c*d**3/(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**
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) + 27*B*c*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) + 18*B*c*d*e**2*x**2*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) + 18*B*c*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) + 6*B*c*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*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*A*b*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*
x**2) + 2*A*c*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*A*c*d**2*e/(2*d**2*e**4 + 4*d*e
**5*x + 2*e**6*x**2) + 4*A*c*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*c*d*e**2*x/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2
) + 2*B*b*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*B*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) + 4*B*b*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*B*b*d*e**2*x/(2*d*
*2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*b*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
6*B*c*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*B*c*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e*
*6*x**2) - 12*B*c*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*c*d**2*e*x/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) - 6*B*c*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*
c*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (2*A*b*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**
5*x) + 2*A*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*A*b*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e*log
(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*c*d*e**2*x*log(d/e + x)/(2*d*e**4 +
2*e**5*x) + 2*A*c*e**3*x**2/(2*d*e**4 + 2*e**5*x) - 4*B*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*B*b*d
**2*e/(2*d*e**4 + 2*e**5*x) - 4*B*b*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*B*b*e**3*x**2/(2*d*e**4 +
2*e**5*x) + 6*B*c*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**2*e*x*
log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*B*c*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*c*e**3*x**3/(2*d*e**4 + 2*e**
5*x), Eq(m, -2)), (-A*b*d*log(d/e + x)/e**2 + A*b*x/e + A*c*d**2*log(d/e + x)/e**3 - A*c*d*x/e**2 + A*c*x**2/(
2*e) + B*b*d**2*log(d/e + x)/e**3 - B*b*d*x/e**2 + B*b*x**2/(2*e) - B*c*d**3*log(d/e + x)/e**4 + B*c*d**2*x/e*
*3 - B*c*d*x**2/(2*e**2) + B*c*x**3/(3*e), Eq(m, -1)), (-A*b*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*A*b*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*A*b*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) + A*b*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*A*b*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*A*b
*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) + A*b*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*A*b*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*A*b*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*A*b*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) + 2*A*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) + 8*A*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) - 2*A*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 + 50*e**4*m + 24*e**4) -
8*A*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) + A*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) + 5*A*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) + 4*A*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) + A*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) + 7*A*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) + 14*A*c*e**4*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*e**4*x**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) + 2*B*b*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) + 8*B*b*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) - 2*B*b*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 + 50*e**4*m + 24*e**4) - 8*B*b*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) + B*b*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) + 5*B*b*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) + 4*B*b*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) + B*b*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) + 7*B*b*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) + 14*B*b*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) + 8*B*b*e**4*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*B*c*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) + 6*B*c*d**3*e*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**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) - 3*B*c*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) + B*c*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) + 3*B*c*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) + 2*B*c*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) + B*c*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) + 6*B*c*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) + 11*B*c*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) +
6*B*c*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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