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

Optimal. Leaf size=153 \[ -\frac {(d+e x)^{m+2} \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{e^4 (m+2)}-\frac {(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)} \]

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Rubi [A]  time = 0.11, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {771} \begin {gather*} -\frac {(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}+\frac {(d+e x)^{m+2} \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*B*c*d^2 - B*e*(2*b*d - a*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 (a+b x+c x^2\right ) \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^3}+\frac {\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (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 {(B d-A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (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 = 0.34, size = 181, normalized size = 1.18 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {(d+e x) \left (B \left (c e (2 a e (m+3)+b d (m-2))-b^2 e^2 (m+2)+6 c^2 d^2\right )-A c e (m+4) (2 c d-b e)\right )}{e^2 (m+2)}-\frac {\left (e (a e-b d)+c d^2\right ) (-2 A c e (m+4)+b B e (m+1)+6 B c d)}{e^2 (m+1)}+(a+x (b+c x)) (A c e (m+4)+B (b e-3 c d)+B c e (m+3) x)\right )}{c e^2 (m+3) (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B 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[(A + B*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

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

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fricas [B]  time = 0.41, size = 538, normalized size = 3.52 \begin {gather*} \frac {{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 24 \, A a d e^{3} + 8 \, {\left (B b + A c\right )} d^{3} e - 12 \, {\left (B a + A b\right )} d^{2} e^{2} + {\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 (9 \, A a d e^{3} - {\left (B a + A b\right )} d^{2} e^{2}\right )} m^{2} + {\left (12 \, {\left (B a + A b\right )} e^{4} + {\left ({\left (B b + A c\right )} d e^{3} + {\left (B a + A b\right )} e^{4}\right )} m^{3} - {\left (3 \, B c d^{2} e^{2} - 5 \, {\left (B b + A c\right )} d e^{3} - 8 \, {\left (B a + A b\right )} e^{4}\right )} m^{2} - {\left (3 \, B c d^{2} e^{2} - 4 \, {\left (B b + A c\right )} d e^{3} - 19 \, {\left (B a + A b\right )} e^{4}\right )} m\right )} x^{2} + {\left (26 \, A a d e^{3} + 2 \, {\left (B b + A c\right )} d^{3} e - 7 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} m + {\left (24 \, A a e^{4} + {\left (A a e^{4} + {\left (B a + A b\right )} d e^{3}\right )} m^{3} + {\left (9 \, A a e^{4} - 2 \, {\left (B b + A c\right )} d^{2} e^{2} + 7 \, {\left (B a + A b\right )} d e^{3}\right )} m^{2} + 2 \, {\left (3 \, B c d^{3} e + 13 \, A a e^{4} - 4 \, {\left (B b + A c\right )} d^{2} e^{2} + 6 \, {\left (B a + A b\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((B*x+A)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(A*a*d*e^3*m^3 - 6*B*c*d^4 + 24*A*a*d*e^3 + 8*(B*b + A*c)*d^3*e - 12*(B*a + A*b)*d^2*e^2 + (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 + (9*A*a*d*e^3 - (B*a + A*b)*d^2*e^2)*
m^2 + (12*(B*a + A*b)*e^4 + ((B*b + A*c)*d*e^3 + (B*a + A*b)*e^4)*m^3 - (3*B*c*d^2*e^2 - 5*(B*b + A*c)*d*e^3 -
8*(B*a + A*b)*e^4)*m^2 - (3*B*c*d^2*e^2 - 4*(B*b + A*c)*d*e^3 - 19*(B*a + A*b)*e^4)*m)*x^2 + (26*A*a*d*e^3 +
2*(B*b + A*c)*d^3*e - 7*(B*a + A*b)*d^2*e^2)*m + (24*A*a*e^4 + (A*a*e^4 + (B*a + A*b)*d*e^3)*m^3 + (9*A*a*e^4
- 2*(B*b + A*c)*d^2*e^2 + 7*(B*a + A*b)*d*e^3)*m^2 + 2*(3*B*c*d^3*e + 13*A*a*e^4 - 4*(B*b + A*c)*d^2*e^2 + 6*(
B*a + A*b)*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 = 0.22, size = 1162, normalized size = 7.59

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+a),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*B*a*m^3*x^2*e^4 + (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*B*a*d*m^3*x*e^3 + (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 + (x*e + d)^m*A*a*m^3*x*e^4 + 8*(x*e
+ d)^m*B*a*m^2*x^2*e^4 + 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 + (x*e + d)^m*A*a*d*m^3*e^3 + 7*(x*e + d)^m*B*a*d*m^2*x*e^3 + 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*B*a*d^2*m^2*e^
2 - (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 + 9*(x*e + d)^m*A*a*m^2*x*e^4 + 19*(x*e + d)
^m*B*a*m*x^2*e^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 + 9*(x
*e + d)^m*A*a*d*m^2*e^3 + 12*(x*e + d)^m*B*a*d*m*x*e^3 + 12*(x*e + d)^m*A*b*d*m*x*e^3 - 7*(x*e + d)^m*B*a*d^2*
m*e^2 - 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 + 26*(x*e + d)^m*A*a*m
*x*e^4 + 12*(x*e + d)^m*B*a*x^2*e^4 + 12*(x*e + d)^m*A*b*x^2*e^4 + 26*(x*e + d)^m*A*a*d*m*e^3 - 12*(x*e + d)^m
*B*a*d^2*e^2 - 12*(x*e + d)^m*A*b*d^2*e^2 + 24*(x*e + d)^m*A*a*x*e^4 + 24*(x*e + d)^m*A*a*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 = 0.01, size = 498, normalized size = 3.25 \begin {gather*} \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}+B a \,e^{3} m^{3} x +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}+A a \,e^{3} m^{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}+8 B a \,e^{3} m^{2} x -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}+9 A a \,e^{3} m^{2}-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}-B a d \,e^{2} m^{2}+19 B a \,e^{3} m x -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}+26 A a \,e^{3} m -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 -7 B a d \,e^{2} m +12 B a \,e^{3} x +2 B b \,d^{2} e m -8 B b d \,e^{2} x +6 B c \,d^{2} e x +24 a A \,e^{3}-12 A b d \,e^{2}+8 A c \,d^{2} e -12 a B d \,e^{2}+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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^m*(c*x^2+b*x+a),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+B*a*e^3*m^3*x+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+A*a*e^3*m^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+8*B*a*e^3*m^2*x-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+9*
A*a*e^3*m^2-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-B*a*d*e^2*m^2+19*B*a*e^3*m*x-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+26*A*a*e^3*m-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-7*B*a*d*e^2*m+12*B*a*e^3*x+2*B*b*d^2*e*m-8*B*b*d*e^2*x+6*B*c*d^2*e*x+24*A*a*e^3-12*A*b*d*e^2+8*A*c
*d^2*e-12*B*a*d*e^2+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 = 0.53, size = 352, normalized size = 2.30 \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 a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \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 (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(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*a/((m^2 + 3*m + 2)*e^2) + (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) + (e*x + d)^(m + 1)*A*a/(e*(m + 1)) + ((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 = 2.97, size = 602, normalized size = 3.93 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (24\,A\,a\,d\,e^3-6\,B\,c\,d^4+8\,A\,c\,d^3\,e+8\,B\,b\,d^3\,e-12\,A\,b\,d^2\,e^2-12\,B\,a\,d^2\,e^2-A\,b\,d^2\,e^2\,m^2-B\,a\,d^2\,e^2\,m^2+26\,A\,a\,d\,e^3\,m+2\,A\,c\,d^3\,e\,m+2\,B\,b\,d^3\,e\,m+9\,A\,a\,d\,e^3\,m^2+A\,a\,d\,e^3\,m^3-7\,A\,b\,d^2\,e^2\,m-7\,B\,a\,d^2\,e^2\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (24\,A\,a\,e^4+26\,A\,a\,e^4\,m+9\,A\,a\,e^4\,m^2+A\,a\,e^4\,m^3-2\,A\,c\,d^2\,e^2\,m^2-2\,B\,b\,d^2\,e^2\,m^2+12\,A\,b\,d\,e^3\,m+12\,B\,a\,d\,e^3\,m+6\,B\,c\,d^3\,e\,m+7\,A\,b\,d\,e^3\,m^2+7\,B\,a\,d\,e^3\,m^2+A\,b\,d\,e^3\,m^3+B\,a\,d\,e^3\,m^3-8\,A\,c\,d^2\,e^2\,m-8\,B\,b\,d^2\,e^2\,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 {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,A\,b\,e^2+12\,B\,a\,e^2+7\,A\,b\,e^2\,m+7\,B\,a\,e^2\,m-3\,B\,c\,d^2\,m+A\,b\,e^2\,m^2+B\,a\,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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^m*(24*A*a*d*e^3 - 6*B*c*d^4 + 8*A*c*d^3*e + 8*B*b*d^3*e - 12*A*b*d^2*e^2 - 12*B*a*d^2*e^2 - A*b*d^2
*e^2*m^2 - B*a*d^2*e^2*m^2 + 26*A*a*d*e^3*m + 2*A*c*d^3*e*m + 2*B*b*d^3*e*m + 9*A*a*d*e^3*m^2 + A*a*d*e^3*m^3
- 7*A*b*d^2*e^2*m - 7*B*a*d^2*e^2*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x*(d + e*x)^m*(24*A*a*e^4 +
26*A*a*e^4*m + 9*A*a*e^4*m^2 + A*a*e^4*m^3 - 2*A*c*d^2*e^2*m^2 - 2*B*b*d^2*e^2*m^2 + 12*A*b*d*e^3*m + 12*B*a*
d*e^3*m + 6*B*c*d^3*e*m + 7*A*b*d*e^3*m^2 + 7*B*a*d*e^3*m^2 + A*b*d*e^3*m^3 + B*a*d*e^3*m^3 - 8*A*c*d^2*e^2*m
- 8*B*b*d^2*e^2*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) + (x^2*(m + 1)*(d + e*x)^m*(12*A*b*e^2 + 12*B*a*e^2 + 7*A
*b*e^2*m + 7*B*a*e^2*m - 3*B*c*d^2*m + A*b*e^2*m^2 + B*a*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))

________________________________________________________________________________________

sympy [A]  time = 5.25, size = 5930, normalized size = 38.76

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+a),x)

[Out]

Piecewise((d**m*(A*a*x + A*b*x**2/2 + A*c*x**3/3 + B*a*x**2/2 + B*b*x**3/3 + B*c*x**4/4), Eq(e, 0)), (-2*A*a*e
**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 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) - B*a*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*a*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*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*a*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 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) - B*a*d*e**2/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2) - 2*B*a*e**3*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*a*e**3/(2*d*e**4 + 2*e**5*x) + 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) + 2*B*a*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2
*B*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*e**3*x*log(d/e + x)/(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*a*log(d/e + x)/e - 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*a*d*log(d/e + x)/e**2 + B*a*x/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*a*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*a*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*a*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*a*d
*e**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) + A*a*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*a*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*a*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*a*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) - 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 + 3
5*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 + 35*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 + 50*e**4*m + 24*e**4) - B*a*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*a*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*a*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*a*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*a*d*e**3*m**2*x*(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) + 12*B*a*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*a*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*a*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*a*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*a*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*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 + 1
0*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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