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

Optimal. Leaf size=144 \[ \frac {(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac {(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac {c g (d+e x)^{m+4}}{e^4 (m+4)} \]

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Rubi [A]  time = 0.11, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 {(e f-d g) (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} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac {(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac {c g (d+e x)^{m+4}}{e^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.33, size = 180, normalized size = 1.25 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {(d+e x) \left (c e (2 a e g (m+3)+b d g (m-2)+b e f (m+4))-b^2 e^2 g (m+2)+2 c^2 d (3 d g-e f (m+4))\right )}{e^2 (m+2)}-\frac {\left (e (a e-b d)+c d^2\right ) (b e g (m+1)+6 c d g-2 c e f (m+4))}{e^2 (m+1)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (m+4)+e g (m+3) x))\right )}{c e^2 (m+3) (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 613, normalized size = 4.26 \begin {gather*} \frac {{\left (a d e^{3} f m^{3} + {\left (c e^{4} g m^{3} + 6 \, c e^{4} g m^{2} + 11 \, c e^{4} g m + 6 \, c e^{4} g\right )} x^{4} + {\left (8 \, c e^{4} f + 8 \, b e^{4} g + {\left (c e^{4} f + {\left (c d e^{3} + b e^{4}\right )} g\right )} m^{3} + {\left (7 \, c e^{4} f + {\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} g\right )} m^{2} + 2 \, {\left (7 \, c e^{4} f + {\left (c d e^{3} + 7 \, b e^{4}\right )} g\right )} m\right )} x^{3} - {\left (a d^{2} e^{2} g + {\left (b d^{2} e^{2} - 9 \, a d e^{3}\right )} f\right )} m^{2} + {\left (12 \, b e^{4} f + 12 \, a e^{4} g + {\left ({\left (c d e^{3} + b e^{4}\right )} f + {\left (b d e^{3} + a e^{4}\right )} g\right )} m^{3} + {\left ({\left (5 \, c d e^{3} + 8 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} g\right )} m^{2} + {\left ({\left (4 \, c d e^{3} + 19 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} g\right )} m\right )} x^{2} + 4 \, {\left (2 \, c d^{3} e - 3 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} f - 2 \, {\left (3 \, c d^{4} - 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} g + {\left ({\left (2 \, c d^{3} e - 7 \, b d^{2} e^{2} + 26 \, a d e^{3}\right )} f + {\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} g\right )} m + {\left (24 \, a e^{4} f + {\left (a d e^{3} g + {\left (b d e^{3} + a e^{4}\right )} f\right )} m^{3} - {\left ({\left (2 \, c d^{2} e^{2} - 7 \, b d e^{3} - 9 \, a e^{4}\right )} f + {\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} g\right )} m^{2} - 2 \, {\left ({\left (4 \, c d^{2} e^{2} - 6 \, b d e^{3} - 13 \, a e^{4}\right )} f - {\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} g\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((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(a*d*e^3*f*m^3 + (c*e^4*g*m^3 + 6*c*e^4*g*m^2 + 11*c*e^4*g*m + 6*c*e^4*g)*x^4 + (8*c*e^4*f + 8*b*e^4*g + (c*e^
4*f + (c*d*e^3 + b*e^4)*g)*m^3 + (7*c*e^4*f + (3*c*d*e^3 + 7*b*e^4)*g)*m^2 + 2*(7*c*e^4*f + (c*d*e^3 + 7*b*e^4
)*g)*m)*x^3 - (a*d^2*e^2*g + (b*d^2*e^2 - 9*a*d*e^3)*f)*m^2 + (12*b*e^4*f + 12*a*e^4*g + ((c*d*e^3 + b*e^4)*f
+ (b*d*e^3 + a*e^4)*g)*m^3 + ((5*c*d*e^3 + 8*b*e^4)*f - (3*c*d^2*e^2 - 5*b*d*e^3 - 8*a*e^4)*g)*m^2 + ((4*c*d*e
^3 + 19*b*e^4)*f - (3*c*d^2*e^2 - 4*b*d*e^3 - 19*a*e^4)*g)*m)*x^2 + 4*(2*c*d^3*e - 3*b*d^2*e^2 + 6*a*d*e^3)*f
- 2*(3*c*d^4 - 4*b*d^3*e + 6*a*d^2*e^2)*g + ((2*c*d^3*e - 7*b*d^2*e^2 + 26*a*d*e^3)*f + (2*b*d^3*e - 7*a*d^2*e
^2)*g)*m + (24*a*e^4*f + (a*d*e^3*g + (b*d*e^3 + a*e^4)*f)*m^3 - ((2*c*d^2*e^2 - 7*b*d*e^3 - 9*a*e^4)*f + (2*b
*d^2*e^2 - 7*a*d*e^3)*g)*m^2 - 2*((4*c*d^2*e^2 - 6*b*d*e^3 - 13*a*e^4)*f - (3*c*d^3*e - 4*b*d^2*e^2 + 6*a*d*e^
3)*g)*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 = 0.20, size = 1162, normalized size = 8.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c*g*m^3*x^4*e^4 + (x*e + d)^m*c*d*g*m^3*x^3*e^3 + (x*e + d)^m*c*f*m^3*x^3*e^4 + (x*e + d)^m*b*g*m
^3*x^3*e^4 + 6*(x*e + d)^m*c*g*m^2*x^4*e^4 + (x*e + d)^m*c*d*f*m^3*x^2*e^3 + (x*e + d)^m*b*d*g*m^3*x^2*e^3 + 3
*(x*e + d)^m*c*d*g*m^2*x^3*e^3 - 3*(x*e + d)^m*c*d^2*g*m^2*x^2*e^2 + (x*e + d)^m*b*f*m^3*x^2*e^4 + (x*e + d)^m
*a*g*m^3*x^2*e^4 + 7*(x*e + d)^m*c*f*m^2*x^3*e^4 + 7*(x*e + d)^m*b*g*m^2*x^3*e^4 + 11*(x*e + d)^m*c*g*m*x^4*e^
4 + (x*e + d)^m*b*d*f*m^3*x*e^3 + (x*e + d)^m*a*d*g*m^3*x*e^3 + 5*(x*e + d)^m*c*d*f*m^2*x^2*e^3 + 5*(x*e + d)^
m*b*d*g*m^2*x^2*e^3 + 2*(x*e + d)^m*c*d*g*m*x^3*e^3 - 2*(x*e + d)^m*c*d^2*f*m^2*x*e^2 - 2*(x*e + d)^m*b*d^2*g*
m^2*x*e^2 - 3*(x*e + d)^m*c*d^2*g*m*x^2*e^2 + 6*(x*e + d)^m*c*d^3*g*m*x*e + (x*e + d)^m*a*f*m^3*x*e^4 + 8*(x*e
+ d)^m*b*f*m^2*x^2*e^4 + 8*(x*e + d)^m*a*g*m^2*x^2*e^4 + 14*(x*e + d)^m*c*f*m*x^3*e^4 + 14*(x*e + d)^m*b*g*m*
x^3*e^4 + 6*(x*e + d)^m*c*g*x^4*e^4 + (x*e + d)^m*a*d*f*m^3*e^3 + 7*(x*e + d)^m*b*d*f*m^2*x*e^3 + 7*(x*e + d)^
m*a*d*g*m^2*x*e^3 + 4*(x*e + d)^m*c*d*f*m*x^2*e^3 + 4*(x*e + d)^m*b*d*g*m*x^2*e^3 - (x*e + d)^m*b*d^2*f*m^2*e^
2 - (x*e + d)^m*a*d^2*g*m^2*e^2 - 8*(x*e + d)^m*c*d^2*f*m*x*e^2 - 8*(x*e + d)^m*b*d^2*g*m*x*e^2 + 2*(x*e + d)^
m*c*d^3*f*m*e + 2*(x*e + d)^m*b*d^3*g*m*e - 6*(x*e + d)^m*c*d^4*g + 9*(x*e + d)^m*a*f*m^2*x*e^4 + 19*(x*e + d)
^m*b*f*m*x^2*e^4 + 19*(x*e + d)^m*a*g*m*x^2*e^4 + 8*(x*e + d)^m*c*f*x^3*e^4 + 8*(x*e + d)^m*b*g*x^3*e^4 + 9*(x
*e + d)^m*a*d*f*m^2*e^3 + 12*(x*e + d)^m*b*d*f*m*x*e^3 + 12*(x*e + d)^m*a*d*g*m*x*e^3 - 7*(x*e + d)^m*b*d^2*f*
m*e^2 - 7*(x*e + d)^m*a*d^2*g*m*e^2 + 8*(x*e + d)^m*c*d^3*f*e + 8*(x*e + d)^m*b*d^3*g*e + 26*(x*e + d)^m*a*f*m
*x*e^4 + 12*(x*e + d)^m*b*f*x^2*e^4 + 12*(x*e + d)^m*a*g*x^2*e^4 + 26*(x*e + d)^m*a*d*f*m*e^3 - 12*(x*e + d)^m
*b*d^2*f*e^2 - 12*(x*e + d)^m*a*d^2*g*e^2 + 24*(x*e + d)^m*a*f*x*e^4 + 24*(x*e + d)^m*a*d*f*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 = 503, normalized size = 3.49 \begin {gather*} -\frac {\left (-c \,e^{3} g \,m^{3} x^{3}-b \,e^{3} g \,m^{3} x^{2}-c \,e^{3} f \,m^{3} x^{2}-6 c \,e^{3} g \,m^{2} x^{3}-a \,e^{3} g \,m^{3} x -b \,e^{3} f \,m^{3} x -7 b \,e^{3} g \,m^{2} x^{2}+3 c d \,e^{2} g \,m^{2} x^{2}-7 c \,e^{3} f \,m^{2} x^{2}-11 c \,e^{3} g m \,x^{3}-a \,e^{3} f \,m^{3}-8 a \,e^{3} g \,m^{2} x +2 b d \,e^{2} g \,m^{2} x -8 b \,e^{3} f \,m^{2} x -14 b \,e^{3} g m \,x^{2}+2 c d \,e^{2} f \,m^{2} x +9 c d \,e^{2} g m \,x^{2}-14 c \,e^{3} f m \,x^{2}-6 c g \,x^{3} e^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}-19 a \,e^{3} g m x +b d \,e^{2} f \,m^{2}+10 b d \,e^{2} g m x -19 b \,e^{3} f m x -8 b \,e^{3} g \,x^{2}-6 c \,d^{2} e g m x +10 c d \,e^{2} f m x +6 c d \,e^{2} g \,x^{2}-8 c \,e^{3} f \,x^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -12 a \,e^{3} g x -2 b \,d^{2} e g m +7 b d \,e^{2} f m +8 b d \,e^{2} g x -12 b \,e^{3} f x -2 c \,d^{2} e f m -6 c \,d^{2} e g x +8 c d \,e^{2} f x +12 a d \,e^{2} g -24 a \,e^{3} f -8 b \,d^{2} e g +12 b d \,e^{2} f +6 c \,d^{3} g -8 c \,d^{2} e f \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((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a),x)

[Out]

-(e*x+d)^(m+1)*(-c*e^3*g*m^3*x^3-b*e^3*g*m^3*x^2-c*e^3*f*m^3*x^2-6*c*e^3*g*m^2*x^3-a*e^3*g*m^3*x-b*e^3*f*m^3*x
-7*b*e^3*g*m^2*x^2+3*c*d*e^2*g*m^2*x^2-7*c*e^3*f*m^2*x^2-11*c*e^3*g*m*x^3-a*e^3*f*m^3-8*a*e^3*g*m^2*x+2*b*d*e^
2*g*m^2*x-8*b*e^3*f*m^2*x-14*b*e^3*g*m*x^2+2*c*d*e^2*f*m^2*x+9*c*d*e^2*g*m*x^2-14*c*e^3*f*m*x^2-6*c*e^3*g*x^3+
a*d*e^2*g*m^2-9*a*e^3*f*m^2-19*a*e^3*g*m*x+b*d*e^2*f*m^2+10*b*d*e^2*g*m*x-19*b*e^3*f*m*x-8*b*e^3*g*x^2-6*c*d^2
*e*g*m*x+10*c*d*e^2*f*m*x+6*c*d*e^2*g*x^2-8*c*e^3*f*x^2+7*a*d*e^2*g*m-26*a*e^3*f*m-12*a*e^3*g*x-2*b*d^2*e*g*m+
7*b*d*e^2*f*m+8*b*d*e^2*g*x-12*b*e^3*f*x-2*c*d^2*e*f*m-6*c*d^2*e*g*x+8*c*d*e^2*f*x+12*a*d*e^2*g-24*a*e^3*f-8*b
*d^2*e*g+12*b*d*e^2*f+6*c*d^3*g-8*c*d^2*e*f)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [B]  time = 0.52, size = 352, normalized size = 2.44 \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 f}{{\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 g}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a f}{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} c f}{{\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} b g}{{\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} c g}{{\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((e*x+d)^m*(g*x+f)*(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*f/((m^2 + 3*m + 2)*e^2) + (e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e
*x + d)^m*a*g/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a*f/(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*c*f/((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*b*g/((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*c
*g/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4)

________________________________________________________________________________________

mupad [B]  time = 3.59, size = 602, normalized size = 4.18 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (24\,a\,d\,e^3\,f-6\,c\,d^4\,g+8\,b\,d^3\,e\,g+8\,c\,d^3\,e\,f-12\,a\,d^2\,e^2\,g-12\,b\,d^2\,e^2\,f+9\,a\,d\,e^3\,f\,m^2+a\,d\,e^3\,f\,m^3-7\,a\,d^2\,e^2\,g\,m-7\,b\,d^2\,e^2\,f\,m-a\,d^2\,e^2\,g\,m^2-b\,d^2\,e^2\,f\,m^2+26\,a\,d\,e^3\,f\,m+2\,b\,d^3\,e\,g\,m+2\,c\,d^3\,e\,f\,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\,e^4\,f+26\,a\,e^4\,f\,m+9\,a\,e^4\,f\,m^2+a\,e^4\,f\,m^3+7\,a\,d\,e^3\,g\,m^2+7\,b\,d\,e^3\,f\,m^2+a\,d\,e^3\,g\,m^3+b\,d\,e^3\,f\,m^3-8\,b\,d^2\,e^2\,g\,m-8\,c\,d^2\,e^2\,f\,m-2\,b\,d^2\,e^2\,g\,m^2-2\,c\,d^2\,e^2\,f\,m^2+12\,a\,d\,e^3\,g\,m+12\,b\,d\,e^3\,f\,m+6\,c\,d^3\,e\,g\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e^2\,g+12\,b\,e^2\,f+7\,a\,e^2\,g\,m+7\,b\,e^2\,f\,m-3\,c\,d^2\,g\,m+a\,e^2\,g\,m^2+b\,e^2\,f\,m^2+4\,b\,d\,e\,g\,m+4\,c\,d\,e\,f\,m+b\,d\,e\,g\,m^2+c\,d\,e\,f\,m^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {c\,g\,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^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,b\,e\,g+4\,c\,e\,f+b\,e\,g\,m+c\,d\,g\,m+c\,e\,f\,m\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((f + g*x)*(d + e*x)^m*(a + b*x + c*x^2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 5.86, size = 5930, normalized size = 41.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a*f*x + a*g*x**2/2 + b*f*x**2/2 + b*g*x**3/3 + c*f*x**3/3 + c*g*x**4/4), Eq(e, 0)), (-a*d*e**
2*g/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*a*e**3*f/(6*d**3*e**4 + 18*d**2*e**5*x +
18*d*e**6*x**2 + 6*e**7*x**3) - 3*a*e**3*g*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*d**2*e*g/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - b*d*e**2*f/(6*d**3*e**4 + 18*d**2
*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*b*d*e**2*g*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e*
*7*x**3) - 3*b*e**3*f*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*e**3*g*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*c*d**3*g*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*c*d**3*g/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3)
- 2*c*d**2*e*f/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c*d**2*e*g*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*c*d**2*e*g*x/(6*d**3*e**4 + 18*d**2*e**5*x +
18*d*e**6*x**2 + 6*e**7*x**3) - 6*c*d*e**2*f*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) +
18*c*d*e**2*g*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*c*d*e**2*g
*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*c*e**3*f*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*c*e**3*g*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*d*e**2*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - a*e**3*f/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) - 2*a*e**3*g*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*d**2*e*g*log(d/e +
x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*b*d**2*e*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - b*d*e**
2*f/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*g*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*
x**2) + 4*b*d*e**2*g*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*b*e**3*f*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e
**6*x**2) + 2*b*e**3*g*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d**3*g*log(d/e + x)/(2
*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*c*d**3*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c*d**2*e*f*lo
g(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*c*d**2*e*f/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
12*c*d**2*e*g*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c*d**2*e*g*x/(2*d**2*e**4 + 4*d*e**
5*x + 2*e**6*x**2) + 4*c*d*e**2*f*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*c*d*e**2*f*x/(2*
d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d*e**2*g*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2
) + 2*c*e**3*f*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c*e**3*g*x**3/(2*d**2*e**4 + 4*d
*e**5*x + 2*e**6*x**2), Eq(m, -3)), (2*a*d*e**2*g*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*a*d*e**2*g/(2*d*e**4
+ 2*e**5*x) - 2*a*e**3*f/(2*d*e**4 + 2*e**5*x) + 2*a*e**3*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e*
g*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e*g/(2*d*e**4 + 2*e**5*x) + 2*b*d*e**2*f*log(d/e + x)/(2*d*e**
4 + 2*e**5*x) + 2*b*d*e**2*f/(2*d*e**4 + 2*e**5*x) - 4*b*d*e**2*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*b*e
**3*f*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*g*x**2/(2*d*e**4 + 2*e**5*x) + 6*c*d**3*g*log(d/e + x)/(
2*d*e**4 + 2*e**5*x) + 6*c*d**3*g/(2*d*e**4 + 2*e**5*x) - 4*c*d**2*e*f*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*
c*d**2*e*f/(2*d*e**4 + 2*e**5*x) + 6*c*d**2*e*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*c*d*e**2*f*x*log(d/e
+ x)/(2*d*e**4 + 2*e**5*x) - 3*c*d*e**2*g*x**2/(2*d*e**4 + 2*e**5*x) + 2*c*e**3*f*x**2/(2*d*e**4 + 2*e**5*x) +
c*e**3*g*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (-a*d*g*log(d/e + x)/e**2 + a*f*log(d/e + x)/e + a*g*x/e + b
*d**2*g*log(d/e + x)/e**3 - b*d*f*log(d/e + x)/e**2 - b*d*g*x/e**2 + b*f*x/e + b*g*x**2/(2*e) - c*d**3*g*log(d
/e + x)/e**4 + c*d**2*f*log(d/e + x)/e**3 + c*d**2*g*x/e**3 - c*d*f*x/e**2 - c*d*g*x**2/(2*e**2) + c*f*x**2/(2
*e) + c*g*x**3/(3*e), Eq(m, -1)), (-a*d**2*e**2*g*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*d**2*e**2*g*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*d**2*e**2*g*(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*d
*e**3*f*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*d*e**3*f*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*d*e**3*f*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*d*e**3*f*(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*d*e**3*g*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*d*e**3*g*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*d*e**3*g*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 2
4*e**4) + a*e**4*f*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*e
**4*f*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*e**4*f*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*e**4*f*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*e**4*g*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*e**4*g*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*e**4*g*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*e**4*g*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*d**3*e*g*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*
d**3*e*g*(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*d**2*e**2*f*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*d**2*e**2*f*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*d**2*e**2*f*(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*d**2*e**2*g*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*d**2*e**2*g*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*d*e**3*f*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*d*e**3*f*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*d*e**3*f*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*d*e**
3*g*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*d*e**3*g*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*d*e**3*g*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*e**4*f*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*e**4*f*m**2*x**2*(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) + 19*b*e**4*f*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*e**4*f*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*e**4*g*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*e**4*g*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*e**4*g*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*e**4
*g*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*d**4*g*(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*d**3*e*f*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*c*d**3*e*f*(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*d**3*e*g*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*c*d**2*e**2*f*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*c*d**2*e**2*f*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*c*d**2*e**2*g*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*c*d
**2*e**2*g*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) + c*d*e**3*f*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*c*d*e**3*f*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*c*d*e**3*f*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) + c*d*e**3*g*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*c*d*e**3*g*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*c*d*e**3*g*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) + c*e**4*f*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*c*e**4*f*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*c*e**4*f*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*c*e**4*f*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) + c*e**4*g*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*c*e**4*g*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*c*e**4*g*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*c*e**4*g*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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