3.6.55 \(\int (d+e x) (f+g x)^n (a+2 c d x+c e x^2) \, dx\)

Optimal. Leaf size=146 \[ \frac {(f+g x)^{n+2} \left (a e g^2+c \left (2 d^2 g^2-6 d e f g+3 e^2 f^2\right )\right )}{g^4 (n+2)}-\frac {(e f-d g) (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^4 (n+1)}-\frac {3 c e (e f-d g) (f+g x)^{n+3}}{g^4 (n+3)}+\frac {c e^2 (f+g x)^{n+4}}{g^4 (n+4)} \]

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Rubi [A]  time = 0.11, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {(f+g x)^{n+2} \left (a e g^2+c \left (2 d^2 g^2-6 d e f g+3 e^2 f^2\right )\right )}{g^4 (n+2)}-\frac {(e f-d g) (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^4 (n+1)}-\frac {3 c e (e f-d g) (f+g x)^{n+3}}{g^4 (n+3)}+\frac {c e^2 (f+g x)^{n+4}}{g^4 (n+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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) (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\int \left (\frac {(e f-d g) \left (-a g^2-c f (e f-2 d g)\right ) (f+g x)^n}{g^3}+\frac {\left (a e g^2+c \left (3 e^2 f^2-6 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{1+n}}{g^3}-\frac {3 c e (e f-d g) (f+g x)^{2+n}}{g^3}+\frac {c e^2 (f+g x)^{3+n}}{g^3}\right ) \, dx\\ &=-\frac {(e f-d g) \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^4 (1+n)}+\frac {\left (a e g^2+c \left (3 e^2 f^2-6 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^4 (2+n)}-\frac {3 c e (e f-d g) (f+g x)^{3+n}}{g^4 (3+n)}+\frac {c e^2 (f+g x)^{4+n}}{g^4 (4+n)}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 141, normalized size = 0.97 \begin {gather*} \frac {(f+g x)^{n+1} \left (\frac {2 (f+g x) \left (a e g^2 (n+3)+c \left (-d^2 g^2 n-6 d e f g+3 e^2 f^2\right )\right )}{g^2 (n+2)}+\frac {6 (d g-e f) \left (a g^2+c f (e f-2 d g)\right )}{g^2 (n+1)}+(a+c x (2 d+e x)) (d g (n+6)-3 e f+e g (n+3) x)\right )}{g^2 (n+3) (n+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 549, normalized size = 3.76 \begin {gather*} \frac {{\left (a d f g^{3} n^{3} - 6 \, c e^{2} f^{4} + 24 \, c d e f^{3} g + 24 \, a d f g^{3} - 12 \, {\left (2 \, c d^{2} + a e\right )} f^{2} g^{2} + {\left (c e^{2} g^{4} n^{3} + 6 \, c e^{2} g^{4} n^{2} + 11 \, c e^{2} g^{4} n + 6 \, c e^{2} g^{4}\right )} x^{4} + {\left (24 \, c d e g^{4} + {\left (c e^{2} f g^{3} + 3 \, c d e g^{4}\right )} n^{3} + 3 \, {\left (c e^{2} f g^{3} + 7 \, c d e g^{4}\right )} n^{2} + 2 \, {\left (c e^{2} f g^{3} + 21 \, c d e g^{4}\right )} n\right )} x^{3} + {\left (9 \, a d f g^{3} - {\left (2 \, c d^{2} + a e\right )} f^{2} g^{2}\right )} n^{2} + {\left (12 \, {\left (2 \, c d^{2} + a e\right )} g^{4} + {\left (3 \, c d e f g^{3} + {\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n^{3} - {\left (3 \, c e^{2} f^{2} g^{2} - 15 \, c d e f g^{3} - 8 \, {\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n^{2} - {\left (3 \, c e^{2} f^{2} g^{2} - 12 \, c d e f g^{3} - 19 \, {\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n\right )} x^{2} + {\left (6 \, c d e f^{3} g + 26 \, a d f g^{3} - 7 \, {\left (2 \, c d^{2} + a e\right )} f^{2} g^{2}\right )} n + {\left (24 \, a d g^{4} + {\left (a d g^{4} + {\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n^{3} - {\left (6 \, c d e f^{2} g^{2} - 9 \, a d g^{4} - 7 \, {\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n^{2} + 2 \, {\left (3 \, c e^{2} f^{3} g - 12 \, c d e f^{2} g^{2} + 13 \, a d g^{4} + 6 \, {\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n\right )} x\right )} {\left (g x + f\right )}^{n}}{g^{4} n^{4} + 10 \, g^{4} n^{3} + 35 \, g^{4} n^{2} + 50 \, g^{4} n + 24 \, g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.39, size = 1018, normalized size = 6.97 \begin {gather*} \frac {{\left (g x + f\right )}^{n} c g^{4} n^{3} x^{4} e^{2} + 3 \, {\left (g x + f\right )}^{n} c d g^{4} n^{3} x^{3} e + 2 \, {\left (g x + f\right )}^{n} c d^{2} g^{4} n^{3} x^{2} + {\left (g x + f\right )}^{n} c f g^{3} n^{3} x^{3} e^{2} + 6 \, {\left (g x + f\right )}^{n} c g^{4} n^{2} x^{4} e^{2} + 3 \, {\left (g x + f\right )}^{n} c d f g^{3} n^{3} x^{2} e + 21 \, {\left (g x + f\right )}^{n} c d g^{4} n^{2} x^{3} e + 2 \, {\left (g x + f\right )}^{n} c d^{2} f g^{3} n^{3} x + 16 \, {\left (g x + f\right )}^{n} c d^{2} g^{4} n^{2} x^{2} + 3 \, {\left (g x + f\right )}^{n} c f g^{3} n^{2} x^{3} e^{2} + 11 \, {\left (g x + f\right )}^{n} c g^{4} n x^{4} e^{2} + 15 \, {\left (g x + f\right )}^{n} c d f g^{3} n^{2} x^{2} e + {\left (g x + f\right )}^{n} a g^{4} n^{3} x^{2} e + 42 \, {\left (g x + f\right )}^{n} c d g^{4} n x^{3} e + 14 \, {\left (g x + f\right )}^{n} c d^{2} f g^{3} n^{2} x + {\left (g x + f\right )}^{n} a d g^{4} n^{3} x + 38 \, {\left (g x + f\right )}^{n} c d^{2} g^{4} n x^{2} - 3 \, {\left (g x + f\right )}^{n} c f^{2} g^{2} n^{2} x^{2} e^{2} + 2 \, {\left (g x + f\right )}^{n} c f g^{3} n x^{3} e^{2} + 6 \, {\left (g x + f\right )}^{n} c g^{4} x^{4} e^{2} - 6 \, {\left (g x + f\right )}^{n} c d f^{2} g^{2} n^{2} x e + {\left (g x + f\right )}^{n} a f g^{3} n^{3} x e + 12 \, {\left (g x + f\right )}^{n} c d f g^{3} n x^{2} e + 8 \, {\left (g x + f\right )}^{n} a g^{4} n^{2} x^{2} e + 24 \, {\left (g x + f\right )}^{n} c d g^{4} x^{3} e - 2 \, {\left (g x + f\right )}^{n} c d^{2} f^{2} g^{2} n^{2} + {\left (g x + f\right )}^{n} a d f g^{3} n^{3} + 24 \, {\left (g x + f\right )}^{n} c d^{2} f g^{3} n x + 9 \, {\left (g x + f\right )}^{n} a d g^{4} n^{2} x + 24 \, {\left (g x + f\right )}^{n} c d^{2} g^{4} x^{2} - 3 \, {\left (g x + f\right )}^{n} c f^{2} g^{2} n x^{2} e^{2} - 24 \, {\left (g x + f\right )}^{n} c d f^{2} g^{2} n x e + 7 \, {\left (g x + f\right )}^{n} a f g^{3} n^{2} x e + 19 \, {\left (g x + f\right )}^{n} a g^{4} n x^{2} e - 14 \, {\left (g x + f\right )}^{n} c d^{2} f^{2} g^{2} n + 9 \, {\left (g x + f\right )}^{n} a d f g^{3} n^{2} + 26 \, {\left (g x + f\right )}^{n} a d g^{4} n x + 6 \, {\left (g x + f\right )}^{n} c f^{3} g n x e^{2} + 6 \, {\left (g x + f\right )}^{n} c d f^{3} g n e - {\left (g x + f\right )}^{n} a f^{2} g^{2} n^{2} e + 12 \, {\left (g x + f\right )}^{n} a f g^{3} n x e + 12 \, {\left (g x + f\right )}^{n} a g^{4} x^{2} e - 24 \, {\left (g x + f\right )}^{n} c d^{2} f^{2} g^{2} + 26 \, {\left (g x + f\right )}^{n} a d f g^{3} n + 24 \, {\left (g x + f\right )}^{n} a d g^{4} x + 24 \, {\left (g x + f\right )}^{n} c d f^{3} g e - 7 \, {\left (g x + f\right )}^{n} a f^{2} g^{2} n e + 24 \, {\left (g x + f\right )}^{n} a d f g^{3} - 6 \, {\left (g x + f\right )}^{n} c f^{4} e^{2} - 12 \, {\left (g x + f\right )}^{n} a f^{2} g^{2} e}{g^{4} n^{4} + 10 \, g^{4} n^{3} + 35 \, g^{4} n^{2} + 50 \, g^{4} n + 24 \, g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((g*x + f)^n*c*g^4*n^3*x^4*e^2 + 3*(g*x + f)^n*c*d*g^4*n^3*x^3*e + 2*(g*x + f)^n*c*d^2*g^4*n^3*x^2 + (g*x + f)
^n*c*f*g^3*n^3*x^3*e^2 + 6*(g*x + f)^n*c*g^4*n^2*x^4*e^2 + 3*(g*x + f)^n*c*d*f*g^3*n^3*x^2*e + 21*(g*x + f)^n*
c*d*g^4*n^2*x^3*e + 2*(g*x + f)^n*c*d^2*f*g^3*n^3*x + 16*(g*x + f)^n*c*d^2*g^4*n^2*x^2 + 3*(g*x + f)^n*c*f*g^3
*n^2*x^3*e^2 + 11*(g*x + f)^n*c*g^4*n*x^4*e^2 + 15*(g*x + f)^n*c*d*f*g^3*n^2*x^2*e + (g*x + f)^n*a*g^4*n^3*x^2
*e + 42*(g*x + f)^n*c*d*g^4*n*x^3*e + 14*(g*x + f)^n*c*d^2*f*g^3*n^2*x + (g*x + f)^n*a*d*g^4*n^3*x + 38*(g*x +
f)^n*c*d^2*g^4*n*x^2 - 3*(g*x + f)^n*c*f^2*g^2*n^2*x^2*e^2 + 2*(g*x + f)^n*c*f*g^3*n*x^3*e^2 + 6*(g*x + f)^n*
c*g^4*x^4*e^2 - 6*(g*x + f)^n*c*d*f^2*g^2*n^2*x*e + (g*x + f)^n*a*f*g^3*n^3*x*e + 12*(g*x + f)^n*c*d*f*g^3*n*x
^2*e + 8*(g*x + f)^n*a*g^4*n^2*x^2*e + 24*(g*x + f)^n*c*d*g^4*x^3*e - 2*(g*x + f)^n*c*d^2*f^2*g^2*n^2 + (g*x +
f)^n*a*d*f*g^3*n^3 + 24*(g*x + f)^n*c*d^2*f*g^3*n*x + 9*(g*x + f)^n*a*d*g^4*n^2*x + 24*(g*x + f)^n*c*d^2*g^4*
x^2 - 3*(g*x + f)^n*c*f^2*g^2*n*x^2*e^2 - 24*(g*x + f)^n*c*d*f^2*g^2*n*x*e + 7*(g*x + f)^n*a*f*g^3*n^2*x*e + 1
9*(g*x + f)^n*a*g^4*n*x^2*e - 14*(g*x + f)^n*c*d^2*f^2*g^2*n + 9*(g*x + f)^n*a*d*f*g^3*n^2 + 26*(g*x + f)^n*a*
d*g^4*n*x + 6*(g*x + f)^n*c*f^3*g*n*x*e^2 + 6*(g*x + f)^n*c*d*f^3*g*n*e - (g*x + f)^n*a*f^2*g^2*n^2*e + 12*(g*
x + f)^n*a*f*g^3*n*x*e + 12*(g*x + f)^n*a*g^4*x^2*e - 24*(g*x + f)^n*c*d^2*f^2*g^2 + 26*(g*x + f)^n*a*d*f*g^3*
n + 24*(g*x + f)^n*a*d*g^4*x + 24*(g*x + f)^n*c*d*f^3*g*e - 7*(g*x + f)^n*a*f^2*g^2*n*e + 24*(g*x + f)^n*a*d*f
*g^3 - 6*(g*x + f)^n*c*f^4*e^2 - 12*(g*x + f)^n*a*f^2*g^2*e)/(g^4*n^4 + 10*g^4*n^3 + 35*g^4*n^2 + 50*g^4*n + 2
4*g^4)

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maple [B]  time = 0.01, size = 449, normalized size = 3.08 \begin {gather*} \frac {\left (c \,e^{2} g^{3} n^{3} x^{3}+3 c d e \,g^{3} n^{3} x^{2}+6 c \,e^{2} g^{3} n^{2} x^{3}+2 c \,d^{2} g^{3} n^{3} x +21 c d e \,g^{3} n^{2} x^{2}-3 c \,e^{2} f \,g^{2} n^{2} x^{2}+11 c \,e^{2} g^{3} n \,x^{3}+a e \,g^{3} n^{3} x +16 c \,d^{2} g^{3} n^{2} x -6 c d e f \,g^{2} n^{2} x +42 c d e \,g^{3} n \,x^{2}-9 c \,e^{2} f \,g^{2} n \,x^{2}+6 c \,e^{2} x^{3} g^{3}+a d \,g^{3} n^{3}+8 a e \,g^{3} n^{2} x -2 c \,d^{2} f \,g^{2} n^{2}+38 c \,d^{2} g^{3} n x -30 c d e f \,g^{2} n x +24 c d e \,g^{3} x^{2}+6 c \,e^{2} f^{2} g n x -6 c \,e^{2} f \,g^{2} x^{2}+9 a d \,g^{3} n^{2}-a e f \,g^{2} n^{2}+19 a e \,g^{3} n x -14 c \,d^{2} f \,g^{2} n +24 c \,d^{2} g^{3} x +6 c d e \,f^{2} g n -24 c d e f \,g^{2} x +6 c \,e^{2} f^{2} g x +26 a d \,g^{3} n -7 a e f \,g^{2} n +12 a e \,g^{3} x -24 c \,d^{2} f \,g^{2}+24 c d e \,f^{2} g -6 c \,e^{2} f^{3}+24 a d \,g^{3}-12 a e f \,g^{2}\right ) \left (g x +f \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)

[Out]

(g*x+f)^(n+1)*(c*e^2*g^3*n^3*x^3+3*c*d*e*g^3*n^3*x^2+6*c*e^2*g^3*n^2*x^3+2*c*d^2*g^3*n^3*x+21*c*d*e*g^3*n^2*x^
2-3*c*e^2*f*g^2*n^2*x^2+11*c*e^2*g^3*n*x^3+a*e*g^3*n^3*x+16*c*d^2*g^3*n^2*x-6*c*d*e*f*g^2*n^2*x+42*c*d*e*g^3*n
*x^2-9*c*e^2*f*g^2*n*x^2+6*c*e^2*g^3*x^3+a*d*g^3*n^3+8*a*e*g^3*n^2*x-2*c*d^2*f*g^2*n^2+38*c*d^2*g^3*n*x-30*c*d
*e*f*g^2*n*x+24*c*d*e*g^3*x^2+6*c*e^2*f^2*g*n*x-6*c*e^2*f*g^2*x^2+9*a*d*g^3*n^2-a*e*f*g^2*n^2+19*a*e*g^3*n*x-1
4*c*d^2*f*g^2*n+24*c*d^2*g^3*x+6*c*d*e*f^2*g*n-24*c*d*e*f*g^2*x+6*c*e^2*f^2*g*x+26*a*d*g^3*n-7*a*e*f*g^2*n+12*
a*e*g^3*x-24*c*d^2*f*g^2+24*c*d*e*f^2*g-6*c*e^2*f^3+24*a*d*g^3-12*a*e*f*g^2)/g^4/(n^4+10*n^3+35*n^2+50*n+24)

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maxima [A]  time = 0.51, size = 289, normalized size = 1.98 \begin {gather*} \frac {2 \, {\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} c d^{2}}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} g^{3} x^{3} + {\left (n^{2} + n\right )} f g^{2} x^{2} - 2 \, f^{2} g n x + 2 \, f^{3}\right )} {\left (g x + f\right )}^{n} c d e}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {{\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} a e}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {{\left (g x + f\right )}^{n + 1} a d}{g {\left (n + 1\right )}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} f g^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} f^{2} g^{2} x^{2} + 6 \, f^{3} g n x - 6 \, f^{4}\right )} {\left (g x + f\right )}^{n} c e^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.29, size = 572, normalized size = 3.92 \begin {gather*} \frac {x\,{\left (f+g\,x\right )}^n\,\left (2\,c\,d^2\,f\,g^3\,n^3+14\,c\,d^2\,f\,g^3\,n^2+24\,c\,d^2\,f\,g^3\,n-6\,c\,d\,e\,f^2\,g^2\,n^2-24\,c\,d\,e\,f^2\,g^2\,n+a\,d\,g^4\,n^3+9\,a\,d\,g^4\,n^2+26\,a\,d\,g^4\,n+24\,a\,d\,g^4+6\,c\,e^2\,f^3\,g\,n+a\,e\,f\,g^3\,n^3+7\,a\,e\,f\,g^3\,n^2+12\,a\,e\,f\,g^3\,n\right )}{g^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {{\left (f+g\,x\right )}^n\,\left (2\,c\,d^2\,f^2\,g^2\,n^2+14\,c\,d^2\,f^2\,g^2\,n+24\,c\,d^2\,f^2\,g^2-6\,c\,d\,e\,f^3\,g\,n-24\,c\,d\,e\,f^3\,g-a\,d\,f\,g^3\,n^3-9\,a\,d\,f\,g^3\,n^2-26\,a\,d\,f\,g^3\,n-24\,a\,d\,f\,g^3+6\,c\,e^2\,f^4+a\,e\,f^2\,g^2\,n^2+7\,a\,e\,f^2\,g^2\,n+12\,a\,e\,f^2\,g^2\right )}{g^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {c\,e^2\,x^4\,{\left (f+g\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {x^2\,{\left (f+g\,x\right )}^n\,\left (n+1\right )\,\left (2\,c\,d^2\,g^2\,n^2+14\,c\,d^2\,g^2\,n+24\,c\,d^2\,g^2+3\,c\,d\,e\,f\,g\,n^2+12\,c\,d\,e\,f\,g\,n-3\,c\,e^2\,f^2\,n+a\,e\,g^2\,n^2+7\,a\,e\,g^2\,n+12\,a\,e\,g^2\right )}{g^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {c\,e\,x^3\,{\left (f+g\,x\right )}^n\,\left (12\,d\,g+3\,d\,g\,n+e\,f\,n\right )\,\left (n^2+3\,n+2\right )}{g\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^n*(d + e*x)*(a + 2*c*d*x + c*e*x^2),x)

[Out]

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

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sympy [A]  time = 5.27, size = 4952, normalized size = 33.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((f**n*(a*d*x + a*e*x**2/2 + c*d**2*x**2 + c*d*e*x**3 + c*e**2*x**4/4), Eq(g, 0)), (-2*a*d*g**3/(6*f*
*3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) - a*e*f*g**2/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**
6*x**2 + 6*g**7*x**3) - 3*a*e*g**3*x/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) - 2*c*d**2*
f*g**2/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) - 6*c*d**2*g**3*x/(6*f**3*g**4 + 18*f**2*
g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) - 6*c*d*e*f**2*g/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**
7*x**3) - 18*c*d*e*f*g**2*x/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) - 18*c*d*e*g**3*x**2
/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) + 6*c*e**2*f**3*log(f/g + x)/(6*f**3*g**4 + 18*
f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) + 11*c*e**2*f**3/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 +
6*g**7*x**3) + 18*c*e**2*f**2*g*x*log(f/g + x)/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) +
27*c*e**2*f**2*g*x/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) + 18*c*e**2*f*g**2*x**2*log(
f/g + x)/(6*f**3*g**4 + 18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) + 18*c*e**2*f*g**2*x**2/(6*f**3*g**4 +
18*f**2*g**5*x + 18*f*g**6*x**2 + 6*g**7*x**3) + 6*c*e**2*g**3*x**3*log(f/g + x)/(6*f**3*g**4 + 18*f**2*g**5*x
+ 18*f*g**6*x**2 + 6*g**7*x**3), Eq(n, -4)), (-a*d*g**3/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - a*e*f*g**2
/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 2*a*e*g**3*x/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 2*c*d**2*f
*g**2/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 4*c*d**2*g**3*x/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) + 6*
c*d*e*f**2*g*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) + 9*c*d*e*f**2*g/(2*f**2*g**4 + 4*f*g**5*x
+ 2*g**6*x**2) + 12*c*d*e*f*g**2*x*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) + 12*c*d*e*f*g**2*x/(
2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) + 6*c*d*e*g**3*x**2*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x*
*2) - 6*c*e**2*f**3*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 9*c*e**2*f**3/(2*f**2*g**4 + 4*f*g
**5*x + 2*g**6*x**2) - 12*c*e**2*f**2*g*x*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 12*c*e**2*f*
*2*g*x/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2) - 6*c*e**2*f*g**2*x**2*log(f/g + x)/(2*f**2*g**4 + 4*f*g**5*x
+ 2*g**6*x**2) + 2*c*e**2*g**3*x**3/(2*f**2*g**4 + 4*f*g**5*x + 2*g**6*x**2), Eq(n, -3)), (-2*a*d*g**3/(2*f*g*
*4 + 2*g**5*x) + 2*a*e*f*g**2*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 2*a*e*f*g**2/(2*f*g**4 + 2*g**5*x) + 2*a*e*
g**3*x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*f*g**2*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*f*g*
*2/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*g**3*x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) - 12*c*d*e*f**2*g*log(f/g + x)/(
2*f*g**4 + 2*g**5*x) - 12*c*d*e*f**2*g/(2*f*g**4 + 2*g**5*x) - 12*c*d*e*f*g**2*x*log(f/g + x)/(2*f*g**4 + 2*g*
*5*x) + 6*c*d*e*g**3*x**2/(2*f*g**4 + 2*g**5*x) + 6*c*e**2*f**3*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 6*c*e**2*
f**3/(2*f*g**4 + 2*g**5*x) + 6*c*e**2*f**2*g*x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) - 3*c*e**2*f*g**2*x**2/(2*f*
g**4 + 2*g**5*x) + c*e**2*g**3*x**3/(2*f*g**4 + 2*g**5*x), Eq(n, -2)), (a*d*log(f/g + x)/g - a*e*f*log(f/g + x
)/g**2 + a*e*x/g - 2*c*d**2*f*log(f/g + x)/g**2 + 2*c*d**2*x/g + 3*c*d*e*f**2*log(f/g + x)/g**3 - 3*c*d*e*f*x/
g**2 + 3*c*d*e*x**2/(2*g) - c*e**2*f**3*log(f/g + x)/g**4 + c*e**2*f**2*x/g**3 - c*e**2*f*x**2/(2*g**2) + c*e*
*2*x**3/(3*g), Eq(n, -1)), (a*d*f*g**3*n**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n
+ 24*g**4) + 9*a*d*f*g**3*n**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) +
26*a*d*f*g**3*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*a*d*f*g**3*(
f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + a*d*g**4*n**3*x*(f + g*x)**n/(g*
*4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 9*a*d*g**4*n**2*x*(f + g*x)**n/(g**4*n**4 + 10*
g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 26*a*d*g**4*n*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*
g**4*n**2 + 50*g**4*n + 24*g**4) + 24*a*d*g**4*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g*
*4*n + 24*g**4) - a*e*f**2*g**2*n**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g*
*4) - 7*a*e*f**2*g**2*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 12*a*e*
f**2*g**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + a*e*f*g**3*n**3*x*(f
+ g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 7*a*e*f*g**3*n**2*x*(f + g*x)**n/(
g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 12*a*e*f*g**3*n*x*(f + g*x)**n/(g**4*n**4 + 1
0*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + a*e*g**4*n**3*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3
+ 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 8*a*e*g**4*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*
n**2 + 50*g**4*n + 24*g**4) + 19*a*e*g**4*n*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g*
*4*n + 24*g**4) + 12*a*e*g**4*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4
) - 2*c*d**2*f**2*g**2*n**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 14*
c*d**2*f**2*g**2*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 24*c*d**2*f*
*2*g**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 2*c*d**2*f*g**3*n**3*x*
(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 14*c*d**2*f*g**3*n**2*x*(f + g*
x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d**2*f*g**3*n*x*(f + g*x)**n/(g**
4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 2*c*d**2*g**4*n**3*x**2*(f + g*x)**n/(g**4*n**4
+ 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 16*c*d**2*g**4*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g
**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 38*c*d**2*g**4*n*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3
+ 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d**2*g**4*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n
**2 + 50*g**4*n + 24*g**4) + 6*c*d*e*f**3*g*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*
n + 24*g**4) + 24*c*d*e*f**3*g*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) -
6*c*d*e*f**2*g**2*n**2*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 24*c*d
*e*f**2*g**2*n*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 3*c*d*e*f*g**3
*n**3*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 15*c*d*e*f*g**3*n**2
*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 12*c*d*e*f*g**3*n*x**2*(f
+ g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 3*c*d*e*g**4*n**3*x**3*(f + g*x)*
*n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 21*c*d*e*g**4*n**2*x**3*(f + g*x)**n/(g**
4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 42*c*d*e*g**4*n*x**3*(f + g*x)**n/(g**4*n**4 + 1
0*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d*e*g**4*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3
+ 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 6*c*e**2*f**4*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 +
50*g**4*n + 24*g**4) + 6*c*e**2*f**3*g*n*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n +
24*g**4) - 3*c*e**2*f**2*g**2*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 2
4*g**4) - 3*c*e**2*f**2*g**2*n*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**
4) + c*e**2*f*g**3*n**3*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 3*
c*e**2*f*g**3*n**2*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 2*c*e**
2*f*g**3*n*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + c*e**2*g**4*n**
3*x**4*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*e**2*g**4*n**2*x**4*
(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 11*c*e**2*g**4*n*x**4*(f + g*x)
**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*e**2*g**4*x**4*(f + g*x)**n/(g**4*n*
*4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4), True))

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