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

Optimal. Leaf size=111 \[ -\frac {3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac {(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac {3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac {b^3 (d+e x)^{m+4}}{e^4 (m+4)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} -\frac {3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac {(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac {3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac {b^3 (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^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 95, normalized size = 0.86 \begin {gather*} \frac {(d+e x)^{m+1} \left (-\frac {3 b^2 (d+e x)^2 (b d-a e)}{m+3}+\frac {3 b (d+e x) (b d-a e)^2}{m+2}-\frac {(b d-a e)^3}{m+1}+\frac {b^3 (d+e x)^3}{m+4}\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 496, normalized size = 4.47 \begin {gather*} \frac {{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} + {\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} + {\left (24 \, a b^{2} e^{4} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \, {\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \, {\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \, {\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \, {\left (12 \, a^{2} b e^{4} + {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} - {\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} - {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} + {\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m + {\left (24 \, a^{3} e^{4} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \, {\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \, {\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\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*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

(a^3*d*e^3*m^3 - 6*b^3*d^4 + 24*a*b^2*d^3*e - 36*a^2*b*d^2*e^2 + 24*a^3*d*e^3 + (b^3*e^4*m^3 + 6*b^3*e^4*m^2 +
11*b^3*e^4*m + 6*b^3*e^4)*x^4 + (24*a*b^2*e^4 + (b^3*d*e^3 + 3*a*b^2*e^4)*m^3 + 3*(b^3*d*e^3 + 7*a*b^2*e^4)*m
^2 + 2*(b^3*d*e^3 + 21*a*b^2*e^4)*m)*x^3 - 3*(a^2*b*d^2*e^2 - 3*a^3*d*e^3)*m^2 + 3*(12*a^2*b*e^4 + (a*b^2*d*e^
3 + a^2*b*e^4)*m^3 - (b^3*d^2*e^2 - 5*a*b^2*d*e^3 - 8*a^2*b*e^4)*m^2 - (b^3*d^2*e^2 - 4*a*b^2*d*e^3 - 19*a^2*b
*e^4)*m)*x^2 + (6*a*b^2*d^3*e - 21*a^2*b*d^2*e^2 + 26*a^3*d*e^3)*m + (24*a^3*e^4 + (3*a^2*b*d*e^3 + a^3*e^4)*m
^3 - 3*(2*a*b^2*d^2*e^2 - 7*a^2*b*d*e^3 - 3*a^3*e^4)*m^2 + 2*(3*b^3*d^3*e - 12*a*b^2*d^2*e^2 + 18*a^2*b*d*e^3
+ 13*a^3*e^4)*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)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 835, normalized size = 7.52 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{3} m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} b^{3} d m^{3} x^{3} e^{3} + 3 \, {\left (x e + d\right )}^{m} a b^{2} m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{3} m^{2} x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} a b^{2} d m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} b^{3} d m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} b^{3} d^{2} m^{2} x^{2} e^{2} + 3 \, {\left (x e + d\right )}^{m} a^{2} b m^{3} x^{2} e^{4} + 21 \, {\left (x e + d\right )}^{m} a b^{2} m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} b^{3} m x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} a^{2} b d m^{3} x e^{3} + 15 \, {\left (x e + d\right )}^{m} a b^{2} d m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{3} d m x^{3} e^{3} - 6 \, {\left (x e + d\right )}^{m} a b^{2} d^{2} m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} b^{3} d^{2} m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} b^{3} d^{3} m x e + {\left (x e + d\right )}^{m} a^{3} m^{3} x e^{4} + 24 \, {\left (x e + d\right )}^{m} a^{2} b m^{2} x^{2} e^{4} + 42 \, {\left (x e + d\right )}^{m} a b^{2} m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} a^{3} d m^{3} e^{3} + 21 \, {\left (x e + d\right )}^{m} a^{2} b d m^{2} x e^{3} + 12 \, {\left (x e + d\right )}^{m} a b^{2} d m x^{2} e^{3} - 3 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} m^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} a b^{2} d^{2} m x e^{2} + 6 \, {\left (x e + d\right )}^{m} a b^{2} d^{3} m e - 6 \, {\left (x e + d\right )}^{m} b^{3} d^{4} + 9 \, {\left (x e + d\right )}^{m} a^{3} m^{2} x e^{4} + 57 \, {\left (x e + d\right )}^{m} a^{2} b m x^{2} e^{4} + 24 \, {\left (x e + d\right )}^{m} a b^{2} x^{3} e^{4} + 9 \, {\left (x e + d\right )}^{m} a^{3} d m^{2} e^{3} + 36 \, {\left (x e + d\right )}^{m} a^{2} b d m x e^{3} - 21 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} m e^{2} + 24 \, {\left (x e + d\right )}^{m} a b^{2} d^{3} e + 26 \, {\left (x e + d\right )}^{m} a^{3} m x e^{4} + 36 \, {\left (x e + d\right )}^{m} a^{2} b x^{2} e^{4} + 26 \, {\left (x e + d\right )}^{m} a^{3} d m e^{3} - 36 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} a^{3} x e^{4} + 24 \, {\left (x e + d\right )}^{m} a^{3} d e^{3}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*b^3*m^3*x^4*e^4 + (x*e + d)^m*b^3*d*m^3*x^3*e^3 + 3*(x*e + d)^m*a*b^2*m^3*x^3*e^4 + 6*(x*e + d)^m
*b^3*m^2*x^4*e^4 + 3*(x*e + d)^m*a*b^2*d*m^3*x^2*e^3 + 3*(x*e + d)^m*b^3*d*m^2*x^3*e^3 - 3*(x*e + d)^m*b^3*d^2
*m^2*x^2*e^2 + 3*(x*e + d)^m*a^2*b*m^3*x^2*e^4 + 21*(x*e + d)^m*a*b^2*m^2*x^3*e^4 + 11*(x*e + d)^m*b^3*m*x^4*e
^4 + 3*(x*e + d)^m*a^2*b*d*m^3*x*e^3 + 15*(x*e + d)^m*a*b^2*d*m^2*x^2*e^3 + 2*(x*e + d)^m*b^3*d*m*x^3*e^3 - 6*
(x*e + d)^m*a*b^2*d^2*m^2*x*e^2 - 3*(x*e + d)^m*b^3*d^2*m*x^2*e^2 + 6*(x*e + d)^m*b^3*d^3*m*x*e + (x*e + d)^m*
a^3*m^3*x*e^4 + 24*(x*e + d)^m*a^2*b*m^2*x^2*e^4 + 42*(x*e + d)^m*a*b^2*m*x^3*e^4 + 6*(x*e + d)^m*b^3*x^4*e^4
+ (x*e + d)^m*a^3*d*m^3*e^3 + 21*(x*e + d)^m*a^2*b*d*m^2*x*e^3 + 12*(x*e + d)^m*a*b^2*d*m*x^2*e^3 - 3*(x*e + d
)^m*a^2*b*d^2*m^2*e^2 - 24*(x*e + d)^m*a*b^2*d^2*m*x*e^2 + 6*(x*e + d)^m*a*b^2*d^3*m*e - 6*(x*e + d)^m*b^3*d^4
+ 9*(x*e + d)^m*a^3*m^2*x*e^4 + 57*(x*e + d)^m*a^2*b*m*x^2*e^4 + 24*(x*e + d)^m*a*b^2*x^3*e^4 + 9*(x*e + d)^m
*a^3*d*m^2*e^3 + 36*(x*e + d)^m*a^2*b*d*m*x*e^3 - 21*(x*e + d)^m*a^2*b*d^2*m*e^2 + 24*(x*e + d)^m*a*b^2*d^3*e
+ 26*(x*e + d)^m*a^3*m*x*e^4 + 36*(x*e + d)^m*a^2*b*x^2*e^4 + 26*(x*e + d)^m*a^3*d*m*e^3 - 36*(x*e + d)^m*a^2*
b*d^2*e^2 + 24*(x*e + d)^m*a^3*x*e^4 + 24*(x*e + d)^m*a^3*d*e^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4
+ 24*e^4)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 386, normalized size = 3.48 \begin {gather*} \frac {\left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 b^{3} e^{3} x^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e m x -6 b^{3} d \,e^{2} x^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 a \,b^{2} d \,e^{2} x +6 b^{3} d^{2} e x +24 a^{3} e^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} 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*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

(e*x+d)^(m+1)*(b^3*e^3*m^3*x^3+3*a*b^2*e^3*m^3*x^2+6*b^3*e^3*m^2*x^3+3*a^2*b*e^3*m^3*x+21*a*b^2*e^3*m^2*x^2-3*
b^3*d*e^2*m^2*x^2+11*b^3*e^3*m*x^3+a^3*e^3*m^3+24*a^2*b*e^3*m^2*x-6*a*b^2*d*e^2*m^2*x+42*a*b^2*e^3*m*x^2-9*b^3
*d*e^2*m*x^2+6*b^3*e^3*x^3+9*a^3*e^3*m^2-3*a^2*b*d*e^2*m^2+57*a^2*b*e^3*m*x-30*a*b^2*d*e^2*m*x+24*a*b^2*e^3*x^
2+6*b^3*d^2*e*m*x-6*b^3*d*e^2*x^2+26*a^3*e^3*m-21*a^2*b*d*e^2*m+36*a^2*b*e^3*x+6*a*b^2*d^2*e*m-24*a*b^2*d*e^2*
x+6*b^3*d^2*e*x+24*a^3*e^3-36*a^2*b*d*e^2+24*a*b^2*d^2*e-6*b^3*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

________________________________________________________________________________________

maxima [B]  time = 0.66, size = 246, normalized size = 2.22 \begin {gather*} \frac {3 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a b^{2}}{{\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^{3}}{{\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*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.43, size = 478, normalized size = 4.31 \begin {gather*} \frac {x\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^4\,m^3+9\,a^3\,e^4\,m^2+26\,a^3\,e^4\,m+24\,a^3\,e^4+3\,a^2\,b\,d\,e^3\,m^3+21\,a^2\,b\,d\,e^3\,m^2+36\,a^2\,b\,d\,e^3\,m-6\,a\,b^2\,d^2\,e^2\,m^2-24\,a\,b^2\,d^2\,e^2\,m+6\,b^3\,d^3\,e\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b^3\,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\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^3\,m^3+9\,a^3\,e^3\,m^2+26\,a^3\,e^3\,m+24\,a^3\,e^3-3\,a^2\,b\,d\,e^2\,m^2-21\,a^2\,b\,d\,e^2\,m-36\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e\,m+24\,a\,b^2\,d^2\,e-6\,b^3\,d^3\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^2\,m^2+7\,a^2\,e^2\,m+12\,a^2\,e^2+a\,b\,d\,e\,m^2+4\,a\,b\,d\,e\,m-b^2\,d^2\,m\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e+3\,a\,e\,m+b\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(d + e*x)^m*(24*a^3*e^4 + 26*a^3*e^4*m + 9*a^3*e^4*m^2 + a^3*e^4*m^3 + 6*b^3*d^3*e*m + 36*a^2*b*d*e^3*m - 2
4*a*b^2*d^2*e^2*m + 21*a^2*b*d*e^3*m^2 + 3*a^2*b*d*e^3*m^3 - 6*a*b^2*d^2*e^2*m^2))/(e^4*(50*m + 35*m^2 + 10*m^
3 + m^4 + 24)) + (b^3*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*(d +
e*x)^m*(24*a^3*e^3 - 6*b^3*d^3 + 26*a^3*e^3*m + 9*a^3*e^3*m^2 + a^3*e^3*m^3 + 24*a*b^2*d^2*e - 36*a^2*b*d*e^2
+ 6*a*b^2*d^2*e*m - 21*a^2*b*d*e^2*m - 3*a^2*b*d*e^2*m^2))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (3*b*x^
2*(m + 1)*(d + e*x)^m*(12*a^2*e^2 + 7*a^2*e^2*m - b^2*d^2*m + a^2*e^2*m^2 + 4*a*b*d*e*m + a*b*d*e*m^2))/(e^2*(
50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b^2*x^3*(d + e*x)^m*(12*a*e + 3*a*e*m + b*d*m)*(3*m + m^2 + 2))/(e*(50*
m + 35*m^2 + 10*m^3 + m^4 + 24))

________________________________________________________________________________________

sympy [A]  time = 4.73, size = 4058, normalized size = 36.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(e, 0)), (-2*a**3*e**3/(6*d**3*e**4
+ 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 3*a**2*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) - 9*a**2*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) - 6*a*b**2*
d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*a*b**2*d*e**2*x/(6*d**3*e**4 + 18*d*
*2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*a*b**2*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*e**3/(2*
d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 3*a**2*b*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*a**2*b*e*
*3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*b**2*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**
6*x**2) + 9*a*b**2*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*b**2*d*e**2*x*log(d/e + x)/(2*d**2*e
**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*b**2*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*b**2*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**3*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) - 9*b**3*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*b**3*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**3*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*b**3*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**3*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e
**6*x**2), Eq(m, -3)), (-2*a**3*e**3/(2*d*e**4 + 2*e**5*x) + 6*a**2*b*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x
) + 6*a**2*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 6*a**2*b*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 12*a*b**2*d**
2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 12*a*b**2*d**2*e/(2*d*e**4 + 2*e**5*x) - 12*a*b**2*d*e**2*x*log(d/e +
x)/(2*d*e**4 + 2*e**5*x) + 6*a*b**2*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*b**3*d**3*log(d/e + x)/(2*d*e**4 + 2*
e**5*x) + 6*b**3*d**3/(2*d*e**4 + 2*e**5*x) + 6*b**3*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*b**3*d*e*
*2*x**2/(2*d*e**4 + 2*e**5*x) + b**3*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (a**3*log(d/e + x)/e - 3*a**
2*b*d*log(d/e + x)/e**2 + 3*a**2*b*x/e + 3*a*b**2*d**2*log(d/e + x)/e**3 - 3*a*b**2*d*x/e**2 + 3*a*b**2*x**2/(
2*e) - b**3*d**3*log(d/e + x)/e**4 + b**3*d**2*x/e**3 - b**3*d*x**2/(2*e**2) + b**3*x**3/(3*e), Eq(m, -1)), (a
**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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) - 3*a
**2*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) - 21*a**2*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) - 36*a**2*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) + 3*a**2*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) + 21*a**2*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) + 36*a**2*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) + 3*a**2*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) + 24*a**2*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) + 57*a**2*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) + 36*a**2*b*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + 6*a*b**2*d**3*e*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24
*e**4) + 24*a*b**2*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*a*b
**2*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) - 24*a*b**2*
d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*a*b**2*d*e**3*m
**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 15*a*b**2*d*e**3*m**2*
x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*a*b**2*d*e**3*m*x**2*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*a*b**2*e**4*m**3*x**3*(d + e*x)
**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 21*a*b**2*e**4*m**2*x**3*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 42*a*b**2*e**4*m*x**3*(d + e*x)**m/(e**4*m**4
+ 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*a*b**2*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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))

________________________________________________________________________________________