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

Optimal. Leaf size=175 \[ -\frac {5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}-\frac {(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac {b^5 (d+e x)^{m+6}}{e^6 (m+6)} \]

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Rubi [A]  time = 0.09, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac {5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}-\frac {(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac {b^5 (d+e x)^{m+6}}{e^6 (m+6)} \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)^2,x]

[Out]

-(((b*d - a*e)^5*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (10
*b^2*(b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m))/(e^6*(4 + m)) -
(5*b^4*(b*d - a*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + 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 )^2 \, dx &=\int (a+b x)^5 (d+e x)^m \, dx\\ &=\int \left (\frac {(-b d+a e)^5 (d+e x)^m}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{1+m}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{2+m}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{3+m}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{4+m}}{e^5}+\frac {b^5 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac {(b d-a e)^5 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m}}{e^6 (2+m)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m}}{e^6 (3+m)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m}}{e^6 (4+m)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {b^5 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 149, normalized size = 0.85 \begin {gather*} \frac {(d+e x)^{m+1} \left (-\frac {5 b^4 (d+e x)^4 (b d-a e)}{m+5}+\frac {10 b^3 (d+e x)^3 (b d-a e)^2}{m+4}-\frac {10 b^2 (d+e x)^2 (b d-a e)^3}{m+3}+\frac {5 b (d+e x) (b d-a e)^4}{m+2}-\frac {(b d-a e)^5}{m+1}+\frac {b^5 (d+e x)^5}{m+6}\right )}{e^6} \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)^2,x]

[Out]

((d + e*x)^(1 + m)*(-((b*d - a*e)^5/(1 + m)) + (5*b*(b*d - a*e)^4*(d + e*x))/(2 + m) - (10*b^2*(b*d - a*e)^3*(
d + e*x)^2)/(3 + m) + (10*b^3*(b*d - a*e)^2*(d + e*x)^3)/(4 + m) - (5*b^4*(b*d - a*e)*(d + e*x)^4)/(5 + m) + (
b^5*(d + e*x)^5)/(6 + m)))/e^6

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IntegrateAlgebraic [F]  time = 0.46, 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 )^2 \, 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)^2,x]

[Out]

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

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fricas [B]  time = 0.47, size = 1460, normalized size = 8.34

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)^2,x, algorithm="fricas")

[Out]

(a^5*d*e^5*m^5 - 120*b^5*d^6 + 720*a*b^4*d^5*e - 1800*a^2*b^3*d^4*e^2 + 2400*a^3*b^2*d^3*e^3 - 1800*a^4*b*d^2*
e^4 + 720*a^5*d*e^5 + (b^5*e^6*m^5 + 15*b^5*e^6*m^4 + 85*b^5*e^6*m^3 + 225*b^5*e^6*m^2 + 274*b^5*e^6*m + 120*b
^5*e^6)*x^6 + (720*a*b^4*e^6 + (b^5*d*e^5 + 5*a*b^4*e^6)*m^5 + 10*(b^5*d*e^5 + 8*a*b^4*e^6)*m^4 + 5*(7*b^5*d*e
^5 + 95*a*b^4*e^6)*m^3 + 50*(b^5*d*e^5 + 26*a*b^4*e^6)*m^2 + 12*(2*b^5*d*e^5 + 135*a*b^4*e^6)*m)*x^5 - 5*(a^4*
b*d^2*e^4 - 4*a^5*d*e^5)*m^4 + 5*(360*a^2*b^3*e^6 + (a*b^4*d*e^5 + 2*a^2*b^3*e^6)*m^5 - (b^5*d^2*e^4 - 12*a*b^
4*d*e^5 - 34*a^2*b^3*e^6)*m^4 - (6*b^5*d^2*e^4 - 47*a*b^4*d*e^5 - 214*a^2*b^3*e^6)*m^3 - (11*b^5*d^2*e^4 - 72*
a*b^4*d*e^5 - 614*a^2*b^3*e^6)*m^2 - 6*(b^5*d^2*e^4 - 6*a*b^4*d*e^5 - 132*a^2*b^3*e^6)*m)*x^4 + 5*(4*a^3*b^2*d
^3*e^3 - 18*a^4*b*d^2*e^4 + 31*a^5*d*e^5)*m^3 + 10*(240*a^3*b^2*e^6 + (a^2*b^3*d*e^5 + a^3*b^2*e^6)*m^5 - 2*(a
*b^4*d^2*e^4 - 7*a^2*b^3*d*e^5 - 9*a^3*b^2*e^6)*m^4 + (2*b^5*d^3*e^3 - 18*a*b^4*d^2*e^4 + 65*a^2*b^3*d*e^5 + 1
21*a^3*b^2*e^6)*m^3 + 2*(3*b^5*d^3*e^3 - 20*a*b^4*d^2*e^4 + 56*a^2*b^3*d*e^5 + 186*a^3*b^2*e^6)*m^2 + 4*(b^5*d
^3*e^3 - 6*a*b^4*d^2*e^4 + 15*a^2*b^3*d*e^5 + 127*a^3*b^2*e^6)*m)*x^3 - 5*(12*a^2*b^3*d^4*e^2 - 60*a^3*b^2*d^3
*e^3 + 119*a^4*b*d^2*e^4 - 116*a^5*d*e^5)*m^2 + 5*(360*a^4*b*e^6 + (2*a^3*b^2*d*e^5 + a^4*b*e^6)*m^5 - (6*a^2*
b^3*d^2*e^4 - 32*a^3*b^2*d*e^5 - 19*a^4*b*e^6)*m^4 + (12*a*b^4*d^3*e^3 - 72*a^2*b^3*d^2*e^4 + 178*a^3*b^2*d*e^
5 + 137*a^4*b*e^6)*m^3 - (12*b^5*d^4*e^2 - 84*a*b^4*d^3*e^3 + 246*a^2*b^3*d^2*e^4 - 388*a^3*b^2*d*e^5 - 461*a^
4*b*e^6)*m^2 - 6*(2*b^5*d^4*e^2 - 12*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 - 40*a^3*b^2*d*e^5 - 117*a^4*b*e^6)*m)
*x^2 + 2*(60*a*b^4*d^5*e - 330*a^2*b^3*d^4*e^2 + 740*a^3*b^2*d^3*e^3 - 855*a^4*b*d^2*e^4 + 522*a^5*d*e^5)*m +
(720*a^5*e^6 + (5*a^4*b*d*e^5 + a^5*e^6)*m^5 - 10*(2*a^3*b^2*d^2*e^4 - 9*a^4*b*d*e^5 - 2*a^5*e^6)*m^4 + 5*(12*
a^2*b^3*d^3*e^3 - 60*a^3*b^2*d^2*e^4 + 119*a^4*b*d*e^5 + 31*a^5*e^6)*m^3 - 10*(12*a*b^4*d^4*e^2 - 66*a^2*b^3*d
^3*e^3 + 148*a^3*b^2*d^2*e^4 - 171*a^4*b*d*e^5 - 58*a^5*e^6)*m^2 + 12*(10*b^5*d^5*e - 60*a*b^4*d^4*e^2 + 150*a
^2*b^3*d^3*e^3 - 200*a^3*b^2*d^2*e^4 + 150*a^4*b*d*e^5 + 87*a^5*e^6)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 +
175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)

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giac [B]  time = 0.25, size = 2525, normalized size = 14.43

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)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*b^5*m^5*x^6*e^6 + (x*e + d)^m*b^5*d*m^5*x^5*e^5 + 5*(x*e + d)^m*a*b^4*m^5*x^5*e^6 + 15*(x*e + d)^
m*b^5*m^4*x^6*e^6 + 5*(x*e + d)^m*a*b^4*d*m^5*x^4*e^5 + 10*(x*e + d)^m*b^5*d*m^4*x^5*e^5 - 5*(x*e + d)^m*b^5*d
^2*m^4*x^4*e^4 + 10*(x*e + d)^m*a^2*b^3*m^5*x^4*e^6 + 80*(x*e + d)^m*a*b^4*m^4*x^5*e^6 + 85*(x*e + d)^m*b^5*m^
3*x^6*e^6 + 10*(x*e + d)^m*a^2*b^3*d*m^5*x^3*e^5 + 60*(x*e + d)^m*a*b^4*d*m^4*x^4*e^5 + 35*(x*e + d)^m*b^5*d*m
^3*x^5*e^5 - 20*(x*e + d)^m*a*b^4*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*b^5*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*b^5*d^
3*m^3*x^3*e^3 + 10*(x*e + d)^m*a^3*b^2*m^5*x^3*e^6 + 170*(x*e + d)^m*a^2*b^3*m^4*x^4*e^6 + 475*(x*e + d)^m*a*b
^4*m^3*x^5*e^6 + 225*(x*e + d)^m*b^5*m^2*x^6*e^6 + 10*(x*e + d)^m*a^3*b^2*d*m^5*x^2*e^5 + 140*(x*e + d)^m*a^2*
b^3*d*m^4*x^3*e^5 + 235*(x*e + d)^m*a*b^4*d*m^3*x^4*e^5 + 50*(x*e + d)^m*b^5*d*m^2*x^5*e^5 - 30*(x*e + d)^m*a^
2*b^3*d^2*m^4*x^2*e^4 - 180*(x*e + d)^m*a*b^4*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*b^5*d^2*m^2*x^4*e^4 + 60*(x*e +
d)^m*a*b^4*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*b^5*d^3*m^2*x^3*e^3 - 60*(x*e + d)^m*b^5*d^4*m^2*x^2*e^2 + 5*(x*e
+ d)^m*a^4*b*m^5*x^2*e^6 + 180*(x*e + d)^m*a^3*b^2*m^4*x^3*e^6 + 1070*(x*e + d)^m*a^2*b^3*m^3*x^4*e^6 + 1300*
(x*e + d)^m*a*b^4*m^2*x^5*e^6 + 274*(x*e + d)^m*b^5*m*x^6*e^6 + 5*(x*e + d)^m*a^4*b*d*m^5*x*e^5 + 160*(x*e + d
)^m*a^3*b^2*d*m^4*x^2*e^5 + 650*(x*e + d)^m*a^2*b^3*d*m^3*x^3*e^5 + 360*(x*e + d)^m*a*b^4*d*m^2*x^4*e^5 + 24*(
x*e + d)^m*b^5*d*m*x^5*e^5 - 20*(x*e + d)^m*a^3*b^2*d^2*m^4*x*e^4 - 360*(x*e + d)^m*a^2*b^3*d^2*m^3*x^2*e^4 -
400*(x*e + d)^m*a*b^4*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*b^5*d^2*m*x^4*e^4 + 60*(x*e + d)^m*a^2*b^3*d^3*m^3*x*e^
3 + 420*(x*e + d)^m*a*b^4*d^3*m^2*x^2*e^3 + 40*(x*e + d)^m*b^5*d^3*m*x^3*e^3 - 120*(x*e + d)^m*a*b^4*d^4*m^2*x
*e^2 - 60*(x*e + d)^m*b^5*d^4*m*x^2*e^2 + 120*(x*e + d)^m*b^5*d^5*m*x*e + (x*e + d)^m*a^5*m^5*x*e^6 + 95*(x*e
+ d)^m*a^4*b*m^4*x^2*e^6 + 1210*(x*e + d)^m*a^3*b^2*m^3*x^3*e^6 + 3070*(x*e + d)^m*a^2*b^3*m^2*x^4*e^6 + 1620*
(x*e + d)^m*a*b^4*m*x^5*e^6 + 120*(x*e + d)^m*b^5*x^6*e^6 + (x*e + d)^m*a^5*d*m^5*e^5 + 90*(x*e + d)^m*a^4*b*d
*m^4*x*e^5 + 890*(x*e + d)^m*a^3*b^2*d*m^3*x^2*e^5 + 1120*(x*e + d)^m*a^2*b^3*d*m^2*x^3*e^5 + 180*(x*e + d)^m*
a*b^4*d*m*x^4*e^5 - 5*(x*e + d)^m*a^4*b*d^2*m^4*e^4 - 300*(x*e + d)^m*a^3*b^2*d^2*m^3*x*e^4 - 1230*(x*e + d)^m
*a^2*b^3*d^2*m^2*x^2*e^4 - 240*(x*e + d)^m*a*b^4*d^2*m*x^3*e^4 + 20*(x*e + d)^m*a^3*b^2*d^3*m^3*e^3 + 660*(x*e
+ d)^m*a^2*b^3*d^3*m^2*x*e^3 + 360*(x*e + d)^m*a*b^4*d^3*m*x^2*e^3 - 60*(x*e + d)^m*a^2*b^3*d^4*m^2*e^2 - 720
*(x*e + d)^m*a*b^4*d^4*m*x*e^2 + 120*(x*e + d)^m*a*b^4*d^5*m*e - 120*(x*e + d)^m*b^5*d^6 + 20*(x*e + d)^m*a^5*
m^4*x*e^6 + 685*(x*e + d)^m*a^4*b*m^3*x^2*e^6 + 3720*(x*e + d)^m*a^3*b^2*m^2*x^3*e^6 + 3960*(x*e + d)^m*a^2*b^
3*m*x^4*e^6 + 720*(x*e + d)^m*a*b^4*x^5*e^6 + 20*(x*e + d)^m*a^5*d*m^4*e^5 + 595*(x*e + d)^m*a^4*b*d*m^3*x*e^5
+ 1940*(x*e + d)^m*a^3*b^2*d*m^2*x^2*e^5 + 600*(x*e + d)^m*a^2*b^3*d*m*x^3*e^5 - 90*(x*e + d)^m*a^4*b*d^2*m^3
*e^4 - 1480*(x*e + d)^m*a^3*b^2*d^2*m^2*x*e^4 - 900*(x*e + d)^m*a^2*b^3*d^2*m*x^2*e^4 + 300*(x*e + d)^m*a^3*b^
2*d^3*m^2*e^3 + 1800*(x*e + d)^m*a^2*b^3*d^3*m*x*e^3 - 660*(x*e + d)^m*a^2*b^3*d^4*m*e^2 + 720*(x*e + d)^m*a*b
^4*d^5*e + 155*(x*e + d)^m*a^5*m^3*x*e^6 + 2305*(x*e + d)^m*a^4*b*m^2*x^2*e^6 + 5080*(x*e + d)^m*a^3*b^2*m*x^3
*e^6 + 1800*(x*e + d)^m*a^2*b^3*x^4*e^6 + 155*(x*e + d)^m*a^5*d*m^3*e^5 + 1710*(x*e + d)^m*a^4*b*d*m^2*x*e^5 +
1200*(x*e + d)^m*a^3*b^2*d*m*x^2*e^5 - 595*(x*e + d)^m*a^4*b*d^2*m^2*e^4 - 2400*(x*e + d)^m*a^3*b^2*d^2*m*x*e
^4 + 1480*(x*e + d)^m*a^3*b^2*d^3*m*e^3 - 1800*(x*e + d)^m*a^2*b^3*d^4*e^2 + 580*(x*e + d)^m*a^5*m^2*x*e^6 + 3
510*(x*e + d)^m*a^4*b*m*x^2*e^6 + 2400*(x*e + d)^m*a^3*b^2*x^3*e^6 + 580*(x*e + d)^m*a^5*d*m^2*e^5 + 1800*(x*e
+ d)^m*a^4*b*d*m*x*e^5 - 1710*(x*e + d)^m*a^4*b*d^2*m*e^4 + 2400*(x*e + d)^m*a^3*b^2*d^3*e^3 + 1044*(x*e + d)
^m*a^5*m*x*e^6 + 1800*(x*e + d)^m*a^4*b*x^2*e^6 + 1044*(x*e + d)^m*a^5*d*m*e^5 - 1800*(x*e + d)^m*a^4*b*d^2*e^
4 + 720*(x*e + d)^m*a^5*x*e^6 + 720*(x*e + d)^m*a^5*d*e^5)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 +
1624*m^2*e^6 + 1764*m*e^6 + 720*e^6)

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maple [B]  time = 0.06, size = 1345, normalized size = 7.69 \begin {gather*} \frac {\left (b^{5} e^{5} m^{5} x^{5}+5 a \,b^{4} e^{5} m^{5} x^{4}+15 b^{5} e^{5} m^{4} x^{5}+10 a^{2} b^{3} e^{5} m^{5} x^{3}+80 a \,b^{4} e^{5} m^{4} x^{4}-5 b^{5} d \,e^{4} m^{4} x^{4}+85 b^{5} e^{5} m^{3} x^{5}+10 a^{3} b^{2} e^{5} m^{5} x^{2}+170 a^{2} b^{3} e^{5} m^{4} x^{3}-20 a \,b^{4} d \,e^{4} m^{4} x^{3}+475 a \,b^{4} e^{5} m^{3} x^{4}-50 b^{5} d \,e^{4} m^{3} x^{4}+225 b^{5} e^{5} m^{2} x^{5}+5 a^{4} b \,e^{5} m^{5} x +180 a^{3} b^{2} e^{5} m^{4} x^{2}-30 a^{2} b^{3} d \,e^{4} m^{4} x^{2}+1070 a^{2} b^{3} e^{5} m^{3} x^{3}-240 a \,b^{4} d \,e^{4} m^{3} x^{3}+1300 a \,b^{4} e^{5} m^{2} x^{4}+20 b^{5} d^{2} e^{3} m^{3} x^{3}-175 b^{5} d \,e^{4} m^{2} x^{4}+274 b^{5} e^{5} m \,x^{5}+a^{5} e^{5} m^{5}+95 a^{4} b \,e^{5} m^{4} x -20 a^{3} b^{2} d \,e^{4} m^{4} x +1210 a^{3} b^{2} e^{5} m^{3} x^{2}-420 a^{2} b^{3} d \,e^{4} m^{3} x^{2}+3070 a^{2} b^{3} e^{5} m^{2} x^{3}+60 a \,b^{4} d^{2} e^{3} m^{3} x^{2}-940 a \,b^{4} d \,e^{4} m^{2} x^{3}+1620 a \,b^{4} e^{5} m \,x^{4}+120 b^{5} d^{2} e^{3} m^{2} x^{3}-250 b^{5} d \,e^{4} m \,x^{4}+120 b^{5} e^{5} x^{5}+20 a^{5} e^{5} m^{4}-5 a^{4} b d \,e^{4} m^{4}+685 a^{4} b \,e^{5} m^{3} x -320 a^{3} b^{2} d \,e^{4} m^{3} x +3720 a^{3} b^{2} e^{5} m^{2} x^{2}+60 a^{2} b^{3} d^{2} e^{3} m^{3} x -1950 a^{2} b^{3} d \,e^{4} m^{2} x^{2}+3960 a^{2} b^{3} e^{5} m \,x^{3}+540 a \,b^{4} d^{2} e^{3} m^{2} x^{2}-1440 a \,b^{4} d \,e^{4} m \,x^{3}+720 a \,b^{4} e^{5} x^{4}-60 b^{5} d^{3} e^{2} m^{2} x^{2}+220 b^{5} d^{2} e^{3} m \,x^{3}-120 b^{5} d \,e^{4} x^{4}+155 a^{5} e^{5} m^{3}-90 a^{4} b d \,e^{4} m^{3}+2305 a^{4} b \,e^{5} m^{2} x +20 a^{3} b^{2} d^{2} e^{3} m^{3}-1780 a^{3} b^{2} d \,e^{4} m^{2} x +5080 a^{3} b^{2} e^{5} m \,x^{2}+720 a^{2} b^{3} d^{2} e^{3} m^{2} x -3360 a^{2} b^{3} d \,e^{4} m \,x^{2}+1800 a^{2} b^{3} e^{5} x^{3}-120 a \,b^{4} d^{3} e^{2} m^{2} x +1200 a \,b^{4} d^{2} e^{3} m \,x^{2}-720 a \,b^{4} d \,e^{4} x^{3}-180 b^{5} d^{3} e^{2} m \,x^{2}+120 b^{5} d^{2} e^{3} x^{3}+580 a^{5} e^{5} m^{2}-595 a^{4} b d \,e^{4} m^{2}+3510 a^{4} b \,e^{5} m x +300 a^{3} b^{2} d^{2} e^{3} m^{2}-3880 a^{3} b^{2} d \,e^{4} m x +2400 a^{3} b^{2} e^{5} x^{2}-60 a^{2} b^{3} d^{3} e^{2} m^{2}+2460 a^{2} b^{3} d^{2} e^{3} m x -1800 a^{2} b^{3} d \,e^{4} x^{2}-840 a \,b^{4} d^{3} e^{2} m x +720 a \,b^{4} d^{2} e^{3} x^{2}+120 b^{5} d^{4} e m x -120 b^{5} d^{3} e^{2} x^{2}+1044 a^{5} e^{5} m -1710 a^{4} b d \,e^{4} m +1800 a^{4} b \,e^{5} x +1480 a^{3} b^{2} d^{2} e^{3} m -2400 a^{3} b^{2} d \,e^{4} x -660 a^{2} b^{3} d^{3} e^{2} m +1800 a^{2} b^{3} d^{2} e^{3} x +120 a \,b^{4} d^{4} e m -720 a \,b^{4} d^{3} e^{2} x +120 b^{5} d^{4} e x +720 a^{5} e^{5}-1800 a^{4} b d \,e^{4}+2400 a^{3} b^{2} d^{2} e^{3}-1800 a^{2} b^{3} d^{3} e^{2}+720 a \,b^{4} d^{4} e -120 b^{5} d^{5}\right ) \left (e x +d \right )^{m +1}}{\left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right ) e^{6}} \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)^2,x)

[Out]

(e*x+d)^(m+1)*(b^5*e^5*m^5*x^5+5*a*b^4*e^5*m^5*x^4+15*b^5*e^5*m^4*x^5+10*a^2*b^3*e^5*m^5*x^3+80*a*b^4*e^5*m^4*
x^4-5*b^5*d*e^4*m^4*x^4+85*b^5*e^5*m^3*x^5+10*a^3*b^2*e^5*m^5*x^2+170*a^2*b^3*e^5*m^4*x^3-20*a*b^4*d*e^4*m^4*x
^3+475*a*b^4*e^5*m^3*x^4-50*b^5*d*e^4*m^3*x^4+225*b^5*e^5*m^2*x^5+5*a^4*b*e^5*m^5*x+180*a^3*b^2*e^5*m^4*x^2-30
*a^2*b^3*d*e^4*m^4*x^2+1070*a^2*b^3*e^5*m^3*x^3-240*a*b^4*d*e^4*m^3*x^3+1300*a*b^4*e^5*m^2*x^4+20*b^5*d^2*e^3*
m^3*x^3-175*b^5*d*e^4*m^2*x^4+274*b^5*e^5*m*x^5+a^5*e^5*m^5+95*a^4*b*e^5*m^4*x-20*a^3*b^2*d*e^4*m^4*x+1210*a^3
*b^2*e^5*m^3*x^2-420*a^2*b^3*d*e^4*m^3*x^2+3070*a^2*b^3*e^5*m^2*x^3+60*a*b^4*d^2*e^3*m^3*x^2-940*a*b^4*d*e^4*m
^2*x^3+1620*a*b^4*e^5*m*x^4+120*b^5*d^2*e^3*m^2*x^3-250*b^5*d*e^4*m*x^4+120*b^5*e^5*x^5+20*a^5*e^5*m^4-5*a^4*b
*d*e^4*m^4+685*a^4*b*e^5*m^3*x-320*a^3*b^2*d*e^4*m^3*x+3720*a^3*b^2*e^5*m^2*x^2+60*a^2*b^3*d^2*e^3*m^3*x-1950*
a^2*b^3*d*e^4*m^2*x^2+3960*a^2*b^3*e^5*m*x^3+540*a*b^4*d^2*e^3*m^2*x^2-1440*a*b^4*d*e^4*m*x^3+720*a*b^4*e^5*x^
4-60*b^5*d^3*e^2*m^2*x^2+220*b^5*d^2*e^3*m*x^3-120*b^5*d*e^4*x^4+155*a^5*e^5*m^3-90*a^4*b*d*e^4*m^3+2305*a^4*b
*e^5*m^2*x+20*a^3*b^2*d^2*e^3*m^3-1780*a^3*b^2*d*e^4*m^2*x+5080*a^3*b^2*e^5*m*x^2+720*a^2*b^3*d^2*e^3*m^2*x-33
60*a^2*b^3*d*e^4*m*x^2+1800*a^2*b^3*e^5*x^3-120*a*b^4*d^3*e^2*m^2*x+1200*a*b^4*d^2*e^3*m*x^2-720*a*b^4*d*e^4*x
^3-180*b^5*d^3*e^2*m*x^2+120*b^5*d^2*e^3*x^3+580*a^5*e^5*m^2-595*a^4*b*d*e^4*m^2+3510*a^4*b*e^5*m*x+300*a^3*b^
2*d^2*e^3*m^2-3880*a^3*b^2*d*e^4*m*x+2400*a^3*b^2*e^5*x^2-60*a^2*b^3*d^3*e^2*m^2+2460*a^2*b^3*d^2*e^3*m*x-1800
*a^2*b^3*d*e^4*x^2-840*a*b^4*d^3*e^2*m*x+720*a*b^4*d^2*e^3*x^2+120*b^5*d^4*e*m*x-120*b^5*d^3*e^2*x^2+1044*a^5*
e^5*m-1710*a^4*b*d*e^4*m+1800*a^4*b*e^5*x+1480*a^3*b^2*d^2*e^3*m-2400*a^3*b^2*d*e^4*x-660*a^2*b^3*d^3*e^2*m+18
00*a^2*b^3*d^2*e^3*x+120*a*b^4*d^4*e*m-720*a*b^4*d^3*e^2*x+120*b^5*d^4*e*x+720*a^5*e^5-1800*a^4*b*d*e^4+2400*a
^3*b^2*d^2*e^3-1800*a^2*b^3*d^3*e^2+720*a*b^4*d^4*e-120*b^5*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764
*m+720)

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maxima [B]  time = 0.67, size = 581, normalized size = 3.32 \begin {gather*} \frac {5 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{4} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{5}}{e {\left (m + 1\right )}} + \frac {10 \, {\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^{3} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {10 \, {\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} a^{2} b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {5 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} a b^{4}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} b^{5}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \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)^2,x, algorithm="maxima")

[Out]

5*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^4*b/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a^5/(e*(m + 1)
) + 10*((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^3*b^2/((m^3 + 6*m^2
+ 11*m + 6)*e^3) + 10*((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*a^2*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 5*((m^4 + 10*m^3 +
35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12
*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*a*b^4/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m +
120)*e^5) + ((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)
*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^
4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*b^5/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m +
720)*e^6)

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mupad [B]  time = 2.78, size = 1291, normalized size = 7.38 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^5\,d\,e^5\,m^5+20\,a^5\,d\,e^5\,m^4+155\,a^5\,d\,e^5\,m^3+580\,a^5\,d\,e^5\,m^2+1044\,a^5\,d\,e^5\,m+720\,a^5\,d\,e^5-5\,a^4\,b\,d^2\,e^4\,m^4-90\,a^4\,b\,d^2\,e^4\,m^3-595\,a^4\,b\,d^2\,e^4\,m^2-1710\,a^4\,b\,d^2\,e^4\,m-1800\,a^4\,b\,d^2\,e^4+20\,a^3\,b^2\,d^3\,e^3\,m^3+300\,a^3\,b^2\,d^3\,e^3\,m^2+1480\,a^3\,b^2\,d^3\,e^3\,m+2400\,a^3\,b^2\,d^3\,e^3-60\,a^2\,b^3\,d^4\,e^2\,m^2-660\,a^2\,b^3\,d^4\,e^2\,m-1800\,a^2\,b^3\,d^4\,e^2+120\,a\,b^4\,d^5\,e\,m+720\,a\,b^4\,d^5\,e-120\,b^5\,d^6\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {b^5\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^5\,e^6\,m^5+20\,a^5\,e^6\,m^4+155\,a^5\,e^6\,m^3+580\,a^5\,e^6\,m^2+1044\,a^5\,e^6\,m+720\,a^5\,e^6+5\,a^4\,b\,d\,e^5\,m^5+90\,a^4\,b\,d\,e^5\,m^4+595\,a^4\,b\,d\,e^5\,m^3+1710\,a^4\,b\,d\,e^5\,m^2+1800\,a^4\,b\,d\,e^5\,m-20\,a^3\,b^2\,d^2\,e^4\,m^4-300\,a^3\,b^2\,d^2\,e^4\,m^3-1480\,a^3\,b^2\,d^2\,e^4\,m^2-2400\,a^3\,b^2\,d^2\,e^4\,m+60\,a^2\,b^3\,d^3\,e^3\,m^3+660\,a^2\,b^3\,d^3\,e^3\,m^2+1800\,a^2\,b^3\,d^3\,e^3\,m-120\,a\,b^4\,d^4\,e^2\,m^2-720\,a\,b^4\,d^4\,e^2\,m+120\,b^5\,d^5\,e\,m\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {5\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^4\,e^4\,m^4+18\,a^4\,e^4\,m^3+119\,a^4\,e^4\,m^2+342\,a^4\,e^4\,m+360\,a^4\,e^4+2\,a^3\,b\,d\,e^3\,m^4+30\,a^3\,b\,d\,e^3\,m^3+148\,a^3\,b\,d\,e^3\,m^2+240\,a^3\,b\,d\,e^3\,m-6\,a^2\,b^2\,d^2\,e^2\,m^3-66\,a^2\,b^2\,d^2\,e^2\,m^2-180\,a^2\,b^2\,d^2\,e^2\,m+12\,a\,b^3\,d^3\,e\,m^2+72\,a\,b^3\,d^3\,e\,m-12\,b^4\,d^4\,m\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {5\,b^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (2\,a^2\,e^2\,m^2+22\,a^2\,e^2\,m+60\,a^2\,e^2+a\,b\,d\,e\,m^2+6\,a\,b\,d\,e\,m-b^2\,d^2\,m\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {10\,b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (a^3\,e^3\,m^3+15\,a^3\,e^3\,m^2+74\,a^3\,e^3\,m+120\,a^3\,e^3+a^2\,b\,d\,e^2\,m^3+11\,a^2\,b\,d\,e^2\,m^2+30\,a^2\,b\,d\,e^2\,m-2\,a\,b^2\,d^2\,e\,m^2-12\,a\,b^2\,d^2\,e\,m+2\,b^3\,d^3\,m\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {b^4\,x^5\,{\left (d+e\,x\right )}^m\,\left (30\,a\,e+5\,a\,e\,m+b\,d\,m\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\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)^2,x)

[Out]

((d + e*x)^m*(720*a^5*d*e^5 - 120*b^5*d^6 - 1800*a^4*b*d^2*e^4 + 580*a^5*d*e^5*m^2 + 155*a^5*d*e^5*m^3 + 20*a^
5*d*e^5*m^4 + a^5*d*e^5*m^5 - 1800*a^2*b^3*d^4*e^2 + 2400*a^3*b^2*d^3*e^3 + 720*a*b^4*d^5*e + 1044*a^5*d*e^5*m
+ 120*a*b^4*d^5*e*m - 60*a^2*b^3*d^4*e^2*m^2 + 300*a^3*b^2*d^3*e^3*m^2 + 20*a^3*b^2*d^3*e^3*m^3 - 1710*a^4*b*
d^2*e^4*m - 660*a^2*b^3*d^4*e^2*m + 1480*a^3*b^2*d^3*e^3*m - 595*a^4*b*d^2*e^4*m^2 - 90*a^4*b*d^2*e^4*m^3 - 5*
a^4*b*d^2*e^4*m^4))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (b^5*x^6*(d + e*x)^m*
(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)
+ (x*(d + e*x)^m*(720*a^5*e^6 + 1044*a^5*e^6*m + 580*a^5*e^6*m^2 + 155*a^5*e^6*m^3 + 20*a^5*e^6*m^4 + a^5*e^6*
m^5 + 120*b^5*d^5*e*m + 1800*a^4*b*d*e^5*m + 660*a^2*b^3*d^3*e^3*m^2 - 1480*a^3*b^2*d^2*e^4*m^2 + 60*a^2*b^3*d
^3*e^3*m^3 - 300*a^3*b^2*d^2*e^4*m^3 - 20*a^3*b^2*d^2*e^4*m^4 - 720*a*b^4*d^4*e^2*m + 1710*a^4*b*d*e^5*m^2 + 5
95*a^4*b*d*e^5*m^3 + 90*a^4*b*d*e^5*m^4 + 5*a^4*b*d*e^5*m^5 + 1800*a^2*b^3*d^3*e^3*m - 2400*a^3*b^2*d^2*e^4*m
- 120*a*b^4*d^4*e^2*m^2))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (5*b*x^2*(m + 1
)*(d + e*x)^m*(360*a^4*e^4 + 342*a^4*e^4*m - 12*b^4*d^4*m + 119*a^4*e^4*m^2 + 18*a^4*e^4*m^3 + a^4*e^4*m^4 + 7
2*a*b^3*d^3*e*m + 240*a^3*b*d*e^3*m - 66*a^2*b^2*d^2*e^2*m^2 - 6*a^2*b^2*d^2*e^2*m^3 + 12*a*b^3*d^3*e*m^2 + 14
8*a^3*b*d*e^3*m^2 + 30*a^3*b*d*e^3*m^3 + 2*a^3*b*d*e^3*m^4 - 180*a^2*b^2*d^2*e^2*m))/(e^4*(1764*m + 1624*m^2 +
735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (5*b^3*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(60*a^2*e^2 + 22*a
^2*e^2*m - b^2*d^2*m + 2*a^2*e^2*m^2 + 6*a*b*d*e*m + a*b*d*e*m^2))/(e^2*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4
+ 21*m^5 + m^6 + 720)) + (10*b^2*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(120*a^3*e^3 + 74*a^3*e^3*m + 2*b^3*d^3*m +
15*a^3*e^3*m^2 + a^3*e^3*m^3 - 12*a*b^2*d^2*e*m + 30*a^2*b*d*e^2*m - 2*a*b^2*d^2*e*m^2 + 11*a^2*b*d*e^2*m^2 +
a^2*b*d*e^2*m^3))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (b^4*x^5*(d + e*x)^m*(3
0*a*e + 5*a*e*m + b*d*m)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(e*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m
^5 + m^6 + 720))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**2,x)

[Out]

Timed out

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