3.1084 \(\int (e x)^m (A+B x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=155 \[ \frac {a^2 A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+4} \left (2 a B c+2 A b c+b^2 B\right )}{e^4 (m+4)}+\frac {(e x)^{m+3} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^3 (m+3)}+\frac {a (e x)^{m+2} (a B+2 A b)}{e^2 (m+2)}+\frac {c (e x)^{m+5} (A c+2 b B)}{e^5 (m+5)}+\frac {B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

[Out]

a^2*A*(e*x)^(1+m)/e/(1+m)+a*(2*A*b+B*a)*(e*x)^(2+m)/e^2/(2+m)+(2*a*b*B+A*(2*a*c+b^2))*(e*x)^(3+m)/e^3/(3+m)+(2
*A*b*c+2*B*a*c+B*b^2)*(e*x)^(4+m)/e^4/(4+m)+c*(A*c+2*B*b)*(e*x)^(5+m)/e^5/(5+m)+B*c^2*(e*x)^(6+m)/e^6/(6+m)

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Rubi [A]  time = 0.10, antiderivative size = 155, 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 = {765} \[ \frac {a^2 A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+3} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^3 (m+3)}+\frac {(e x)^{m+4} \left (2 a B c+2 A b c+b^2 B\right )}{e^4 (m+4)}+\frac {a (e x)^{m+2} (a B+2 A b)}{e^2 (m+2)}+\frac {c (e x)^{m+5} (A c+2 b B)}{e^5 (m+5)}+\frac {B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a*(2*A*b + a*B)*(e*x)^(2 + m))/(e^2*(2 + m)) + ((2*a*b*B + A*(b^2 + 2*a*c
))*(e*x)^(3 + m))/(e^3*(3 + m)) + ((b^2*B + 2*A*b*c + 2*a*B*c)*(e*x)^(4 + m))/(e^4*(4 + m)) + (c*(2*b*B + A*c)
*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(6 + m))

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A (e x)^m+\frac {a (2 A b+a B) (e x)^{1+m}}{e}+\frac {\left (2 a b B+A \left (b^2+2 a c\right )\right ) (e x)^{2+m}}{e^2}+\frac {\left (b^2 B+2 A b c+2 a B c\right ) (e x)^{3+m}}{e^3}+\frac {c (2 b B+A c) (e x)^{4+m}}{e^4}+\frac {B c^2 (e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac {a^2 A (e x)^{1+m}}{e (1+m)}+\frac {a (2 A b+a B) (e x)^{2+m}}{e^2 (2+m)}+\frac {\left (2 a b B+A \left (b^2+2 a c\right )\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {\left (b^2 B+2 A b c+2 a B c\right ) (e x)^{4+m}}{e^4 (4+m)}+\frac {c (2 b B+A c) (e x)^{5+m}}{e^5 (5+m)}+\frac {B c^2 (e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 289, normalized size = 1.86 \[ \frac {(e x)^m \left (\frac {2 x \left (-\frac {2 a^2 c (m+4) (b B (m+1)-2 A c (m+6))}{m+1}+\frac {x \left (b^2 (m+2)-2 a c (m+3)\right ) \left (-2 a B c (m+5)-A b c (m+6)+b^2 B (m+3)\right )}{m+2}-(a+x (b+c x)) \left (c (m+3) x \left (-2 a B c (m+5)-A b c (m+6)+b^2 B (m+3)\right )+b \left (-2 a B c (m+5)-A b c (m+6)+b^2 B (m+3)\right )+a c (m+4) (b B (m+1)-2 A c (m+6))\right )+a b \left (-2 a B c (m+5)-A b c (m+6)+b^2 B (m+3)\right )-\frac {a b c (m+4) x (b B (m+1)-2 A c (m+6))}{m+2}\right )}{c (m+3) (m+4)}+x (a+x (b+c x))^2 (A c (m+6)+2 b B+B c (m+5) x)\right )}{c (m+5) (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^m*(x*(2*b*B + A*c*(6 + m) + B*c*(5 + m)*x)*(a + x*(b + c*x))^2 + (2*x*((-2*a^2*c*(4 + m)*(b*B*(1 + m) -
2*A*c*(6 + m)))/(1 + m) + a*b*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m)) - (a*b*c*(4 + m)*(b*B*(1 + m)
- 2*A*c*(6 + m))*x)/(2 + m) + ((b^2*(2 + m) - 2*a*c*(3 + m))*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m)
)*x)/(2 + m) - (a*c*(4 + m)*(b*B*(1 + m) - 2*A*c*(6 + m)) + b*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m)
) + c*(3 + m)*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m))*x)*(a + x*(b + c*x))))/(c*(3 + m)*(4 + m))))/(
c*(5 + m)*(6 + m))

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fricas [B]  time = 0.91, size = 573, normalized size = 3.70 \[ \frac {{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} + {\left ({\left (2 \, B b c + A c^{2}\right )} m^{5} + 16 \, {\left (2 \, B b c + A c^{2}\right )} m^{4} + 95 \, {\left (2 \, B b c + A c^{2}\right )} m^{3} + 288 \, B b c + 144 \, A c^{2} + 260 \, {\left (2 \, B b c + A c^{2}\right )} m^{2} + 324 \, {\left (2 \, B b c + A c^{2}\right )} m\right )} x^{5} + {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{5} + 17 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{4} + 107 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{3} + 180 \, B b^{2} + 307 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{2} + 360 \, {\left (B a + A b\right )} c + 396 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{5} + 18 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{4} + 121 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{3} + 480 \, B a b + 240 \, A b^{2} + 480 \, A a c + 372 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{2} + 508 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{5} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m^{4} + 137 \, {\left (B a^{2} + 2 \, A a b\right )} m^{3} + 360 \, B a^{2} + 720 \, A a b + 461 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 702 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

((B*c^2*m^5 + 15*B*c^2*m^4 + 85*B*c^2*m^3 + 225*B*c^2*m^2 + 274*B*c^2*m + 120*B*c^2)*x^6 + ((2*B*b*c + A*c^2)*
m^5 + 16*(2*B*b*c + A*c^2)*m^4 + 95*(2*B*b*c + A*c^2)*m^3 + 288*B*b*c + 144*A*c^2 + 260*(2*B*b*c + A*c^2)*m^2
+ 324*(2*B*b*c + A*c^2)*m)*x^5 + ((B*b^2 + 2*(B*a + A*b)*c)*m^5 + 17*(B*b^2 + 2*(B*a + A*b)*c)*m^4 + 107*(B*b^
2 + 2*(B*a + A*b)*c)*m^3 + 180*B*b^2 + 307*(B*b^2 + 2*(B*a + A*b)*c)*m^2 + 360*(B*a + A*b)*c + 396*(B*b^2 + 2*
(B*a + A*b)*c)*m)*x^4 + ((2*B*a*b + A*b^2 + 2*A*a*c)*m^5 + 18*(2*B*a*b + A*b^2 + 2*A*a*c)*m^4 + 121*(2*B*a*b +
A*b^2 + 2*A*a*c)*m^3 + 480*B*a*b + 240*A*b^2 + 480*A*a*c + 372*(2*B*a*b + A*b^2 + 2*A*a*c)*m^2 + 508*(2*B*a*b
+ A*b^2 + 2*A*a*c)*m)*x^3 + ((B*a^2 + 2*A*a*b)*m^5 + 19*(B*a^2 + 2*A*a*b)*m^4 + 137*(B*a^2 + 2*A*a*b)*m^3 + 3
60*B*a^2 + 720*A*a*b + 461*(B*a^2 + 2*A*a*b)*m^2 + 702*(B*a^2 + 2*A*a*b)*m)*x^2 + (A*a^2*m^5 + 20*A*a^2*m^4 +
155*A*a^2*m^3 + 580*A*a^2*m^2 + 1044*A*a^2*m + 720*A*a^2)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*
m^2 + 1764*m + 720)

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giac [B]  time = 0.25, size = 1142, normalized size = 7.37 \[ \frac {B c^{2} m^{5} x^{6} x^{m} e^{m} + 2 \, B b c m^{5} x^{5} x^{m} e^{m} + A c^{2} m^{5} x^{5} x^{m} e^{m} + 15 \, B c^{2} m^{4} x^{6} x^{m} e^{m} + B b^{2} m^{5} x^{4} x^{m} e^{m} + 2 \, B a c m^{5} x^{4} x^{m} e^{m} + 2 \, A b c m^{5} x^{4} x^{m} e^{m} + 32 \, B b c m^{4} x^{5} x^{m} e^{m} + 16 \, A c^{2} m^{4} x^{5} x^{m} e^{m} + 85 \, B c^{2} m^{3} x^{6} x^{m} e^{m} + 2 \, B a b m^{5} x^{3} x^{m} e^{m} + A b^{2} m^{5} x^{3} x^{m} e^{m} + 2 \, A a c m^{5} x^{3} x^{m} e^{m} + 17 \, B b^{2} m^{4} x^{4} x^{m} e^{m} + 34 \, B a c m^{4} x^{4} x^{m} e^{m} + 34 \, A b c m^{4} x^{4} x^{m} e^{m} + 190 \, B b c m^{3} x^{5} x^{m} e^{m} + 95 \, A c^{2} m^{3} x^{5} x^{m} e^{m} + 225 \, B c^{2} m^{2} x^{6} x^{m} e^{m} + B a^{2} m^{5} x^{2} x^{m} e^{m} + 2 \, A a b m^{5} x^{2} x^{m} e^{m} + 36 \, B a b m^{4} x^{3} x^{m} e^{m} + 18 \, A b^{2} m^{4} x^{3} x^{m} e^{m} + 36 \, A a c m^{4} x^{3} x^{m} e^{m} + 107 \, B b^{2} m^{3} x^{4} x^{m} e^{m} + 214 \, B a c m^{3} x^{4} x^{m} e^{m} + 214 \, A b c m^{3} x^{4} x^{m} e^{m} + 520 \, B b c m^{2} x^{5} x^{m} e^{m} + 260 \, A c^{2} m^{2} x^{5} x^{m} e^{m} + 274 \, B c^{2} m x^{6} x^{m} e^{m} + A a^{2} m^{5} x x^{m} e^{m} + 19 \, B a^{2} m^{4} x^{2} x^{m} e^{m} + 38 \, A a b m^{4} x^{2} x^{m} e^{m} + 242 \, B a b m^{3} x^{3} x^{m} e^{m} + 121 \, A b^{2} m^{3} x^{3} x^{m} e^{m} + 242 \, A a c m^{3} x^{3} x^{m} e^{m} + 307 \, B b^{2} m^{2} x^{4} x^{m} e^{m} + 614 \, B a c m^{2} x^{4} x^{m} e^{m} + 614 \, A b c m^{2} x^{4} x^{m} e^{m} + 648 \, B b c m x^{5} x^{m} e^{m} + 324 \, A c^{2} m x^{5} x^{m} e^{m} + 120 \, B c^{2} x^{6} x^{m} e^{m} + 20 \, A a^{2} m^{4} x x^{m} e^{m} + 137 \, B a^{2} m^{3} x^{2} x^{m} e^{m} + 274 \, A a b m^{3} x^{2} x^{m} e^{m} + 744 \, B a b m^{2} x^{3} x^{m} e^{m} + 372 \, A b^{2} m^{2} x^{3} x^{m} e^{m} + 744 \, A a c m^{2} x^{3} x^{m} e^{m} + 396 \, B b^{2} m x^{4} x^{m} e^{m} + 792 \, B a c m x^{4} x^{m} e^{m} + 792 \, A b c m x^{4} x^{m} e^{m} + 288 \, B b c x^{5} x^{m} e^{m} + 144 \, A c^{2} x^{5} x^{m} e^{m} + 155 \, A a^{2} m^{3} x x^{m} e^{m} + 461 \, B a^{2} m^{2} x^{2} x^{m} e^{m} + 922 \, A a b m^{2} x^{2} x^{m} e^{m} + 1016 \, B a b m x^{3} x^{m} e^{m} + 508 \, A b^{2} m x^{3} x^{m} e^{m} + 1016 \, A a c m x^{3} x^{m} e^{m} + 180 \, B b^{2} x^{4} x^{m} e^{m} + 360 \, B a c x^{4} x^{m} e^{m} + 360 \, A b c x^{4} x^{m} e^{m} + 580 \, A a^{2} m^{2} x x^{m} e^{m} + 702 \, B a^{2} m x^{2} x^{m} e^{m} + 1404 \, A a b m x^{2} x^{m} e^{m} + 480 \, B a b x^{3} x^{m} e^{m} + 240 \, A b^{2} x^{3} x^{m} e^{m} + 480 \, A a c x^{3} x^{m} e^{m} + 1044 \, A a^{2} m x x^{m} e^{m} + 360 \, B a^{2} x^{2} x^{m} e^{m} + 720 \, A a b x^{2} x^{m} e^{m} + 720 \, A a^{2} x x^{m} e^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c^2*m^5*x^6*x^m*e^m + 2*B*b*c*m^5*x^5*x^m*e^m + A*c^2*m^5*x^5*x^m*e^m + 15*B*c^2*m^4*x^6*x^m*e^m + B*b^2*m^
5*x^4*x^m*e^m + 2*B*a*c*m^5*x^4*x^m*e^m + 2*A*b*c*m^5*x^4*x^m*e^m + 32*B*b*c*m^4*x^5*x^m*e^m + 16*A*c^2*m^4*x^
5*x^m*e^m + 85*B*c^2*m^3*x^6*x^m*e^m + 2*B*a*b*m^5*x^3*x^m*e^m + A*b^2*m^5*x^3*x^m*e^m + 2*A*a*c*m^5*x^3*x^m*e
^m + 17*B*b^2*m^4*x^4*x^m*e^m + 34*B*a*c*m^4*x^4*x^m*e^m + 34*A*b*c*m^4*x^4*x^m*e^m + 190*B*b*c*m^3*x^5*x^m*e^
m + 95*A*c^2*m^3*x^5*x^m*e^m + 225*B*c^2*m^2*x^6*x^m*e^m + B*a^2*m^5*x^2*x^m*e^m + 2*A*a*b*m^5*x^2*x^m*e^m + 3
6*B*a*b*m^4*x^3*x^m*e^m + 18*A*b^2*m^4*x^3*x^m*e^m + 36*A*a*c*m^4*x^3*x^m*e^m + 107*B*b^2*m^3*x^4*x^m*e^m + 21
4*B*a*c*m^3*x^4*x^m*e^m + 214*A*b*c*m^3*x^4*x^m*e^m + 520*B*b*c*m^2*x^5*x^m*e^m + 260*A*c^2*m^2*x^5*x^m*e^m +
274*B*c^2*m*x^6*x^m*e^m + A*a^2*m^5*x*x^m*e^m + 19*B*a^2*m^4*x^2*x^m*e^m + 38*A*a*b*m^4*x^2*x^m*e^m + 242*B*a*
b*m^3*x^3*x^m*e^m + 121*A*b^2*m^3*x^3*x^m*e^m + 242*A*a*c*m^3*x^3*x^m*e^m + 307*B*b^2*m^2*x^4*x^m*e^m + 614*B*
a*c*m^2*x^4*x^m*e^m + 614*A*b*c*m^2*x^4*x^m*e^m + 648*B*b*c*m*x^5*x^m*e^m + 324*A*c^2*m*x^5*x^m*e^m + 120*B*c^
2*x^6*x^m*e^m + 20*A*a^2*m^4*x*x^m*e^m + 137*B*a^2*m^3*x^2*x^m*e^m + 274*A*a*b*m^3*x^2*x^m*e^m + 744*B*a*b*m^2
*x^3*x^m*e^m + 372*A*b^2*m^2*x^3*x^m*e^m + 744*A*a*c*m^2*x^3*x^m*e^m + 396*B*b^2*m*x^4*x^m*e^m + 792*B*a*c*m*x
^4*x^m*e^m + 792*A*b*c*m*x^4*x^m*e^m + 288*B*b*c*x^5*x^m*e^m + 144*A*c^2*x^5*x^m*e^m + 155*A*a^2*m^3*x*x^m*e^m
+ 461*B*a^2*m^2*x^2*x^m*e^m + 922*A*a*b*m^2*x^2*x^m*e^m + 1016*B*a*b*m*x^3*x^m*e^m + 508*A*b^2*m*x^3*x^m*e^m
+ 1016*A*a*c*m*x^3*x^m*e^m + 180*B*b^2*x^4*x^m*e^m + 360*B*a*c*x^4*x^m*e^m + 360*A*b*c*x^4*x^m*e^m + 580*A*a^2
*m^2*x*x^m*e^m + 702*B*a^2*m*x^2*x^m*e^m + 1404*A*a*b*m*x^2*x^m*e^m + 480*B*a*b*x^3*x^m*e^m + 240*A*b^2*x^3*x^
m*e^m + 480*A*a*c*x^3*x^m*e^m + 1044*A*a^2*m*x*x^m*e^m + 360*B*a^2*x^2*x^m*e^m + 720*A*a*b*x^2*x^m*e^m + 720*A
*a^2*x*x^m*e^m)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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maple [B]  time = 0.05, size = 759, normalized size = 4.90 \[ \frac {\left (B \,c^{2} m^{5} x^{5}+A \,c^{2} m^{5} x^{4}+2 B b c \,m^{5} x^{4}+15 B \,c^{2} m^{4} x^{5}+2 A b c \,m^{5} x^{3}+16 A \,c^{2} m^{4} x^{4}+2 B a c \,m^{5} x^{3}+B \,b^{2} m^{5} x^{3}+32 B b c \,m^{4} x^{4}+85 B \,c^{2} m^{3} x^{5}+2 A a c \,m^{5} x^{2}+A \,b^{2} m^{5} x^{2}+34 A b c \,m^{4} x^{3}+95 A \,c^{2} m^{3} x^{4}+2 B a b \,m^{5} x^{2}+34 B a c \,m^{4} x^{3}+17 B \,b^{2} m^{4} x^{3}+190 B b c \,m^{3} x^{4}+225 B \,c^{2} m^{2} x^{5}+2 A a b \,m^{5} x +36 A a c \,m^{4} x^{2}+18 A \,b^{2} m^{4} x^{2}+214 A b c \,m^{3} x^{3}+260 A \,c^{2} m^{2} x^{4}+B \,a^{2} m^{5} x +36 B a b \,m^{4} x^{2}+214 B a c \,m^{3} x^{3}+107 B \,b^{2} m^{3} x^{3}+520 B b c \,m^{2} x^{4}+274 B \,c^{2} m \,x^{5}+A \,a^{2} m^{5}+38 A a b \,m^{4} x +242 A a c \,m^{3} x^{2}+121 A \,b^{2} m^{3} x^{2}+614 A b c \,m^{2} x^{3}+324 A \,c^{2} m \,x^{4}+19 B \,a^{2} m^{4} x +242 B a b \,m^{3} x^{2}+614 B a c \,m^{2} x^{3}+307 B \,b^{2} m^{2} x^{3}+648 B b c m \,x^{4}+120 B \,c^{2} x^{5}+20 A \,a^{2} m^{4}+274 A a b \,m^{3} x +744 A a c \,m^{2} x^{2}+372 A \,b^{2} m^{2} x^{2}+792 A b c m \,x^{3}+144 A \,c^{2} x^{4}+137 B \,a^{2} m^{3} x +744 B a b \,m^{2} x^{2}+792 B a c m \,x^{3}+396 B \,b^{2} m \,x^{3}+288 B b c \,x^{4}+155 A \,a^{2} m^{3}+922 A a b \,m^{2} x +1016 A a c m \,x^{2}+508 A \,b^{2} m \,x^{2}+360 x^{3} A b c +461 B \,a^{2} m^{2} x +1016 B a b m \,x^{2}+360 B a c \,x^{3}+180 B \,b^{2} x^{3}+580 A \,a^{2} m^{2}+1404 A a b m x +480 A a c \,x^{2}+240 A \,b^{2} x^{2}+702 B \,a^{2} m x +480 B a b \,x^{2}+1044 A \,a^{2} m +720 A a b x +360 B \,a^{2} x +720 A \,a^{2}\right ) x \left (e x \right )^{m}}{\left (m +6\right ) \left (m +5\right ) \left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*c^2*m^5*x^5+A*c^2*m^5*x^4+2*B*b*c*m^5*x^4+15*B*c^2*m^4*x^5+2*A*b*c*m^5*x^3+16*A*c^2*m^4*x^4+2*B*a*c*m^5*x
^3+B*b^2*m^5*x^3+32*B*b*c*m^4*x^4+85*B*c^2*m^3*x^5+2*A*a*c*m^5*x^2+A*b^2*m^5*x^2+34*A*b*c*m^4*x^3+95*A*c^2*m^3
*x^4+2*B*a*b*m^5*x^2+34*B*a*c*m^4*x^3+17*B*b^2*m^4*x^3+190*B*b*c*m^3*x^4+225*B*c^2*m^2*x^5+2*A*a*b*m^5*x+36*A*
a*c*m^4*x^2+18*A*b^2*m^4*x^2+214*A*b*c*m^3*x^3+260*A*c^2*m^2*x^4+B*a^2*m^5*x+36*B*a*b*m^4*x^2+214*B*a*c*m^3*x^
3+107*B*b^2*m^3*x^3+520*B*b*c*m^2*x^4+274*B*c^2*m*x^5+A*a^2*m^5+38*A*a*b*m^4*x+242*A*a*c*m^3*x^2+121*A*b^2*m^3
*x^2+614*A*b*c*m^2*x^3+324*A*c^2*m*x^4+19*B*a^2*m^4*x+242*B*a*b*m^3*x^2+614*B*a*c*m^2*x^3+307*B*b^2*m^2*x^3+64
8*B*b*c*m*x^4+120*B*c^2*x^5+20*A*a^2*m^4+274*A*a*b*m^3*x+744*A*a*c*m^2*x^2+372*A*b^2*m^2*x^2+792*A*b*c*m*x^3+1
44*A*c^2*x^4+137*B*a^2*m^3*x+744*B*a*b*m^2*x^2+792*B*a*c*m*x^3+396*B*b^2*m*x^3+288*B*b*c*x^4+155*A*a^2*m^3+922
*A*a*b*m^2*x+1016*A*a*c*m*x^2+508*A*b^2*m*x^2+360*A*b*c*x^3+461*B*a^2*m^2*x+1016*B*a*b*m*x^2+360*B*a*c*x^3+180
*B*b^2*x^3+580*A*a^2*m^2+1404*A*a*b*m*x+480*A*a*c*x^2+240*A*b^2*x^2+702*B*a^2*m*x+480*B*a*b*x^2+1044*A*a^2*m+7
20*A*a*b*x+360*B*a^2*x+720*A*a^2)*(e*x)^m/(m+6)/(m+5)/(m+4)/(m+3)/(m+2)/(m+1)

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maxima [A]  time = 0.79, size = 230, normalized size = 1.48 \[ \frac {B c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, B b c e^{m} x^{5} x^{m}}{m + 5} + \frac {A c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B b^{2} e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, B a c e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, A b c e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, B a b e^{m} x^{3} x^{m}}{m + 3} + \frac {A b^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {2 \, A a b e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{2}}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

B*c^2*e^m*x^6*x^m/(m + 6) + 2*B*b*c*e^m*x^5*x^m/(m + 5) + A*c^2*e^m*x^5*x^m/(m + 5) + B*b^2*e^m*x^4*x^m/(m + 4
) + 2*B*a*c*e^m*x^4*x^m/(m + 4) + 2*A*b*c*e^m*x^4*x^m/(m + 4) + 2*B*a*b*e^m*x^3*x^m/(m + 3) + A*b^2*e^m*x^3*x^
m/(m + 3) + 2*A*a*c*e^m*x^3*x^m/(m + 3) + B*a^2*e^m*x^2*x^m/(m + 2) + 2*A*a*b*e^m*x^2*x^m/(m + 2) + (e*x)^(m +
1)*A*a^2/(e*(m + 1))

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mupad [B]  time = 1.66, size = 405, normalized size = 2.61 \[ {\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {x^4\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {A\,a^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,x^2\,\left (2\,A\,b+B\,a\right )\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {c\,x^5\,\left (A\,c+2\,B\,b\right )\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {B\,c^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x)^m*((x^3*(A*b^2 + 2*A*a*c + 2*B*a*b)*(508*m + 372*m^2 + 121*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2
+ 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (x^4*(B*b^2 + 2*A*b*c + 2*B*a*c)*(396*m + 307*m^2 + 107*m^3 + 17*
m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (A*a^2*x*(1044*m + 580*m^2 +
155*m^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a*x^2*(2*A*b +
B*a)*(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 +
720) + (c*x^5*(A*c + 2*B*b)*(324*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^3 + 1
75*m^4 + 21*m^5 + m^6 + 720) + (B*c^2*x^6*(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))

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sympy [A]  time = 2.63, size = 4150, normalized size = 26.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**2,x)

[Out]

Piecewise(((-A*a**2/(5*x**5) - A*a*b/(2*x**4) - 2*A*a*c/(3*x**3) - A*b**2/(3*x**3) - A*b*c/x**2 - A*c**2/x - B
*a**2/(4*x**4) - 2*B*a*b/(3*x**3) - B*a*c/x**2 - B*b**2/(2*x**2) - 2*B*b*c/x + B*c**2*log(x))/e**6, Eq(m, -6))
, ((-A*a**2/(4*x**4) - 2*A*a*b/(3*x**3) - A*a*c/x**2 - A*b**2/(2*x**2) - 2*A*b*c/x + A*c**2*log(x) - B*a**2/(3
*x**3) - B*a*b/x**2 - 2*B*a*c/x - B*b**2/x + 2*B*b*c*log(x) + B*c**2*x)/e**5, Eq(m, -5)), ((-A*a**2/(3*x**3) -
A*a*b/x**2 - 2*A*a*c/x - A*b**2/x + 2*A*b*c*log(x) + A*c**2*x - B*a**2/(2*x**2) - 2*B*a*b/x + 2*B*a*c*log(x)
+ B*b**2*log(x) + 2*B*b*c*x + B*c**2*x**2/2)/e**4, Eq(m, -4)), ((-A*a**2/(2*x**2) - 2*A*a*b/x + 2*A*a*c*log(x)
+ A*b**2*log(x) + 2*A*b*c*x + A*c**2*x**2/2 - B*a**2/x + 2*B*a*b*log(x) + 2*B*a*c*x + B*b**2*x + B*b*c*x**2 +
B*c**2*x**3/3)/e**3, Eq(m, -3)), ((-A*a**2/x + 2*A*a*b*log(x) + 2*A*a*c*x + A*b**2*x + A*b*c*x**2 + A*c**2*x*
*3/3 + B*a**2*log(x) + 2*B*a*b*x + B*a*c*x**2 + B*b**2*x**2/2 + 2*B*b*c*x**3/3 + B*c**2*x**4/4)/e**2, Eq(m, -2
)), ((A*a**2*log(x) + 2*A*a*b*x + A*a*c*x**2 + A*b**2*x**2/2 + 2*A*b*c*x**3/3 + A*c**2*x**4/4 + B*a**2*x + B*a
*b*x**2 + 2*B*a*c*x**3/3 + B*b**2*x**3/3 + B*b*c*x**4/2 + B*c**2*x**5/5)/e, Eq(m, -1)), (A*a**2*e**m*m**5*x*x*
*m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*A*a**2*e**m*m**4*x*x**m/(m**6 + 21*m
**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*A*a**2*e**m*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*A*a**2*e**m*m**2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + 1044*A*a**2*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764
*m + 720) + 720*A*a**2*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*A*a*b
*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 38*A*a*b*e**m*m**4*x*
*2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 274*A*a*b*e**m*m**3*x**2*x**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 922*A*a*b*e**m*m**2*x**2*x**m/(m**6 + 21*m**5
+ 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1404*A*a*b*e**m*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 7
35*m**3 + 1624*m**2 + 1764*m + 720) + 720*A*a*b*e**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m*
*2 + 1764*m + 720) + 2*A*a*c*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 36*A*a*c*e**m*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*A*
a*c*e**m*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 744*A*a*c*e**m*m**
2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1016*A*a*c*e**m*m*x**3*x**m/(m
**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 480*A*a*c*e**m*x**3*x**m/(m**6 + 21*m**5 + 1
75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*b**2*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m
**3 + 1624*m**2 + 1764*m + 720) + 18*A*b**2*e**m*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m
**2 + 1764*m + 720) + 121*A*b**2*e**m*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 372*A*b**2*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
508*A*b**2*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 240*A*b**2*e**
m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*A*b*c*e**m*m**5*x**4*x**m/(m
**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*A*b*c*e**m*m**4*x**4*x**m/(m**6 + 21*m**5
+ 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 214*A*b*c*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) + 614*A*b*c*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + 792*A*b*c*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) + 360*A*b*c*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*c
**2*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 16*A*c**2*e**m*m**
4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 95*A*c**2*e**m*m**3*x**5*x**m/
(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 260*A*c**2*e**m*m**2*x**5*x**m/(m**6 + 21*
m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 324*A*c**2*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**
4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*A*c**2*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1
624*m**2 + 1764*m + 720) + B*a**2*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764
*m + 720) + 19*B*a**2*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
137*B*a**2*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 461*B*a**2*
e**m*m**2*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 702*B*a**2*e**m*m*x**2
*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*B*a**2*e**m*x**2*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*B*a*b*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 36*B*a*b*e**m*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 242*B*a*b*e**m*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 744*B*a*b*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 1016*B*a*b*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 480*B*a
*b*e**m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*B*a*c*e**m*m**5*x**4*x
**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*B*a*c*e**m*m**4*x**4*x**m/(m**6 + 2
1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 214*B*a*c*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 614*B*a*c*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720) + 792*B*a*c*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 360*B*a*c*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
+ B*b**2*e**m*m**5*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 17*B*b**2*e*
*m*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 107*B*b**2*e**m*m**3*x**
4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 307*B*b**2*e**m*m**2*x**4*x**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 396*B*b**2*e**m*m*x**4*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 180*B*b**2*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m
**3 + 1624*m**2 + 1764*m + 720) + 2*B*b*c*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 32*B*b*c*e**m*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 190*B*b*c*e**m*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 520*B
*b*c*e**m*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 648*B*b*c*e**m*m*
x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 288*B*b*c*e**m*x**5*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + B*c**2*e**m*m**5*x**6*x**m/(m**6 + 21*m**5 + 175*
m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*B*c**2*e**m*m**4*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m
**3 + 1624*m**2 + 1764*m + 720) + 85*B*c**2*e**m*m**3*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m
**2 + 1764*m + 720) + 225*B*c**2*e**m*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 274*B*c**2*e**m*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 120
*B*c**2*e**m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720), True))

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