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

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

[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))

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Rubi [A]  time = 0.0974905, antiderivative size = 155, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.47889, 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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Maple [B]  time = 0.007, size = 759, normalized size = 4.9 \begin{align*}{\frac{ \left ( B{c}^{2}{m}^{5}{x}^{5}+A{c}^{2}{m}^{5}{x}^{4}+2\,Bbc{m}^{5}{x}^{4}+15\,B{c}^{2}{m}^{4}{x}^{5}+2\,Abc{m}^{5}{x}^{3}+16\,A{c}^{2}{m}^{4}{x}^{4}+2\,Bac{m}^{5}{x}^{3}+B{b}^{2}{m}^{5}{x}^{3}+32\,Bbc{m}^{4}{x}^{4}+85\,B{c}^{2}{m}^{3}{x}^{5}+2\,Aac{m}^{5}{x}^{2}+A{b}^{2}{m}^{5}{x}^{2}+34\,Abc{m}^{4}{x}^{3}+95\,A{c}^{2}{m}^{3}{x}^{4}+2\,Bab{m}^{5}{x}^{2}+34\,Bac{m}^{4}{x}^{3}+17\,B{b}^{2}{m}^{4}{x}^{3}+190\,Bbc{m}^{3}{x}^{4}+225\,B{c}^{2}{m}^{2}{x}^{5}+2\,Aab{m}^{5}x+36\,Aac{m}^{4}{x}^{2}+18\,A{b}^{2}{m}^{4}{x}^{2}+214\,Abc{m}^{3}{x}^{3}+260\,A{c}^{2}{m}^{2}{x}^{4}+B{a}^{2}{m}^{5}x+36\,Bab{m}^{4}{x}^{2}+214\,Bac{m}^{3}{x}^{3}+107\,B{b}^{2}{m}^{3}{x}^{3}+520\,Bbc{m}^{2}{x}^{4}+274\,B{c}^{2}m{x}^{5}+A{a}^{2}{m}^{5}+38\,Aab{m}^{4}x+242\,Aac{m}^{3}{x}^{2}+121\,A{b}^{2}{m}^{3}{x}^{2}+614\,Abc{m}^{2}{x}^{3}+324\,A{c}^{2}m{x}^{4}+19\,B{a}^{2}{m}^{4}x+242\,Bab{m}^{3}{x}^{2}+614\,Bac{m}^{2}{x}^{3}+307\,B{b}^{2}{m}^{2}{x}^{3}+648\,Bbcm{x}^{4}+120\,B{c}^{2}{x}^{5}+20\,A{a}^{2}{m}^{4}+274\,Aab{m}^{3}x+744\,Aac{m}^{2}{x}^{2}+372\,A{b}^{2}{m}^{2}{x}^{2}+792\,Abcm{x}^{3}+144\,A{c}^{2}{x}^{4}+137\,B{a}^{2}{m}^{3}x+744\,Bab{m}^{2}{x}^{2}+792\,Bacm{x}^{3}+396\,B{b}^{2}m{x}^{3}+288\,B{x}^{4}bc+155\,A{a}^{2}{m}^{3}+922\,Aab{m}^{2}x+1016\,Aacm{x}^{2}+508\,A{b}^{2}m{x}^{2}+360\,A{x}^{3}bc+461\,B{a}^{2}{m}^{2}x+1016\,Babm{x}^{2}+360\,aBc{x}^{3}+180\,{b}^{2}B{x}^{3}+580\,A{a}^{2}{m}^{2}+1404\,Aabmx+480\,aAc{x}^{2}+240\,A{b}^{2}{x}^{2}+702\,B{a}^{2}mx+480\,B{x}^{2}ab+1044\,A{a}^{2}m+720\,aAbx+360\,{a}^{2}Bx+720\,A{a}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

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/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.66176, size = 1365, normalized size = 8.81 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 3.52484, size = 4150, normalized size = 26.77 \begin{align*} \text{result too large to display} \end{align*}

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))

________________________________________________________________________________________

Giac [B]  time = 1.4341, size = 1542, normalized size = 9.95 \begin{align*} \text{result too large to display} \end{align*}

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)