3.2148 \(\int (a+b x) (d+e x)^m (a^2+2 a b x+b^2 x^2) \, dx\)

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

[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))

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Rubi [A]  time = 0.0509776, antiderivative size = 111, normalized size of antiderivative = 1., 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} \[ -\frac{3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac{(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac{3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac{b^3 (d+e x)^{m+4}}{e^4 (m+4)} \]

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 = 0.0685515, size = 95, normalized size = 0.86 \[ \frac{(d+e x)^{m+1} \left (-\frac{3 b^2 (d+e x)^2 (b d-a e)}{m+3}+\frac{3 b (d+e x) (b d-a e)^2}{m+2}-\frac{(b d-a e)^3}{m+1}+\frac{b^3 (d+e x)^3}{m+4}\right )}{e^4} \]

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

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Maple [B]  time = 0.007, size = 386, normalized size = 3.5 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \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\,{x}^{3}{b}^{3}{e}^{3}+9\,{a}^{3}{e}^{3}{m}^{2}-3\,{a}^{2}bd{e}^{2}{m}^{2}+57\,{a}^{2}b{e}^{3}mx-30\,a{b}^{2}d{e}^{2}mx+24\,{x}^{2}a{b}^{2}{e}^{3}+6\,{b}^{3}{d}^{2}emx-6\,{x}^{2}{b}^{3}d{e}^{2}+26\,{a}^{3}{e}^{3}m-21\,{a}^{2}bd{e}^{2}m+36\,x{a}^{2}b{e}^{3}+6\,a{b}^{2}{d}^{2}em-24\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e+24\,{e}^{3}{a}^{3}-36\,d{e}^{2}{a}^{2}b+24\,a{d}^{2}e{b}^{2}-6\,{d}^{3}{b}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \end{align*}

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)^(1+m)*(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)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.02863, size = 1011, normalized size = 9.11 \begin{align*} \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{align*}

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)

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Sympy [A]  time = 4.50024, size = 4058, normalized size = 36.56 \begin{align*} \text{result too large to display} \end{align*}

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))

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Giac [B]  time = 1.20421, size = 1127, normalized size = 10.15 \begin{align*} \text{result too large to display} \end{align*}

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)