### 3.2084 $$\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx$$

Optimal. Leaf size=130 $-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac{c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)}$

[Out]

-(((c*d^2 - a*e^2)^3*(d + e*x)^(4 + m))/(e^4*(4 + m))) + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(5 + m))/(e^4*(5 +
m)) - (3*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(6 + m))/(e^4*(6 + m)) + (c^3*d^3*(d + e*x)^(7 + m))/(e^4*(7 + m))

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Rubi [A]  time = 0.0765587, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac{c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-(((c*d^2 - a*e^2)^3*(d + e*x)^(4 + m))/(e^4*(4 + m))) + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(5 + m))/(e^4*(5 +
m)) - (3*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(6 + m))/(e^4*(6 + m)) + (c^3*d^3*(d + e*x)^(7 + m))/(e^4*(7 + m))

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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 (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^{3+m} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^{3+m}}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{4+m}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5+m}}{e^3}+\frac{c^3 d^3 (d+e x)^{6+m}}{e^3}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{6+m}}{e^4 (6+m)}+\frac{c^3 d^3 (d+e x)^{7+m}}{e^4 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0909152, size = 114, normalized size = 0.88 $\frac{(d+e x)^{m+4} \left (-\frac{3 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )}{m+6}+\frac{3 c d (d+e x) \left (c d^2-a e^2\right )^2}{m+5}-\frac{\left (c d^2-a e^2\right )^3}{m+4}+\frac{c^3 d^3 (d+e x)^3}{m+7}\right )}{e^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

((d + e*x)^(4 + m)*(-((c*d^2 - a*e^2)^3/(4 + m)) + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x))/(5 + m) - (3*c^2*d^2*(c
*d^2 - a*e^2)*(d + e*x)^2)/(6 + m) + (c^3*d^3*(d + e*x)^3)/(7 + m)))/e^4

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Maple [B]  time = 0.047, size = 436, normalized size = 3.4 \begin{align*}{\frac{ \left ( ex+d \right ) ^{4+m} \left ({c}^{3}{d}^{3}{e}^{3}{m}^{3}{x}^{3}+3\,a{c}^{2}{d}^{2}{e}^{4}{m}^{3}{x}^{2}+15\,{c}^{3}{d}^{3}{e}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}cd{e}^{5}{m}^{3}x+48\,a{c}^{2}{d}^{2}{e}^{4}{m}^{2}{x}^{2}-3\,{c}^{3}{d}^{4}{e}^{2}{m}^{2}{x}^{2}+74\,{c}^{3}{d}^{3}{e}^{3}m{x}^{3}+{a}^{3}{e}^{6}{m}^{3}+51\,{a}^{2}cd{e}^{5}{m}^{2}x-6\,a{c}^{2}{d}^{3}{e}^{3}{m}^{2}x+249\,a{c}^{2}{d}^{2}{e}^{4}m{x}^{2}-27\,{c}^{3}{d}^{4}{e}^{2}m{x}^{2}+120\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+18\,{a}^{3}{e}^{6}{m}^{2}-3\,{a}^{2}c{d}^{2}{e}^{4}{m}^{2}+282\,{a}^{2}cd{e}^{5}mx-66\,a{c}^{2}{d}^{3}{e}^{3}mx+420\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+6\,{c}^{3}{d}^{5}emx-60\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+107\,{a}^{3}{e}^{6}m-39\,{a}^{2}c{d}^{2}{e}^{4}m+504\,{a}^{2}cd{e}^{5}x+6\,a{c}^{2}{d}^{4}{e}^{2}m-168\,a{c}^{2}{d}^{3}{e}^{3}x+24\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-126\,{a}^{2}c{d}^{2}{e}^{4}+42\,a{c}^{2}{d}^{4}{e}^{2}-6\,{c}^{3}{d}^{6} \right ) }{{e}^{4} \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

(e*x+d)^(4+m)*(c^3*d^3*e^3*m^3*x^3+3*a*c^2*d^2*e^4*m^3*x^2+15*c^3*d^3*e^3*m^2*x^3+3*a^2*c*d*e^5*m^3*x+48*a*c^2
*d^2*e^4*m^2*x^2-3*c^3*d^4*e^2*m^2*x^2+74*c^3*d^3*e^3*m*x^3+a^3*e^6*m^3+51*a^2*c*d*e^5*m^2*x-6*a*c^2*d^3*e^3*m
^2*x+249*a*c^2*d^2*e^4*m*x^2-27*c^3*d^4*e^2*m*x^2+120*c^3*d^3*e^3*x^3+18*a^3*e^6*m^2-3*a^2*c*d^2*e^4*m^2+282*a
^2*c*d*e^5*m*x-66*a*c^2*d^3*e^3*m*x+420*a*c^2*d^2*e^4*x^2+6*c^3*d^5*e*m*x-60*c^3*d^4*e^2*x^2+107*a^3*e^6*m-39*
a^2*c*d^2*e^4*m+504*a^2*c*d*e^5*x+6*a*c^2*d^4*e^2*m-168*a*c^2*d^3*e^3*x+24*c^3*d^5*e*x+210*a^3*e^6-126*a^2*c*d
^2*e^4+42*a*c^2*d^4*e^2-6*c^3*d^6)/e^4/(m^4+22*m^3+179*m^2+638*m+840)

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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+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03512, size = 2418, normalized size = 18.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

(a^3*d^4*e^6*m^3 - 6*c^3*d^10 + 42*a*c^2*d^8*e^2 - 126*a^2*c*d^6*e^4 + 210*a^3*d^4*e^6 + (c^3*d^3*e^7*m^3 + 15
*c^3*d^3*e^7*m^2 + 74*c^3*d^3*e^7*m + 120*c^3*d^3*e^7)*x^7 + (420*c^3*d^4*e^6 + 420*a*c^2*d^2*e^8 + (4*c^3*d^4
*e^6 + 3*a*c^2*d^2*e^8)*m^3 + 3*(19*c^3*d^4*e^6 + 16*a*c^2*d^2*e^8)*m^2 + (269*c^3*d^4*e^6 + 249*a*c^2*d^2*e^8
)*m)*x^6 + 3*(168*c^3*d^5*e^5 + 504*a*c^2*d^3*e^7 + 168*a^2*c*d*e^9 + (2*c^3*d^5*e^5 + 4*a*c^2*d^3*e^7 + a^2*c
*d*e^9)*m^3 + (26*c^3*d^5*e^5 + 62*a*c^2*d^3*e^7 + 17*a^2*c*d*e^9)*m^2 + 2*(57*c^3*d^5*e^5 + 155*a*c^2*d^3*e^7
+ 47*a^2*c*d*e^9)*m)*x^5 + (210*c^3*d^6*e^4 + 1890*a*c^2*d^4*e^6 + 1890*a^2*c*d^2*e^8 + 210*a^3*e^10 + (4*c^3
*d^6*e^4 + 18*a*c^2*d^4*e^6 + 12*a^2*c*d^2*e^8 + a^3*e^10)*m^3 + 3*(14*c^3*d^6*e^4 + 88*a*c^2*d^4*e^6 + 67*a^2
*c*d^2*e^8 + 6*a^3*e^10)*m^2 + (158*c^3*d^6*e^4 + 1236*a*c^2*d^4*e^6 + 1089*a^2*c*d^2*e^8 + 107*a^3*e^10)*m)*x
^4 + (840*a*c^2*d^5*e^5 + 2520*a^2*c*d^3*e^7 + 840*a^3*d*e^9 + (c^3*d^7*e^3 + 12*a*c^2*d^5*e^5 + 18*a^2*c*d^3*
e^7 + 4*a^3*d*e^9)*m^3 + 3*(c^3*d^7*e^3 + 52*a*c^2*d^5*e^5 + 98*a^2*c*d^3*e^7 + 24*a^3*d*e^9)*m^2 + 2*(c^3*d^7
*e^3 + 312*a*c^2*d^5*e^5 + 768*a^2*c*d^3*e^7 + 214*a^3*d*e^9)*m)*x^3 - 3*(a^2*c*d^6*e^4 - 6*a^3*d^4*e^6)*m^2 +
3*(420*a^2*c*d^4*e^6 + 420*a^3*d^2*e^8 + (a*c^2*d^6*e^4 + 4*a^2*c*d^4*e^6 + 2*a^3*d^2*e^8)*m^3 - (c^3*d^8*e^2
- 8*a*c^2*d^6*e^4 - 62*a^2*c*d^4*e^6 - 36*a^3*d^2*e^8)*m^2 - (c^3*d^8*e^2 - 7*a*c^2*d^6*e^4 - 298*a^2*c*d^4*e
^6 - 214*a^3*d^2*e^8)*m)*x^2 + (6*a*c^2*d^8*e^2 - 39*a^2*c*d^6*e^4 + 107*a^3*d^4*e^6)*m + (840*a^3*d^3*e^7 + (
3*a^2*c*d^5*e^5 + 4*a^3*d^3*e^7)*m^3 - 3*(2*a*c^2*d^7*e^3 - 13*a^2*c*d^5*e^5 - 24*a^3*d^3*e^7)*m^2 + 2*(3*c^3*
d^9*e - 21*a*c^2*d^7*e^3 + 63*a^2*c*d^5*e^5 + 214*a^3*d^3*e^7)*m)*x)*(e*x + d)^m/(e^4*m^4 + 22*e^4*m^3 + 179*e
^4*m^2 + 638*e^4*m + 840*e^4)

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Sympy [A]  time = 13.7935, size = 7844, normalized size = 60.34 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Piecewise((c**3*d**6*d**m*x**4/4, Eq(e, 0)), (-8*a**3*d**3*e**6/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*
x**2 + 30*d**3*e**7*x**3) + 6*a**3*d**2*e**7*x/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e*
*7*x**3) + 6*a**3*d*e**8*x**2/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 2*a**3
*e**9*x**3/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) - 3*a**2*c*d**5*e**4/(30*d*
*6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) - 9*a**2*c*d**4*e**5*x/(30*d**6*e**4 + 90*d*
*5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 36*a**2*c*d**3*e**6*x**2/(30*d**6*e**4 + 90*d**5*e**5*x +
90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 12*a**2*c*d**2*e**7*x**3/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e*
*6*x**2 + 30*d**3*e**7*x**3) + 30*a*c**2*d**4*e**5*x**3/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 3
0*d**3*e**7*x**3) + 30*c**3*d**9*log(d/e + x)/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**
7*x**3) + 11*c**3*d**9/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 90*c**3*d**8*
e*x*log(d/e + x)/(30*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 3*c**3*d**8*e*x/(30
*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 90*c**3*d**7*e**2*x**2*log(d/e + x)/(30
*d**6*e**4 + 90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) - 42*c**3*d**7*e**2*x**2/(30*d**6*e**4 +
90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) + 30*c**3*d**6*e**3*x**3*log(d/e + x)/(30*d**6*e**4 +
90*d**5*e**5*x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3) - 44*c**3*d**6*e**3*x**3/(30*d**6*e**4 + 90*d**5*e**5*
x + 90*d**4*e**6*x**2 + 30*d**3*e**7*x**3), Eq(m, -7)), (-7*a**3*d**2*e**6/(20*d**4*e**4 + 40*d**3*e**5*x + 20
*d**2*e**6*x**2) + 6*a**3*d*e**7*x/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 3*a**3*e**8*x**2/(20*
d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) - 3*a**2*c*d**4*e**4/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*
e**6*x**2) - 6*a**2*c*d**3*e**5*x/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 27*a**2*c*d**2*e**6*x*
*2/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 60*a*c**2*d**6*e**2*log(d/e + x)/(20*d**4*e**4 + 40*d
**3*e**5*x + 20*d**2*e**6*x**2) + 27*a*c**2*d**6*e**2/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 12
0*a*c**2*d**5*e**3*x*log(d/e + x)/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) - 6*a*c**2*d**5*e**3*x/(
20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 60*a*c**2*d**4*e**4*x**2*log(d/e + x)/(20*d**4*e**4 + 40*
d**3*e**5*x + 20*d**2*e**6*x**2) - 63*a*c**2*d**4*e**4*x**2/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2
) - 60*c**3*d**8*log(d/e + x)/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) - 27*c**3*d**8/(20*d**4*e**4
+ 40*d**3*e**5*x + 20*d**2*e**6*x**2) - 120*c**3*d**7*e*x*log(d/e + x)/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d*
*2*e**6*x**2) + 6*c**3*d**7*e*x/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) - 60*c**3*d**6*e**2*x**2*l
og(d/e + x)/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2) + 63*c**3*d**6*e**2*x**2/(20*d**4*e**4 + 40*d*
*3*e**5*x + 20*d**2*e**6*x**2) + 20*c**3*d**5*e**3*x**3/(20*d**4*e**4 + 40*d**3*e**5*x + 20*d**2*e**6*x**2), E
q(m, -6)), (-2*a**3*d*e**6/(4*d**2*e**4 + 4*d*e**5*x) + 2*a**3*e**7*x/(4*d**2*e**4 + 4*d*e**5*x) + 12*a**2*c*d
**3*e**4*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) + 6*a**2*c*d**3*e**4/(4*d**2*e**4 + 4*d*e**5*x) + 12*a**2*c*d
**2*e**5*x*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) - 6*a**2*c*d**2*e**5*x/(4*d**2*e**4 + 4*d*e**5*x) - 24*a*c*
*2*d**5*e**2*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) - 14*a*c**2*d**5*e**2/(4*d**2*e**4 + 4*d*e**5*x) - 24*a*c
**2*d**4*e**3*x*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) + 10*a*c**2*d**4*e**3*x/(4*d**2*e**4 + 4*d*e**5*x) + 1
2*a*c**2*d**3*e**4*x**2/(4*d**2*e**4 + 4*d*e**5*x) + 12*c**3*d**7*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) + 7*
c**3*d**7/(4*d**2*e**4 + 4*d*e**5*x) + 12*c**3*d**6*e*x*log(d/e + x)/(4*d**2*e**4 + 4*d*e**5*x) - 5*c**3*d**6*
e*x/(4*d**2*e**4 + 4*d*e**5*x) - 6*c**3*d**5*e**2*x**2/(4*d**2*e**4 + 4*d*e**5*x) + 2*c**3*d**4*e**3*x**3/(4*d
**2*e**4 + 4*d*e**5*x), Eq(m, -5)), (a**3*e**2*log(d/e + x) - 3*a**2*c*d**2*log(d/e + x) + 3*a**2*c*d*e*x + 3*
a*c**2*d**4*log(d/e + x)/e**2 - 3*a*c**2*d**3*x/e + 3*a*c**2*d**2*x**2/2 - c**3*d**6*log(d/e + x)/e**4 + c**3*
d**5*x/e**3 - c**3*d**4*x**2/(2*e**2) + c**3*d**3*x**3/(3*e), Eq(m, -4)), (a**3*d**4*e**6*m**3*(d + e*x)**m/(e
**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 18*a**3*d**4*e**6*m**2*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 107*a**3*d**4*e**6*m*(d + e*x)**m/(e**4*m**4 + 2
2*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*a**3*d**4*e**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m*
*3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*a**3*d**3*e**7*m**3*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 +
179*e**4*m**2 + 638*e**4*m + 840*e**4) + 72*a**3*d**3*e**7*m**2*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 17
9*e**4*m**2 + 638*e**4*m + 840*e**4) + 428*a**3*d**3*e**7*m*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**
4*m**2 + 638*e**4*m + 840*e**4) + 840*a**3*d**3*e**7*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 6*a**3*d**2*e**8*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 +
638*e**4*m + 840*e**4) + 108*a**3*d**2*e**8*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 642*a**3*d**2*e**8*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 +
638*e**4*m + 840*e**4) + 1260*a**3*d**2*e**8*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638
*e**4*m + 840*e**4) + 4*a**3*d*e**9*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**
4*m + 840*e**4) + 72*a**3*d*e**9*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m
+ 840*e**4) + 428*a**3*d*e**9*m*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 84
0*e**4) + 840*a**3*d*e**9*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4)
+ a**3*e**10*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 18*a
**3*e**10*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 107*a**3
*e**10*m*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*a**3*e**10
*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 3*a**2*c*d**6*e**4*m**
2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 39*a**2*c*d**6*e**4*m*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 126*a**2*c*d**6*e**4*(d + e*x)**
m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*a**2*c*d**5*e**5*m**3*x*(d + e*x)**m/
(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 39*a**2*c*d**5*e**5*m**2*x*(d + e*x)**m/(
e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 126*a**2*c*d**5*e**5*m*x*(d + e*x)**m/(e**
4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a**2*c*d**4*e**6*m**3*x**2*(d + e*x)**m/(e
**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 186*a**2*c*d**4*e**6*m**2*x**2*(d + e*x)**m
/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 894*a**2*c*d**4*e**6*m*x**2*(d + e*x)**m
/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1260*a**2*c*d**4*e**6*x**2*(d + e*x)**m/
(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 18*a**2*c*d**3*e**7*m**3*x**3*(d + e*x)**
m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 294*a**2*c*d**3*e**7*m**2*x**3*(d + e*x
)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1536*a**2*c*d**3*e**7*m*x**3*(d + e*
x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 2520*a**2*c*d**3*e**7*x**3*(d + e*x
)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a**2*c*d**2*e**8*m**3*x**4*(d + e
*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 201*a**2*c*d**2*e**8*m**2*x**4*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1089*a**2*c*d**2*e**8*m*x**4*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1890*a**2*c*d**2*e**8*x**4*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*a**2*c*d*e**9*m**3*x**5*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 51*a**2*c*d*e**9*m**2*x**5*(d + e
*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 282*a**2*c*d*e**9*m*x**5*(d + e*x)
**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 504*a**2*c*d*e**9*x**5*(d + e*x)**m/(
e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*a*c**2*d**8*e**2*m*(d + e*x)**m/(e**4*m*
*4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 42*a*c**2*d**8*e**2*(d + e*x)**m/(e**4*m**4 + 22*
e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 6*a*c**2*d**7*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 22*e*
*4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 42*a*c**2*d**7*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m
**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*a*c**2*d**6*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*
m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 24*a*c**2*d**6*e**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**
4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 21*a*c**2*d**6*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4
*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a*c**2*d**5*e**5*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 22*e*
*4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 156*a*c**2*d**5*e**5*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22
*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 624*a*c**2*d**5*e**5*m*x**3*(d + e*x)**m/(e**4*m**4 + 22
*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 840*a*c**2*d**5*e**5*x**3*(d + e*x)**m/(e**4*m**4 + 22*e
**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 18*a*c**2*d**4*e**6*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22
*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 264*a*c**2*d**4*e**6*m**2*x**4*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1236*a*c**2*d**4*e**6*m*x**4*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1890*a*c**2*d**4*e**6*x**4*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a*c**2*d**3*e**7*m**3*x**5*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 186*a*c**2*d**3*e**7*m**2*x**5*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 930*a*c**2*d**3*e**7*m*x**5*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1512*a*c**2*d**3*e**7*x**5*(d + e*x)**m/(e**4*m*
*4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*a*c**2*d**2*e**8*m**3*x**6*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 48*a*c**2*d**2*e**8*m**2*x**6*(d + e*x)**m/(e**4
*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 249*a*c**2*d**2*e**8*m*x**6*(d + e*x)**m/(e**4
*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 420*a*c**2*d**2*e**8*x**6*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 6*c**3*d**10*(d + e*x)**m/(e**4*m**4 + 22*e**4*m
**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*c**3*d**9*e*m*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*
e**4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**8*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e
**4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**8*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*
m**2 + 638*e**4*m + 840*e**4) + c**3*d**7*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**
2 + 638*e**4*m + 840*e**4) + 3*c**3*d**7*e**3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 2*c**3*d**7*e**3*m*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 6
38*e**4*m + 840*e**4) + 4*c**3*d**6*e**4*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 63
8*e**4*m + 840*e**4) + 42*c**3*d**6*e**4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 63
8*e**4*m + 840*e**4) + 158*c**3*d**6*e**4*m*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*
e**4*m + 840*e**4) + 210*c**3*d**6*e**4*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4
*m + 840*e**4) + 6*c**3*d**5*e**5*m**3*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*
m + 840*e**4) + 78*c**3*d**5*e**5*m**2*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*
m + 840*e**4) + 342*c**3*d**5*e**5*m*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m
+ 840*e**4) + 504*c**3*d**5*e**5*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 84
0*e**4) + 4*c**3*d**4*e**6*m**3*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840
*e**4) + 57*c**3*d**4*e**6*m**2*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840
*e**4) + 269*c**3*d**4*e**6*m*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e
**4) + 420*c**3*d**4*e**6*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4)
+ c**3*d**3*e**7*m**3*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) +
15*c**3*d**3*e**7*m**2*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) +
74*c**3*d**3*e**7*m*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 120
*c**3*d**3*e**7*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4), True))

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Giac [B]  time = 1.25022, size = 2696, normalized size = 20.74 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*c^3*d^3*m^3*x^7*e^7 + 4*(x*e + d)^m*c^3*d^4*m^3*x^6*e^6 + 6*(x*e + d)^m*c^3*d^5*m^3*x^5*e^5 + 4*(
x*e + d)^m*c^3*d^6*m^3*x^4*e^4 + (x*e + d)^m*c^3*d^7*m^3*x^3*e^3 + 15*(x*e + d)^m*c^3*d^3*m^2*x^7*e^7 + 57*(x*
e + d)^m*c^3*d^4*m^2*x^6*e^6 + 78*(x*e + d)^m*c^3*d^5*m^2*x^5*e^5 + 42*(x*e + d)^m*c^3*d^6*m^2*x^4*e^4 + 3*(x*
e + d)^m*c^3*d^7*m^2*x^3*e^3 - 3*(x*e + d)^m*c^3*d^8*m^2*x^2*e^2 + 3*(x*e + d)^m*a*c^2*d^2*m^3*x^6*e^8 + 12*(x
*e + d)^m*a*c^2*d^3*m^3*x^5*e^7 + 74*(x*e + d)^m*c^3*d^3*m*x^7*e^7 + 18*(x*e + d)^m*a*c^2*d^4*m^3*x^4*e^6 + 26
9*(x*e + d)^m*c^3*d^4*m*x^6*e^6 + 12*(x*e + d)^m*a*c^2*d^5*m^3*x^3*e^5 + 342*(x*e + d)^m*c^3*d^5*m*x^5*e^5 + 3
*(x*e + d)^m*a*c^2*d^6*m^3*x^2*e^4 + 158*(x*e + d)^m*c^3*d^6*m*x^4*e^4 + 2*(x*e + d)^m*c^3*d^7*m*x^3*e^3 - 3*(
x*e + d)^m*c^3*d^8*m*x^2*e^2 + 6*(x*e + d)^m*c^3*d^9*m*x*e + 48*(x*e + d)^m*a*c^2*d^2*m^2*x^6*e^8 + 186*(x*e +
d)^m*a*c^2*d^3*m^2*x^5*e^7 + 120*(x*e + d)^m*c^3*d^3*x^7*e^7 + 264*(x*e + d)^m*a*c^2*d^4*m^2*x^4*e^6 + 420*(x
*e + d)^m*c^3*d^4*x^6*e^6 + 156*(x*e + d)^m*a*c^2*d^5*m^2*x^3*e^5 + 504*(x*e + d)^m*c^3*d^5*x^5*e^5 + 24*(x*e
+ d)^m*a*c^2*d^6*m^2*x^2*e^4 + 210*(x*e + d)^m*c^3*d^6*x^4*e^4 - 6*(x*e + d)^m*a*c^2*d^7*m^2*x*e^3 - 6*(x*e +
d)^m*c^3*d^10 + 3*(x*e + d)^m*a^2*c*d*m^3*x^5*e^9 + 12*(x*e + d)^m*a^2*c*d^2*m^3*x^4*e^8 + 249*(x*e + d)^m*a*c
^2*d^2*m*x^6*e^8 + 18*(x*e + d)^m*a^2*c*d^3*m^3*x^3*e^7 + 930*(x*e + d)^m*a*c^2*d^3*m*x^5*e^7 + 12*(x*e + d)^m
*a^2*c*d^4*m^3*x^2*e^6 + 1236*(x*e + d)^m*a*c^2*d^4*m*x^4*e^6 + 3*(x*e + d)^m*a^2*c*d^5*m^3*x*e^5 + 624*(x*e +
d)^m*a*c^2*d^5*m*x^3*e^5 + 21*(x*e + d)^m*a*c^2*d^6*m*x^2*e^4 - 42*(x*e + d)^m*a*c^2*d^7*m*x*e^3 + 6*(x*e + d
)^m*a*c^2*d^8*m*e^2 + 51*(x*e + d)^m*a^2*c*d*m^2*x^5*e^9 + 201*(x*e + d)^m*a^2*c*d^2*m^2*x^4*e^8 + 420*(x*e +
d)^m*a*c^2*d^2*x^6*e^8 + 294*(x*e + d)^m*a^2*c*d^3*m^2*x^3*e^7 + 1512*(x*e + d)^m*a*c^2*d^3*x^5*e^7 + 186*(x*e
+ d)^m*a^2*c*d^4*m^2*x^2*e^6 + 1890*(x*e + d)^m*a*c^2*d^4*x^4*e^6 + 39*(x*e + d)^m*a^2*c*d^5*m^2*x*e^5 + 840*
(x*e + d)^m*a*c^2*d^5*x^3*e^5 - 3*(x*e + d)^m*a^2*c*d^6*m^2*e^4 + 42*(x*e + d)^m*a*c^2*d^8*e^2 + (x*e + d)^m*a
^3*m^3*x^4*e^10 + 4*(x*e + d)^m*a^3*d*m^3*x^3*e^9 + 282*(x*e + d)^m*a^2*c*d*m*x^5*e^9 + 6*(x*e + d)^m*a^3*d^2*
m^3*x^2*e^8 + 1089*(x*e + d)^m*a^2*c*d^2*m*x^4*e^8 + 4*(x*e + d)^m*a^3*d^3*m^3*x*e^7 + 1536*(x*e + d)^m*a^2*c*
d^3*m*x^3*e^7 + (x*e + d)^m*a^3*d^4*m^3*e^6 + 894*(x*e + d)^m*a^2*c*d^4*m*x^2*e^6 + 126*(x*e + d)^m*a^2*c*d^5*
m*x*e^5 - 39*(x*e + d)^m*a^2*c*d^6*m*e^4 + 18*(x*e + d)^m*a^3*m^2*x^4*e^10 + 72*(x*e + d)^m*a^3*d*m^2*x^3*e^9
+ 504*(x*e + d)^m*a^2*c*d*x^5*e^9 + 108*(x*e + d)^m*a^3*d^2*m^2*x^2*e^8 + 1890*(x*e + d)^m*a^2*c*d^2*x^4*e^8 +
72*(x*e + d)^m*a^3*d^3*m^2*x*e^7 + 2520*(x*e + d)^m*a^2*c*d^3*x^3*e^7 + 18*(x*e + d)^m*a^3*d^4*m^2*e^6 + 1260
*(x*e + d)^m*a^2*c*d^4*x^2*e^6 - 126*(x*e + d)^m*a^2*c*d^6*e^4 + 107*(x*e + d)^m*a^3*m*x^4*e^10 + 428*(x*e + d
)^m*a^3*d*m*x^3*e^9 + 642*(x*e + d)^m*a^3*d^2*m*x^2*e^8 + 428*(x*e + d)^m*a^3*d^3*m*x*e^7 + 107*(x*e + d)^m*a^
3*d^4*m*e^6 + 210*(x*e + d)^m*a^3*x^4*e^10 + 840*(x*e + d)^m*a^3*d*x^3*e^9 + 1260*(x*e + d)^m*a^3*d^2*x^2*e^8
+ 840*(x*e + d)^m*a^3*d^3*x*e^7 + 210*(x*e + d)^m*a^3*d^4*e^6)/(m^4*e^4 + 22*m^3*e^4 + 179*m^2*e^4 + 638*m*e^4
+ 840*e^4)