\(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\) [2085]

Optimal result

Integrand size = 35, antiderivative size = 90 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]

[In]

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

[Out]

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

Rule 45

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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]

Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^2 (d+e x)^{2+m} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3+m}}{e^2}+\frac {c^2 d^2 (d+e x)^{4+m}}{e^2}\right ) \, dx \\ & = \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {(d+e x)^{3+m} \left (\frac {\left (c d^2-a e^2\right )^2}{3+m}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)}{4+m}+\frac {c^2 d^2 (d+e x)^2}{5+m}\right )}{e^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(90)=180\).

Time = 2.98 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.03

method	result	size
gosper	\(\frac {\left (e x +d \right )^{3+m} \left (c^{2} d^{2} e^{2} m^{2} x^{2}+2 a c d \,e^{3} m^{2} x +7 c^{2} d^{2} e^{2} m \,x^{2}+a^{2} e^{4} m^{2}+16 a c d \,e^{3} m x -2 c^{2} d^{3} e m x +12 x^{2} c^{2} d^{2} e^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +30 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right )}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}\)	\(183\)
norman	\(\frac {\left (a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+3 c^{2} d^{4} m^{2}+9 a^{2} e^{4} m +46 a c \,d^{2} e^{2} m +15 c^{2} d^{4} m +20 a^{2} e^{4}+80 a c \,d^{2} e^{2}+20 c^{2} d^{4}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+12 m^{2}+47 m +60}+\frac {d^{3} \left (a^{2} e^{4} m^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c^{2} d^{2} e^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d^{2} \left (3 a^{2} e^{4} m^{2}+2 a c \,d^{2} e^{2} m^{2}+27 a^{2} e^{4} m +10 a c \,d^{2} e^{2} m -2 c^{2} d^{4} m +60 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {d \left (3 a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+c^{2} d^{4} m^{2}+27 a^{2} e^{4} m +42 a c \,d^{2} e^{2} m +c^{2} d^{4} m +60 a^{2} e^{4}+60 a c \,d^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c d e \left (2 a \,e^{2} m +3 c \,d^{2} m +10 e^{2} a +10 c \,d^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+9 m +20}\)	\(495\)
risch	\(\frac {\left (c^{2} d^{2} e^{5} m^{2} x^{5}+2 a c d \,e^{6} m^{2} x^{4}+3 c^{2} d^{3} e^{4} m^{2} x^{4}+7 c^{2} d^{2} e^{5} m \,x^{5}+a^{2} e^{7} m^{2} x^{3}+6 a c \,d^{2} e^{5} m^{2} x^{3}+16 a c d \,e^{6} m \,x^{4}+3 c^{2} d^{4} e^{3} m^{2} x^{3}+19 c^{2} d^{3} e^{4} m \,x^{4}+12 c^{2} d^{2} e^{5} x^{5}+3 a^{2} d \,e^{6} m^{2} x^{2}+9 a^{2} e^{7} m \,x^{3}+6 a c \,d^{3} e^{4} m^{2} x^{2}+46 a c \,d^{2} e^{5} m \,x^{3}+30 a c d \,e^{6} x^{4}+c^{2} d^{5} e^{2} m^{2} x^{2}+15 c^{2} d^{4} e^{3} m \,x^{3}+30 c^{2} d^{3} e^{4} x^{4}+3 a^{2} d^{2} e^{5} m^{2} x +27 a^{2} d \,e^{6} m \,x^{2}+20 a^{2} e^{7} x^{3}+2 a c \,d^{4} e^{3} m^{2} x +42 a c \,d^{3} e^{4} m \,x^{2}+80 a c \,d^{2} e^{5} x^{3}+c^{2} d^{5} e^{2} m \,x^{2}+20 c^{2} d^{4} e^{3} x^{3}+a^{2} d^{3} e^{4} m^{2}+27 a^{2} d^{2} e^{5} m x +60 a^{2} d \,e^{6} x^{2}+10 a c \,d^{4} e^{3} m x +60 a c \,d^{3} e^{4} x^{2}-2 c^{2} d^{6} e m x +9 a^{2} d^{3} e^{4} m +60 a^{2} d^{2} e^{5} x -2 a c \,d^{5} e^{2} m +20 a^{2} d^{3} e^{4}-10 a c \,d^{5} e^{2}+2 c^{2} d^{7}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) e^{3}}\)	\(536\)
parallelrisch	\(\frac {20 \left (e x +d \right )^{m} a^{2} d^{4} e^{4}+2 \left (e x +d \right )^{m} c^{2} d^{8}+9 x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7} m +15 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3} m +3 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6} m^{2}+x^{2} \left (e x +d \right )^{m} c^{2} d^{6} e^{2} m^{2}+80 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5}+27 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6} m +x^{2} \left (e x +d \right )^{m} c^{2} d^{6} e^{2} m +3 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5} m^{2}+60 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4}+27 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5} m -2 x \left (e x +d \right )^{m} c^{2} d^{7} e m -2 \left (e x +d \right )^{m} a c \,d^{6} e^{2} m +30 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4}+20 x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7}+20 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3}+60 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6}+\left (e x +d \right )^{m} a^{2} d^{4} e^{4} m^{2}+60 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5}+9 \left (e x +d \right )^{m} a^{2} d^{4} e^{4} m -10 \left (e x +d \right )^{m} a c \,d^{6} e^{2}+6 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4} m^{2}+42 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4} m +2 x \left (e x +d \right )^{m} a c \,d^{5} e^{3} m^{2}+10 x \left (e x +d \right )^{m} a c \,d^{5} e^{3} m +2 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6} m^{2}+16 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6} m +6 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5} m^{2}+46 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5} m +12 x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5}+x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5} m^{2}+7 x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5} m +3 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4} m^{2}+19 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4} m +x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7} m^{2}+3 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3} m^{2}+30 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6}}{\left (m^{2}+9 m +20\right ) d \left (3+m \right ) e^{3}}\)	\(813\)

[In]

int((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/e^3*(e*x+d)^(3+m)/(m^3+12*m^2+47*m+60)*(c^2*d^2*e^2*m^2*x^2+2*a*c*d*e^3*m^2*x+7*c^2*d^2*e^2*m*x^2+a^2*e^4*m^
2+16*a*c*d*e^3*m*x-2*c^2*d^3*e*m*x+12*c^2*d^2*e^2*x^2+9*a^2*e^4*m-2*a*c*d^2*e^2*m+30*a*c*d*e^3*x-6*c^2*d^3*e*x
+20*a^2*e^4-10*a*c*d^2*e^2+2*c^2*d^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (90) = 180\).

Time = 0.38 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.32 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {{\left (a^{2} d^{3} e^{4} m^{2} + 2 \, c^{2} d^{7} - 10 \, a c d^{5} e^{2} + 20 \, a^{2} d^{3} e^{4} + {\left (c^{2} d^{2} e^{5} m^{2} + 7 \, c^{2} d^{2} e^{5} m + 12 \, c^{2} d^{2} e^{5}\right )} x^{5} + {\left (30 \, c^{2} d^{3} e^{4} + 30 \, a c d e^{6} + {\left (3 \, c^{2} d^{3} e^{4} + 2 \, a c d e^{6}\right )} m^{2} + {\left (19 \, c^{2} d^{3} e^{4} + 16 \, a c d e^{6}\right )} m\right )} x^{4} + {\left (20 \, c^{2} d^{4} e^{3} + 80 \, a c d^{2} e^{5} + 20 \, a^{2} e^{7} + {\left (3 \, c^{2} d^{4} e^{3} + 6 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} m^{2} + {\left (15 \, c^{2} d^{4} e^{3} + 46 \, a c d^{2} e^{5} + 9 \, a^{2} e^{7}\right )} m\right )} x^{3} + {\left (60 \, a c d^{3} e^{4} + 60 \, a^{2} d e^{6} + {\left (c^{2} d^{5} e^{2} + 6 \, a c d^{3} e^{4} + 3 \, a^{2} d e^{6}\right )} m^{2} + {\left (c^{2} d^{5} e^{2} + 42 \, a c d^{3} e^{4} + 27 \, a^{2} d e^{6}\right )} m\right )} x^{2} - {\left (2 \, a c d^{5} e^{2} - 9 \, a^{2} d^{3} e^{4}\right )} m + {\left (60 \, a^{2} d^{2} e^{5} + {\left (2 \, a c d^{4} e^{3} + 3 \, a^{2} d^{2} e^{5}\right )} m^{2} - {\left (2 \, c^{2} d^{6} e - 10 \, a c d^{4} e^{3} - 27 \, a^{2} d^{2} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]

[In]

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

[Out]

(a^2*d^3*e^4*m^2 + 2*c^2*d^7 - 10*a*c*d^5*e^2 + 20*a^2*d^3*e^4 + (c^2*d^2*e^5*m^2 + 7*c^2*d^2*e^5*m + 12*c^2*d
^2*e^5)*x^5 + (30*c^2*d^3*e^4 + 30*a*c*d*e^6 + (3*c^2*d^3*e^4 + 2*a*c*d*e^6)*m^2 + (19*c^2*d^3*e^4 + 16*a*c*d*
e^6)*m)*x^4 + (20*c^2*d^4*e^3 + 80*a*c*d^2*e^5 + 20*a^2*e^7 + (3*c^2*d^4*e^3 + 6*a*c*d^2*e^5 + a^2*e^7)*m^2 +
(15*c^2*d^4*e^3 + 46*a*c*d^2*e^5 + 9*a^2*e^7)*m)*x^3 + (60*a*c*d^3*e^4 + 60*a^2*d*e^6 + (c^2*d^5*e^2 + 6*a*c*d
^3*e^4 + 3*a^2*d*e^6)*m^2 + (c^2*d^5*e^2 + 42*a*c*d^3*e^4 + 27*a^2*d*e^6)*m)*x^2 - (2*a*c*d^5*e^2 - 9*a^2*d^3*
e^4)*m + (60*a^2*d^2*e^5 + (2*a*c*d^4*e^3 + 3*a^2*d^2*e^5)*m^2 - (2*c^2*d^6*e - 10*a*c*d^4*e^3 - 27*a^2*d^2*e^
5)*m)*x)*(e*x + d)^m/(e^3*m^3 + 12*e^3*m^2 + 47*e^3*m + 60*e^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2494 vs. \(2 (78) = 156\).

Time = 1.11 (sec) , antiderivative size = 2494, normalized size of antiderivative = 27.71 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((c**2*d**4*d**m*x**3/3, Eq(e, 0)), (-a**2*e**4/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) - 2*a*c*d**2
*e**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) - 4*a*c*d*e**3*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c
**2*d**4*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 3*c**2*d**4/(2*d**2*e**3 + 4*d*e**4*x + 2*e**
5*x**2) + 4*c**2*d**3*e*x*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c**2*d**3*e*x/(2*d**2*e**3
+ 4*d*e**4*x + 2*e**5*x**2) + 2*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2), Eq
(m, -5)), (-a**2*e**4/(d*e**3 + e**4*x) + 2*a*c*d**2*e**2*log(d/e + x)/(d*e**3 + e**4*x) + 2*a*c*d**2*e**2/(d*
e**3 + e**4*x) + 2*a*c*d*e**3*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*c**2*d**4*log(d/e + x)/(d*e**3 + e**4*x) -
4*c**2*d**4/(d*e**3 + e**4*x) - 2*c**2*d**3*e*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*c**2*d**3*e*x/(d*e**3 + e**
4*x) + c**2*d**2*e**2*x**2/(d*e**3 + e**4*x), Eq(m, -4)), (a**2*e*log(d/e + x) - 2*a*c*d**2*log(d/e + x)/e + 2
*a*c*d*x + c**2*d**4*log(d/e + x)/e**3 - c**2*d**3*x/e**2 + c**2*d**2*x**2/(2*e), Eq(m, -3)), (a**2*d**3*e**4*
m**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 9*a**2*d**3*e**4*m*(d + e*x)**m/(e**3*m**
3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*a**2*d**3*e**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m
+ 60*e**3) + 3*a**2*d**2*e**5*m**2*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 27*a**2*
d**2*e**5*m*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 60*a**2*d**2*e**5*x*(d + e*x)**m
/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 3*a**2*d*e**6*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*
m**2 + 47*e**3*m + 60*e**3) + 27*a**2*d*e**6*m*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e*
*3) + 60*a**2*d*e**6*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + a**2*e**7*m**2*x**3*
(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 9*a**2*e**7*m*x**3*(d + e*x)**m/(e**3*m**3 + 1
2*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*a**2*e**7*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60
*e**3) - 2*a*c*d**5*e**2*m*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) - 10*a*c*d**5*e**2*(d
+ e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 2*a*c*d**4*e**3*m**2*x*(d + e*x)**m/(e**3*m**3 +
12*e**3*m**2 + 47*e**3*m + 60*e**3) + 10*a*c*d**4*e**3*m*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m
+ 60*e**3) + 6*a*c*d**3*e**4*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 42*a*c
*d**3*e**4*m*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 60*a*c*d**3*e**4*x**2*(d + e
*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 6*a*c*d**2*e**5*m**2*x**3*(d + e*x)**m/(e**3*m**3 +
12*e**3*m**2 + 47*e**3*m + 60*e**3) + 46*a*c*d**2*e**5*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3
*m + 60*e**3) + 80*a*c*d**2*e**5*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 2*a*c*d*
e**6*m**2*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 16*a*c*d*e**6*m*x**4*(d + e*x)*
*m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 30*a*c*d*e**6*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**
2 + 47*e**3*m + 60*e**3) + 2*c**2*d**7*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) - 2*c**2*
d**6*e*m*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*d**5*e**2*m**2*x**2*(d + e*x)*
*m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*d**5*e**2*m*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*
m**2 + 47*e**3*m + 60*e**3) + 3*c**2*d**4*e**3*m**2*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m +
60*e**3) + 15*c**2*d**4*e**3*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*c**2*d*
*4*e**3*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 3*c**2*d**3*e**4*m**2*x**4*(d + e
*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 19*c**2*d**3*e**4*m*x**4*(d + e*x)**m/(e**3*m**3 + 1
2*e**3*m**2 + 47*e**3*m + 60*e**3) + 30*c**2*d**3*e**4*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m
+ 60*e**3) + c**2*d**2*e**5*m**2*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 7*c**2*
d**2*e**5*m*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 12*c**2*d**2*e**5*x**5*(d + e
*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 691 vs. \(2 (90) = 180\).

Time = 0.22 (sec) , antiderivative size = 691, normalized size of antiderivative = 7.68 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a c d^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} d e}{m^{2} + 3 \, m + 2} + \frac {{\left (e x + d\right )}^{m + 1} a^{2} d^{2} e}{m + 1} + \frac {{\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} c^{2} d^{4}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {4 \, {\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 c d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} + \frac {{\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^{2} e}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\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} c^{2} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac {2 \, {\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 c d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e} + \frac {{\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} c^{2} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]

[In]

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

[Out]

2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a*c*d^3/((m^2 + 3*m + 2)*e) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d
^2)*(e*x + d)^m*a^2*d*e/(m^2 + 3*m + 2) + (e*x + d)^(m + 1)*a^2*d^2*e/(m + 1) + ((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*c^2*d^4/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 4*((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*c*d^2/((m^3 + 6*m^2 + 11*m + 6)*e) + ((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^2*e/(m^3 + 6*m^2 + 11*m + 6) +
2*((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*c^2*d^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^3) + 2*((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*c*d/((m^4 + 10*m^
3 + 35*m^2 + 50*m + 24)*e) + ((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*c^2*d
^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (90) = 180\).

Time = 0.28 (sec) , antiderivative size = 805, normalized size of antiderivative = 8.94 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {{\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} m^{2} x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} m^{2} x^{4} + 2 \, {\left (e x + d\right )}^{m} a c d e^{6} m^{2} x^{4} + 7 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} m x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} m^{2} x^{3} + 6 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} m^{2} x^{3} + {\left (e x + d\right )}^{m} a^{2} e^{7} m^{2} x^{3} + 19 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} m x^{4} + 16 \, {\left (e x + d\right )}^{m} a c d e^{6} m x^{4} + 12 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} x^{5} + {\left (e x + d\right )}^{m} c^{2} d^{5} e^{2} m^{2} x^{2} + 6 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} m^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} m^{2} x^{2} + 15 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} m x^{3} + 46 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} m x^{3} + 9 \, {\left (e x + d\right )}^{m} a^{2} e^{7} m x^{3} + 30 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} x^{4} + 30 \, {\left (e x + d\right )}^{m} a c d e^{6} x^{4} + 2 \, {\left (e x + d\right )}^{m} a c d^{4} e^{3} m^{2} x + 3 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} m^{2} x + {\left (e x + d\right )}^{m} c^{2} d^{5} e^{2} m x^{2} + 42 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} m x^{2} + 27 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} m x^{2} + 20 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} x^{3} + 80 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} x^{3} + 20 \, {\left (e x + d\right )}^{m} a^{2} e^{7} x^{3} + {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4} m^{2} - 2 \, {\left (e x + d\right )}^{m} c^{2} d^{6} e m x + 10 \, {\left (e x + d\right )}^{m} a c d^{4} e^{3} m x + 27 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} m x + 60 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} x^{2} + 60 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} x^{2} - 2 \, {\left (e x + d\right )}^{m} a c d^{5} e^{2} m + 9 \, {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4} m + 60 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} x + 2 \, {\left (e x + d\right )}^{m} c^{2} d^{7} - 10 \, {\left (e x + d\right )}^{m} a c d^{5} e^{2} + 20 \, {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]

[In]

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

[Out]

((e*x + d)^m*c^2*d^2*e^5*m^2*x^5 + 3*(e*x + d)^m*c^2*d^3*e^4*m^2*x^4 + 2*(e*x + d)^m*a*c*d*e^6*m^2*x^4 + 7*(e*
x + d)^m*c^2*d^2*e^5*m*x^5 + 3*(e*x + d)^m*c^2*d^4*e^3*m^2*x^3 + 6*(e*x + d)^m*a*c*d^2*e^5*m^2*x^3 + (e*x + d)
^m*a^2*e^7*m^2*x^3 + 19*(e*x + d)^m*c^2*d^3*e^4*m*x^4 + 16*(e*x + d)^m*a*c*d*e^6*m*x^4 + 12*(e*x + d)^m*c^2*d^
2*e^5*x^5 + (e*x + d)^m*c^2*d^5*e^2*m^2*x^2 + 6*(e*x + d)^m*a*c*d^3*e^4*m^2*x^2 + 3*(e*x + d)^m*a^2*d*e^6*m^2*
x^2 + 15*(e*x + d)^m*c^2*d^4*e^3*m*x^3 + 46*(e*x + d)^m*a*c*d^2*e^5*m*x^3 + 9*(e*x + d)^m*a^2*e^7*m*x^3 + 30*(
e*x + d)^m*c^2*d^3*e^4*x^4 + 30*(e*x + d)^m*a*c*d*e^6*x^4 + 2*(e*x + d)^m*a*c*d^4*e^3*m^2*x + 3*(e*x + d)^m*a^
2*d^2*e^5*m^2*x + (e*x + d)^m*c^2*d^5*e^2*m*x^2 + 42*(e*x + d)^m*a*c*d^3*e^4*m*x^2 + 27*(e*x + d)^m*a^2*d*e^6*
m*x^2 + 20*(e*x + d)^m*c^2*d^4*e^3*x^3 + 80*(e*x + d)^m*a*c*d^2*e^5*x^3 + 20*(e*x + d)^m*a^2*e^7*x^3 + (e*x +
d)^m*a^2*d^3*e^4*m^2 - 2*(e*x + d)^m*c^2*d^6*e*m*x + 10*(e*x + d)^m*a*c*d^4*e^3*m*x + 27*(e*x + d)^m*a^2*d^2*e
^5*m*x + 60*(e*x + d)^m*a*c*d^3*e^4*x^2 + 60*(e*x + d)^m*a^2*d*e^6*x^2 - 2*(e*x + d)^m*a*c*d^5*e^2*m + 9*(e*x
+ d)^m*a^2*d^3*e^4*m + 60*(e*x + d)^m*a^2*d^2*e^5*x + 2*(e*x + d)^m*c^2*d^7 - 10*(e*x + d)^m*a*c*d^5*e^2 + 20*
(e*x + d)^m*a^2*d^3*e^4)/(e^3*m^3 + 12*e^3*m^2 + 47*e^3*m + 60*e^3)

Mupad [B] (verification not implemented)

Time = 10.13 (sec) , antiderivative size = 486, normalized size of antiderivative = 5.40 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x^3\,\left (a^2\,e^7\,m^2+9\,a^2\,e^7\,m+20\,a^2\,e^7+6\,a\,c\,d^2\,e^5\,m^2+46\,a\,c\,d^2\,e^5\,m+80\,a\,c\,d^2\,e^5+3\,c^2\,d^4\,e^3\,m^2+15\,c^2\,d^4\,e^3\,m+20\,c^2\,d^4\,e^3\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^3\,\left (a^2\,e^4\,m^2+9\,a^2\,e^4\,m+20\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,m-10\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^2\,x\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+2\,a\,c\,d^2\,e^2\,m^2+10\,a\,c\,d^2\,e^2\,m-2\,c^2\,d^4\,m\right )}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d\,x^2\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+6\,a\,c\,d^2\,e^2\,m^2+42\,a\,c\,d^2\,e^2\,m+60\,a\,c\,d^2\,e^2+c^2\,d^4\,m^2+c^2\,d^4\,m\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c\,d\,e\,x^4\,\left (m+3\right )\,\left (10\,a\,e^2+10\,c\,d^2+2\,a\,e^2\,m+3\,c\,d^2\,m\right )}{m^3+12\,m^2+47\,m+60}\right ) \]

[In]

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

[Out]

(d + e*x)^m*((x^3*(20*a^2*e^7 + 9*a^2*e^7*m + 20*c^2*d^4*e^3 + a^2*e^7*m^2 + 15*c^2*d^4*e^3*m + 3*c^2*d^4*e^3*
m^2 + 80*a*c*d^2*e^5 + 46*a*c*d^2*e^5*m + 6*a*c*d^2*e^5*m^2))/(e^3*(47*m + 12*m^2 + m^3 + 60)) + (d^3*(20*a^2*
e^4 + 2*c^2*d^4 + 9*a^2*e^4*m + a^2*e^4*m^2 - 10*a*c*d^2*e^2 - 2*a*c*d^2*e^2*m))/(e^3*(47*m + 12*m^2 + m^3 + 6
0)) + (d^2*x*(60*a^2*e^4 + 27*a^2*e^4*m - 2*c^2*d^4*m + 3*a^2*e^4*m^2 + 10*a*c*d^2*e^2*m + 2*a*c*d^2*e^2*m^2))
/(e^2*(47*m + 12*m^2 + m^3 + 60)) + (d*x^2*(60*a^2*e^4 + 27*a^2*e^4*m + c^2*d^4*m + 3*a^2*e^4*m^2 + c^2*d^4*m^
2 + 60*a*c*d^2*e^2 + 42*a*c*d^2*e^2*m + 6*a*c*d^2*e^2*m^2))/(e*(47*m + 12*m^2 + m^3 + 60)) + (c^2*d^2*e^2*x^5*
(7*m + m^2 + 12))/(47*m + 12*m^2 + m^3 + 60) + (c*d*e*x^4*(m + 3)*(10*a*e^2 + 10*c*d^2 + 2*a*e^2*m + 3*c*d^2*m
))/(47*m + 12*m^2 + m^3 + 60))