\(\int (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\) [1841]

Optimal result

Integrand size = 27, antiderivative size = 77 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]

[Out]

1/3*(-a*e^2+c*d^2)^2*(e*x+d)^3/e^3-1/2*c*d*(-a*e^2+c*d^2)*(e*x+d)^4/e^3+1/5*c^2*d^2*(e*x+d)^5/e^3

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {624, 45} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=-\frac {c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]

[In]

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

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 624

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{30} x \left (10 a^2 e^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a c d e x \left (6 d^2+8 d e x+3 e^2 x^2\right )+c^2 d^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right ) \]

[In]

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

[Out]

(x*(10*a^2*e^2*(3*d^2 + 3*d*e*x + e^2*x^2) + 5*a*c*d*e*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + c^2*d^2*x^2*(10*d^2 +
15*d*e*x + 6*e^2*x^2)))/30

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\frac {\left (e^{2} a +c \,d^{2}\right ) c d e \,x^{4}}{2}+\frac {\left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right ) x^{3}}{3}+a d e \left (e^{2} a +c \,d^{2}\right ) x^{2}+a^{2} d^{2} e^{2} x\)	\(93\)
norman	\(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\left (\frac {1}{2} a c d \,e^{3}+\frac {1}{2} d^{3} e \,c^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} e^{4}+\frac {4}{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4}\right ) x^{3}+\left (a^{2} d \,e^{3}+a c \,d^{3} e \right ) x^{2}+a^{2} d^{2} e^{2} x\)	\(100\)
risch	\(\frac {1}{5} c^{2} d^{2} e^{2} x^{5}+\frac {1}{2} a c d \,e^{3} x^{4}+\frac {1}{2} c^{2} d^{3} e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{4}+\frac {4}{3} x^{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4} x^{3}+a^{2} d \,e^{3} x^{2}+a c \,d^{3} e \,x^{2}+a^{2} d^{2} e^{2} x\)	\(106\)
parallelrisch	\(\frac {1}{5} c^{2} d^{2} e^{2} x^{5}+\frac {1}{2} a c d \,e^{3} x^{4}+\frac {1}{2} c^{2} d^{3} e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{4}+\frac {4}{3} x^{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4} x^{3}+a^{2} d \,e^{3} x^{2}+a c \,d^{3} e \,x^{2}+a^{2} d^{2} e^{2} x\)	\(106\)
gosper	\(\frac {x \left (6 c^{2} d^{2} e^{2} x^{4}+15 x^{3} a c d \,e^{3}+15 x^{3} d^{3} e \,c^{2}+10 x^{2} a^{2} e^{4}+40 x^{2} a c \,d^{2} e^{2}+10 x^{2} c^{2} d^{4}+30 a^{2} d \,e^{3} x +30 a c \,d^{3} e x +30 a^{2} d^{2} e^{2}\right )}{30}\)	\(107\)

[In]

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

[Out]

1/5*c^2*d^2*e^2*x^5+1/2*(a*e^2+c*d^2)*c*d*e*x^4+1/3*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)*x^3+a*d*e*(a*e^2+c*d^2)*x^
2+a^2*d^2*e^2*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + a^{2} d^{2} e^{2} x + \frac {1}{2} \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{3} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2} \]

[In]

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

[Out]

1/5*c^2*d^2*e^2*x^5 + a^2*d^2*e^2*x + 1/2*(c^2*d^3*e + a*c*d*e^3)*x^4 + 1/3*(c^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4
)*x^3 + (a*c*d^3*e + a^2*d*e^3)*x^2

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=a^{2} d^{2} e^{2} x + \frac {c^{2} d^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac {a c d e^{3}}{2} + \frac {c^{2} d^{3} e}{2}\right ) + x^{3} \left (\frac {a^{2} e^{4}}{3} + \frac {4 a c d^{2} e^{2}}{3} + \frac {c^{2} d^{4}}{3}\right ) + x^{2} \left (a^{2} d e^{3} + a c d^{3} e\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c d e x^{4} + a^{2} d^{2} e^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac {1}{3} \, {\left (2 \, c d e x^{3} + 3 \, {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a d e \]

[In]

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

[Out]

1/5*c^2*d^2*e^2*x^5 + 1/2*(c*d^2 + a*e^2)*c*d*e*x^4 + a^2*d^2*e^2*x + 1/3*(c*d^2 + a*e^2)^2*x^3 + 1/3*(2*c*d*e
*x^3 + 3*(c*d^2 + a*e^2)*x^2)*a*d*e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, c^{2} d^{3} e x^{4} + \frac {1}{2} \, a c d e^{3} x^{4} + \frac {1}{3} \, c^{2} d^{4} x^{3} + \frac {4}{3} \, a c d^{2} e^{2} x^{3} + \frac {1}{3} \, a^{2} e^{4} x^{3} + a c d^{3} e x^{2} + a^{2} d e^{3} x^{2} + a^{2} d^{2} e^{2} x \]

[In]

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

[Out]

1/5*c^2*d^2*e^2*x^5 + 1/2*c^2*d^3*e*x^4 + 1/2*a*c*d*e^3*x^4 + 1/3*c^2*d^4*x^3 + 4/3*a*c*d^2*e^2*x^3 + 1/3*a^2*
e^4*x^3 + a*c*d^3*e*x^2 + a^2*d*e^3*x^2 + a^2*d^2*e^2*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=x^3\,\left (\frac {a^2\,e^4}{3}+\frac {4\,a\,c\,d^2\,e^2}{3}+\frac {c^2\,d^4}{3}\right )+x^2\,\left (a^2\,d\,e^3+c\,a\,d^3\,e\right )+x^4\,\left (\frac {c^2\,d^3\,e}{2}+\frac {a\,c\,d\,e^3}{2}\right )+a^2\,d^2\,e^2\,x+\frac {c^2\,d^2\,e^2\,x^5}{5} \]

[In]

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

[Out]

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