\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1831]

Optimal result

Integrand size = 31, antiderivative size = 39 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {640, 45} \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} (d+e x)^3 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^4}{4 e^2} \]

[In]

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

[Out]

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

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) (d+e x)^2 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^2}{e}+\frac {c d (d+e x)^3}{e}\right ) \, dx \\ & = \frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{12} x \left (4 a e \left (3 d^2+3 d e x+e^2 x^2\right )+c d x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right ) \]

[In]

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

[Out]

(x*(4*a*e*(3*d^2 + 3*d*e*x + e^2*x^2) + c*d*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/12

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38

method	result	size
norman	\(\frac {c d \,e^{2} x^{4}}{4}+\left (\frac {1}{3} a \,e^{3}+\frac {2}{3} d^{2} e c \right ) x^{3}+\left (a d \,e^{2}+\frac {1}{2} d^{3} c \right ) x^{2}+x a e \,d^{2}\)	\(54\)
gosper	\(\frac {x \left (3 c d \,e^{2} x^{3}+4 a \,e^{3} x^{2}+8 c \,d^{2} e \,x^{2}+12 a d \,e^{2} x +6 c \,d^{3} x +12 a e \,d^{2}\right )}{12}\)	\(56\)
risch	\(\frac {1}{4} c d \,e^{2} x^{4}+\frac {1}{3} x^{3} a \,e^{3}+\frac {2}{3} x^{3} d^{2} e c +a d \,e^{2} x^{2}+\frac {1}{2} c \,d^{3} x^{2}+x a e \,d^{2}\)	\(56\)
parallelrisch	\(\frac {1}{4} c d \,e^{2} x^{4}+\frac {1}{3} x^{3} a \,e^{3}+\frac {2}{3} x^{3} d^{2} e c +a d \,e^{2} x^{2}+\frac {1}{2} c \,d^{3} x^{2}+x a e \,d^{2}\)	\(56\)
default	\(\frac {c d \,e^{2} x^{4}}{4}+\frac {\left (d^{2} e c +e \left (e^{2} a +c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (d \left (e^{2} a +c \,d^{2}\right )+a d \,e^{2}\right ) x^{2}}{2}+x a e \,d^{2}\)	\(69\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac {1}{3} \, {\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d^{2} e x + \frac {c d e^{2} x^{4}}{4} + x^{3} \left (\frac {a e^{3}}{3} + \frac {2 c d^{2} e}{3}\right ) + x^{2} \left (a d e^{2} + \frac {c d^{3}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac {1}{3} \, {\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.41 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + \frac {2}{3} \, c d^{2} e x^{3} + \frac {1}{3} \, a e^{3} x^{3} + \frac {1}{2} \, c d^{3} x^{2} + a d e^{2} x^{2} + a d^{2} e x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=x^2\,\left (\frac {c\,d^3}{2}+a\,d\,e^2\right )+x^3\,\left (\frac {2\,c\,d^2\,e}{3}+\frac {a\,e^3}{3}\right )+a\,d^2\,e\,x+\frac {c\,d\,e^2\,x^4}{4} \]

[In]

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

[Out]

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