\(\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1829]

Optimal result

Integrand size = 33, antiderivative size = 39 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^5+\frac {c d (d+e x)^6}{6 e^2} \]

[Out]

1/5*(a-c*d^2/e^2)*(e*x+d)^5+1/6*c*d*(e*x+d)^6/e^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45} \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{5} (d+e x)^5 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^6}{6 e^2} \]

[In]

Int[(d + e*x)^3*(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)^5)/5 + (c*d*(d + e*x)^6)/(6*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)^4 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^4}{e}+\frac {c d (d+e x)^5}{e}\right ) \, dx \\ & = \frac {1}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^5+\frac {c d (d+e x)^6}{6 e^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(39)=78\).

Time = 0.01 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.44 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{30} x \left (6 a e \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+c d x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right ) \]

[In]

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

[Out]

(x*(6*a*e*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + c*d*x*(15*d^4 + 40*d^3*e*x + 45*d^2*
e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)))/30

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(35)=70\).

Time = 2.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.56

method	result	size
norman	\(\frac {c d \,e^{4} x^{6}}{6}+\left (\frac {1}{5} a \,e^{5}+\frac {4}{5} d^{2} e^{3} c \right ) x^{5}+\left (a d \,e^{4}+\frac {3}{2} d^{3} e^{2} c \right ) x^{4}+\left (2 a \,e^{3} d^{2}+\frac {4}{3} c \,d^{4} e \right ) x^{3}+\left (2 a \,d^{3} e^{2}+\frac {1}{2} c \,d^{5}\right ) x^{2}+a e \,d^{4} x\)	\(100\)
gosper	\(\frac {x \left (5 c d \,e^{4} x^{5}+6 x^{4} a \,e^{5}+24 x^{4} d^{2} e^{3} c +30 x^{3} a d \,e^{4}+45 x^{3} d^{3} e^{2} c +60 x^{2} a \,e^{3} d^{2}+40 x^{2} c \,d^{4} e +60 x a \,d^{3} e^{2}+15 x c \,d^{5}+30 a e \,d^{4}\right )}{30}\)	\(104\)
risch	\(\frac {1}{6} c d \,e^{4} x^{6}+\frac {1}{5} x^{5} a \,e^{5}+\frac {4}{5} x^{5} d^{2} e^{3} c +x^{4} a d \,e^{4}+\frac {3}{2} x^{4} d^{3} e^{2} c +2 x^{3} a \,e^{3} d^{2}+\frac {4}{3} x^{3} c \,d^{4} e +2 x^{2} a \,d^{3} e^{2}+\frac {1}{2} x^{2} c \,d^{5}+a e \,d^{4} x\)	\(104\)
parallelrisch	\(\frac {1}{6} c d \,e^{4} x^{6}+\frac {1}{5} x^{5} a \,e^{5}+\frac {4}{5} x^{5} d^{2} e^{3} c +x^{4} a d \,e^{4}+\frac {3}{2} x^{4} d^{3} e^{2} c +2 x^{3} a \,e^{3} d^{2}+\frac {4}{3} x^{3} c \,d^{4} e +2 x^{2} a \,d^{3} e^{2}+\frac {1}{2} x^{2} c \,d^{5}+a e \,d^{4} x\)	\(104\)
default	\(\frac {c d \,e^{4} x^{6}}{6}+\frac {\left (3 d^{2} e^{3} c +e^{3} \left (e^{2} a +c \,d^{2}\right )\right ) x^{5}}{5}+\frac {\left (3 d^{3} e^{2} c +3 d \,e^{2} \left (e^{2} a +c \,d^{2}\right )+a d \,e^{4}\right ) x^{4}}{4}+\frac {\left (c \,d^{4} e +3 d^{2} e \left (e^{2} a +c \,d^{2}\right )+3 a \,e^{3} d^{2}\right ) x^{3}}{3}+\frac {\left (d^{3} \left (e^{2} a +c \,d^{2}\right )+3 a \,d^{3} e^{2}\right ) x^{2}}{2}+a e \,d^{4} x\)	\(155\)

[In]

int((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

1/6*c*d*e^4*x^6+(1/5*a*e^5+4/5*d^2*e^3*c)*x^5+(a*d*e^4+3/2*d^3*e^2*c)*x^4+(2*a*e^3*d^2+4/3*c*d^4*e)*x^3+(2*a*d
^3*e^2+1/2*c*d^5)*x^2+a*e*d^4*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).

Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.62 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{6} \, c d e^{4} x^{6} + a d^{4} e x + \frac {1}{5} \, {\left (4 \, c d^{2} e^{3} + a e^{5}\right )} x^{5} + \frac {1}{2} \, {\left (3 \, c d^{3} e^{2} + 2 \, a d e^{4}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, c d^{4} e + 3 \, a d^{2} e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{5} + 4 \, a d^{3} e^{2}\right )} x^{2} \]

[In]

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

[Out]

1/6*c*d*e^4*x^6 + a*d^4*e*x + 1/5*(4*c*d^2*e^3 + a*e^5)*x^5 + 1/2*(3*c*d^3*e^2 + 2*a*d*e^4)*x^4 + 2/3*(2*c*d^4
*e + 3*a*d^2*e^3)*x^3 + 1/2*(c*d^5 + 4*a*d^3*e^2)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (34) = 68\).

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.74 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d^{4} e x + \frac {c d e^{4} x^{6}}{6} + x^{5} \left (\frac {a e^{5}}{5} + \frac {4 c d^{2} e^{3}}{5}\right ) + x^{4} \left (a d e^{4} + \frac {3 c d^{3} e^{2}}{2}\right ) + x^{3} \cdot \left (2 a d^{2} e^{3} + \frac {4 c d^{4} e}{3}\right ) + x^{2} \cdot \left (2 a d^{3} e^{2} + \frac {c d^{5}}{2}\right ) \]

[In]

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

[Out]

a*d**4*e*x + c*d*e**4*x**6/6 + x**5*(a*e**5/5 + 4*c*d**2*e**3/5) + x**4*(a*d*e**4 + 3*c*d**3*e**2/2) + x**3*(2
*a*d**2*e**3 + 4*c*d**4*e/3) + x**2*(2*a*d**3*e**2 + c*d**5/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.62 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{6} \, c d e^{4} x^{6} + a d^{4} e x + \frac {1}{5} \, {\left (4 \, c d^{2} e^{3} + a e^{5}\right )} x^{5} + \frac {1}{2} \, {\left (3 \, c d^{3} e^{2} + 2 \, a d e^{4}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, c d^{4} e + 3 \, a d^{2} e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{5} + 4 \, a d^{3} e^{2}\right )} x^{2} \]

[In]

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

[Out]

1/6*c*d*e^4*x^6 + a*d^4*e*x + 1/5*(4*c*d^2*e^3 + a*e^5)*x^5 + 1/2*(3*c*d^3*e^2 + 2*a*d*e^4)*x^4 + 2/3*(2*c*d^4
*e + 3*a*d^2*e^3)*x^3 + 1/2*(c*d^5 + 4*a*d^3*e^2)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (35) = 70\).

Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.64 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{6} \, c d e^{4} x^{6} + \frac {4}{5} \, c d^{2} e^{3} x^{5} + \frac {1}{5} \, a e^{5} x^{5} + \frac {3}{2} \, c d^{3} e^{2} x^{4} + a d e^{4} x^{4} + \frac {4}{3} \, c d^{4} e x^{3} + 2 \, a d^{2} e^{3} x^{3} + \frac {1}{2} \, c d^{5} x^{2} + 2 \, a d^{3} e^{2} x^{2} + a d^{4} e x \]

[In]

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

[Out]

1/6*c*d*e^4*x^6 + 4/5*c*d^2*e^3*x^5 + 1/5*a*e^5*x^5 + 3/2*c*d^3*e^2*x^4 + a*d*e^4*x^4 + 4/3*c*d^4*e*x^3 + 2*a*
d^2*e^3*x^3 + 1/2*c*d^5*x^2 + 2*a*d^3*e^2*x^2 + a*d^4*e*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.54 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=x^2\,\left (\frac {c\,d^5}{2}+2\,a\,d^3\,e^2\right )+x^5\,\left (\frac {4\,c\,d^2\,e^3}{5}+\frac {a\,e^5}{5}\right )+x^4\,\left (\frac {3\,c\,d^3\,e^2}{2}+a\,d\,e^4\right )+x^3\,\left (\frac {4\,c\,d^4\,e}{3}+2\,a\,d^2\,e^3\right )+a\,d^4\,e\,x+\frac {c\,d\,e^4\,x^6}{6} \]

[In]

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

[Out]

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