\(\int (d+e x)^3 (a+b (d+e x)^2+c (d+e x)^4) \, dx\) [607]

Optimal result

Integrand size = 28, antiderivative size = 46 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e} \]

[Out]

1/4*a*(e*x+d)^4/e+1/6*b*(e*x+d)^6/e+1/8*c*(e*x+d)^8/e

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1156, 14} \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e} \]

[In]

Int[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

(a*(d + e*x)^4)/(4*e) + (b*(d + e*x)^6)/(6*e) + (c*(d + e*x)^8)/(8*e)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (a x^3+b x^5+c x^7\right ) \, dx,x,d+e x\right )}{e} \\ & = \frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(46)=92\).

Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.26 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=d^3 \left (a+b d^2+c d^4\right ) x+\frac {1}{2} d^2 \left (3 a+5 b d^2+7 c d^4\right ) e x^2+\frac {1}{3} d \left (3 a+10 b d^2+21 c d^4\right ) e^2 x^3+\frac {1}{4} \left (a+10 b d^2+35 c d^4\right ) e^3 x^4+d \left (b+7 c d^2\right ) e^4 x^5+\frac {1}{6} \left (b+21 c d^2\right ) e^5 x^6+c d e^6 x^7+\frac {1}{8} c e^7 x^8 \]

[In]

Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3*a + 10*b*d^2 + 21*c*d^4)*e^2*x^3)/
3 + ((a + 10*b*d^2 + 35*c*d^4)*e^3*x^4)/4 + d*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^6*x^7
+ (c*e^7*x^8)/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs. \(2(40)=80\).

Time = 0.60 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.52

method	result	size
norman	\(\frac {e^{7} c \,x^{8}}{8}+d \,e^{6} c \,x^{7}+\left (\frac {7}{2} d^{2} e^{5} c +\frac {1}{6} b \,e^{5}\right ) x^{6}+\left (7 d^{3} c \,e^{4}+b d \,e^{4}\right ) x^{5}+\left (\frac {35}{4} d^{4} c \,e^{3}+\frac {5}{2} b \,d^{2} e^{3}+\frac {1}{4} a \,e^{3}\right ) x^{4}+\left (7 c \,d^{5} e^{2}+\frac {10}{3} b \,d^{3} e^{2}+d \,e^{2} a \right ) x^{3}+\left (\frac {7}{2} c \,d^{6} e +\frac {5}{2} b \,d^{4} e +\frac {3}{2} a \,d^{2} e \right ) x^{2}+\left (c \,d^{7}+b \,d^{5}+a \,d^{3}\right ) x\)	\(162\)
gosper	\(\frac {x \left (3 e^{7} c \,x^{7}+24 d \,e^{6} c \,x^{6}+84 x^{5} d^{2} e^{5} c +168 c \,d^{3} e^{4} x^{4}+4 x^{5} b \,e^{5}+210 x^{3} d^{4} c \,e^{3}+24 b d \,e^{4} x^{4}+168 x^{2} c \,d^{5} e^{2}+60 x^{3} b \,d^{2} e^{3}+84 x c \,d^{6} e +80 x^{2} b \,d^{3} e^{2}+24 c \,d^{7}+6 a \,e^{3} x^{3}+60 x b \,d^{4} e +24 x^{2} d \,e^{2} a +24 b \,d^{5}+36 x a \,d^{2} e +24 a \,d^{3}\right )}{24}\)	\(176\)
risch	\(\frac {1}{8} e^{7} c \,x^{8}+d \,e^{6} c \,x^{7}+\frac {7}{2} x^{6} d^{2} e^{5} c +\frac {1}{6} x^{6} b \,e^{5}+7 c \,d^{3} e^{4} x^{5}+b d \,e^{4} x^{5}+\frac {35}{4} x^{4} d^{4} c \,e^{3}+\frac {5}{2} x^{4} b \,d^{2} e^{3}+\frac {1}{4} x^{4} a \,e^{3}+7 x^{3} c \,d^{5} e^{2}+\frac {10}{3} x^{3} b \,d^{3} e^{2}+x^{3} d \,e^{2} a +\frac {7}{2} x^{2} c \,d^{6} e +\frac {5}{2} x^{2} b \,d^{4} e +\frac {3}{2} x^{2} a \,d^{2} e +c \,d^{7} x +b \,d^{5} x +a \,d^{3} x\)	\(176\)
parallelrisch	\(\frac {1}{8} e^{7} c \,x^{8}+d \,e^{6} c \,x^{7}+\frac {7}{2} x^{6} d^{2} e^{5} c +\frac {1}{6} x^{6} b \,e^{5}+7 c \,d^{3} e^{4} x^{5}+b d \,e^{4} x^{5}+\frac {35}{4} x^{4} d^{4} c \,e^{3}+\frac {5}{2} x^{4} b \,d^{2} e^{3}+\frac {1}{4} x^{4} a \,e^{3}+7 x^{3} c \,d^{5} e^{2}+\frac {10}{3} x^{3} b \,d^{3} e^{2}+x^{3} d \,e^{2} a +\frac {7}{2} x^{2} c \,d^{6} e +\frac {5}{2} x^{2} b \,d^{4} e +\frac {3}{2} x^{2} a \,d^{2} e +c \,d^{7} x +b \,d^{5} x +a \,d^{3} x\)	\(176\)
default	\(\frac {e^{7} c \,x^{8}}{8}+d \,e^{6} c \,x^{7}+\frac {\left (15 d^{2} e^{5} c +e^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )\right ) x^{6}}{6}+\frac {\left (13 d^{3} c \,e^{4}+3 d \,e^{2} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+e^{3} \left (4 d^{3} e c +2 b d e \right )\right ) x^{5}}{5}+\frac {\left (4 d^{4} c \,e^{3}+3 d^{2} e \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d \,e^{2} \left (4 d^{3} e c +2 b d e \right )+e^{3} \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{4}}{4}+\frac {\left (d^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d^{2} e \left (4 d^{3} e c +2 b d e \right )+3 d \,e^{2} \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{3}}{3}+\frac {\left (d^{3} \left (4 d^{3} e c +2 b d e \right )+3 d^{2} e \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{2}}{2}+d^{3} \left (d^{4} c +b \,d^{2}+a \right ) x\)	\(298\)

[In]

int((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)

[Out]

1/8*e^7*c*x^8+d*e^6*c*x^7+(7/2*d^2*e^5*c+1/6*b*e^5)*x^6+(7*c*d^3*e^4+b*d*e^4)*x^5+(35/4*d^4*c*e^3+5/2*b*d^2*e^
3+1/4*a*e^3)*x^4+(7*c*d^5*e^2+10/3*b*d^3*e^2+d*e^2*a)*x^3+(7/2*c*d^6*e+5/2*b*d^4*e+3/2*a*d^2*e)*x^2+(c*d^7+b*d
^5+a*d^3)*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (40) = 80\).

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.09 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} x^{8} + c d e^{6} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} x \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

1/8*c*e^7*x^8 + c*d*e^6*x^7 + 1/6*(21*c*d^2 + b)*e^5*x^6 + (7*c*d^3 + b*d)*e^4*x^5 + 1/4*(35*c*d^4 + 10*b*d^2
+ a)*e^3*x^4 + 1/3*(21*c*d^5 + 10*b*d^3 + 3*a*d)*e^2*x^3 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*a*d^2)*e*x^2 + (c*d^7 +
b*d^5 + a*d^3)*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (34) = 68\).

Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.87 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=c d e^{6} x^{7} + \frac {c e^{7} x^{8}}{8} + x^{6} \left (\frac {b e^{5}}{6} + \frac {7 c d^{2} e^{5}}{2}\right ) + x^{5} \left (b d e^{4} + 7 c d^{3} e^{4}\right ) + x^{4} \left (\frac {a e^{3}}{4} + \frac {5 b d^{2} e^{3}}{2} + \frac {35 c d^{4} e^{3}}{4}\right ) + x^{3} \left (a d e^{2} + \frac {10 b d^{3} e^{2}}{3} + 7 c d^{5} e^{2}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e}{2} + \frac {5 b d^{4} e}{2} + \frac {7 c d^{6} e}{2}\right ) + x \left (a d^{3} + b d^{5} + c d^{7}\right ) \]

[In]

integrate((e*x+d)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

c*d*e**6*x**7 + c*e**7*x**8/8 + x**6*(b*e**5/6 + 7*c*d**2*e**5/2) + x**5*(b*d*e**4 + 7*c*d**3*e**4) + x**4*(a*
e**3/4 + 5*b*d**2*e**3/2 + 35*c*d**4*e**3/4) + x**3*(a*d*e**2 + 10*b*d**3*e**2/3 + 7*c*d**5*e**2) + x**2*(3*a*
d**2*e/2 + 5*b*d**4*e/2 + 7*c*d**6*e/2) + x*(a*d**3 + b*d**5 + c*d**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (40) = 80\).

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.09 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} x^{8} + c d e^{6} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} x \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

1/8*c*e^7*x^8 + c*d*e^6*x^7 + 1/6*(21*c*d^2 + b)*e^5*x^6 + (7*c*d^3 + b*d)*e^4*x^5 + 1/4*(35*c*d^4 + 10*b*d^2
+ a)*e^3*x^4 + 1/3*(21*c*d^5 + 10*b*d^3 + 3*a*d)*e^2*x^3 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*a*d^2)*e*x^2 + (c*d^7 +
b*d^5 + a*d^3)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (40) = 80\).

Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.48 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} c d^{6} + \frac {3}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{2} c d^{4} e + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )}^{3} c d^{2} e^{2} + \frac {1}{8} \, {\left (e x^{2} + 2 \, d x\right )}^{4} c e^{3} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} b d^{4} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )}^{2} b d^{2} e + \frac {1}{6} \, {\left (e x^{2} + 2 \, d x\right )}^{3} b e^{2} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} a d^{2} + \frac {1}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{2} a e \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

1/2*(e*x^2 + 2*d*x)*c*d^6 + 3/4*(e*x^2 + 2*d*x)^2*c*d^4*e + 1/2*(e*x^2 + 2*d*x)^3*c*d^2*e^2 + 1/8*(e*x^2 + 2*d
*x)^4*c*e^3 + 1/2*(e*x^2 + 2*d*x)*b*d^4 + 1/2*(e*x^2 + 2*d*x)^2*b*d^2*e + 1/6*(e*x^2 + 2*d*x)^3*b*e^2 + 1/2*(e
*x^2 + 2*d*x)*a*d^2 + 1/4*(e*x^2 + 2*d*x)^2*a*e

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.07 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=x\,\left (c\,d^7+b\,d^5+a\,d^3\right )+\frac {e^5\,x^6\,\left (21\,c\,d^2+b\right )}{6}+\frac {c\,e^7\,x^8}{8}+\frac {e^3\,x^4\,\left (35\,c\,d^4+10\,b\,d^2+a\right )}{4}+\frac {d^2\,e\,x^2\,\left (7\,c\,d^4+5\,b\,d^2+3\,a\right )}{2}+\frac {d\,e^2\,x^3\,\left (21\,c\,d^4+10\,b\,d^2+3\,a\right )}{3}+d\,e^4\,x^5\,\left (7\,c\,d^2+b\right )+c\,d\,e^6\,x^7 \]

[In]

int((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x)

[Out]

x*(a*d^3 + b*d^5 + c*d^7) + (e^5*x^6*(b + 21*c*d^2))/6 + (c*e^7*x^8)/8 + (e^3*x^4*(a + 10*b*d^2 + 35*c*d^4))/4
+ (d^2*e*x^2*(3*a + 5*b*d^2 + 7*c*d^4))/2 + (d*e^2*x^3*(3*a + 10*b*d^2 + 21*c*d^4))/3 + d*e^4*x^5*(b + 7*c*d^
2) + c*d*e^6*x^7