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

Optimal. Leaf size=46 \[ \frac{a (d+e x)^4}{4 e}+\frac{b (d+e x)^6}{6 e}+\frac{c (d+e x)^8}{8 e} \]

[Out]

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

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Rubi [A]  time = 0.0536108, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1142, 14} \[ \frac{a (d+e x)^4}{4 e}+\frac{b (d+e x)^6}{6 e}+\frac{c (d+e x)^8}{8 e} \]

Antiderivative was successfully verified.

[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 1142

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]

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]]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{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]  time = 0.0383303, size = 150, normalized size = 3.26 \[ \frac{1}{4} e^3 x^4 \left (a+10 b d^2+35 c d^4\right )+\frac{1}{3} d e^2 x^3 \left (3 a+10 b d^2+21 c d^4\right )+\frac{1}{2} d^2 e x^2 \left (3 a+5 b d^2+7 c d^4\right )+d^3 x \left (a+b d^2+c d^4\right )+\frac{1}{6} e^5 x^6 \left (b+21 c d^2\right )+d e^4 x^5 \left (b+7 c d^2\right )+c d e^6 x^7+\frac{1}{8} c e^7 x^8 \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.001, size = 298, normalized size = 6.5 \begin{align*}{\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\,c{d}^{3}e+2\,bde \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\,c{d}^{3}e+2\,bde \right ) +{e}^{3} \left ( c{d}^{4}+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\,c{d}^{3}e+2\,bde \right ) +3\,d{e}^{2} \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3} \left ( 4\,c{d}^{3}e+2\,bde \right ) +3\,{d}^{2}e \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{2}}{2}}+{d}^{3} \left ( c{d}^{4}+b{d}^{2}+a \right ) x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^7*c*x^8+d*e^6*c*x^7+1/6*(15*d^2*e^5*c+e^3*(6*c*d^2*e^2+b*e^2))*x^6+1/5*(13*d^3*c*e^4+3*d*e^2*(6*c*d^2*e^
2+b*e^2)+e^3*(4*c*d^3*e+2*b*d*e))*x^5+1/4*(4*d^4*c*e^3+3*d^2*e*(6*c*d^2*e^2+b*e^2)+3*d*e^2*(4*c*d^3*e+2*b*d*e)
+e^3*(c*d^4+b*d^2+a))*x^4+1/3*(d^3*(6*c*d^2*e^2+b*e^2)+3*d^2*e*(4*c*d^3*e+2*b*d*e)+3*d*e^2*(c*d^4+b*d^2+a))*x^
3+1/2*(d^3*(4*c*d^3*e+2*b*d*e)+3*d^2*e*(c*d^4+b*d^2+a))*x^2+d^3*(c*d^4+b*d^2+a)*x

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Maxima [B]  time = 1.03065, size = 192, normalized size = 4.17 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.48395, size = 393, normalized size = 8.54 \begin{align*} \frac{1}{8} x^{8} e^{7} c + x^{7} e^{6} d c + \frac{7}{2} x^{6} e^{5} d^{2} c + 7 x^{5} e^{4} d^{3} c + \frac{35}{4} x^{4} e^{3} d^{4} c + \frac{1}{6} x^{6} e^{5} b + 7 x^{3} e^{2} d^{5} c + x^{5} e^{4} d b + \frac{7}{2} x^{2} e d^{6} c + \frac{5}{2} x^{4} e^{3} d^{2} b + x d^{7} c + \frac{10}{3} x^{3} e^{2} d^{3} b + \frac{5}{2} x^{2} e d^{4} b + \frac{1}{4} x^{4} e^{3} a + x d^{5} b + x^{3} e^{2} d a + \frac{3}{2} x^{2} e d^{2} a + x d^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.093475, size = 178, normalized size = 3.87 \begin{align*} 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} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.07922, size = 224, normalized size = 4.87 \begin{align*} \frac{1}{8} \, c x^{8} e^{7} + c d x^{7} e^{6} + \frac{7}{2} \, c d^{2} x^{6} e^{5} + 7 \, c d^{3} x^{5} e^{4} + \frac{35}{4} \, c d^{4} x^{4} e^{3} + 7 \, c d^{5} x^{3} e^{2} + \frac{7}{2} \, c d^{6} x^{2} e + c d^{7} x + \frac{1}{6} \, b x^{6} e^{5} + b d x^{5} e^{4} + \frac{5}{2} \, b d^{2} x^{4} e^{3} + \frac{10}{3} \, b d^{3} x^{3} e^{2} + \frac{5}{2} \, b d^{4} x^{2} e + b d^{5} x + \frac{1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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