3.1828 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{6} (d+e x)^6 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^7}{7 e^2} \]

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^6)/6 + (c*d*(d + e*x)^7)/(7*e^2)

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Rubi [A]  time = 0.0226517, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 43} \[ \frac{1}{6} (d+e x)^6 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^7}{7 e^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(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)^6)/6 + (c*d*(d + e*x)^7)/(7*e^2)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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]

Rule 43

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])

Rubi steps

\begin{align*} \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^5 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^5}{e}+\frac{c d (d+e x)^6}{e}\right ) \, dx\\ &=\frac{1}{6} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^6+\frac{c d (d+e x)^7}{7 e^2}\\ \end{align*}

Mathematica [B]  time = 0.0310917, size = 117, normalized size = 3. \[ \frac{1}{42} x \left (7 a e \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+c d x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(7*a*e*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + c*d*x*(21*d^5 + 70*
d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)))/42

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Maple [B]  time = 0.041, size = 198, normalized size = 5.1 \begin{align*}{\frac{{e}^{5}dc{x}^{7}}{7}}+{\frac{ \left ( 4\,{d}^{2}{e}^{4}c+{e}^{4} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{d}^{3}{e}^{3}c+4\,d{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{5}ad \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{4}{e}^{2}c+6\,{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +4\,{d}^{2}{e}^{4}a \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{5}ec+4\,{d}^{3}e \left ( a{e}^{2}+c{d}^{2} \right ) +6\,{d}^{3}{e}^{3}a \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{4} \left ( a{e}^{2}+c{d}^{2} \right ) +4\,{d}^{4}{e}^{2}a \right ){x}^{2}}{2}}+{d}^{5}aex \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08482, size = 163, normalized size = 4.18 \begin{align*} \frac{1}{7} \, c d e^{5} x^{7} + a d^{5} e x + \frac{1}{6} \,{\left (5 \, c d^{2} e^{4} + a e^{6}\right )} x^{6} +{\left (2 \, c d^{3} e^{3} + a d e^{5}\right )} x^{5} + \frac{5}{2} \,{\left (c d^{4} e^{2} + a d^{2} e^{4}\right )} x^{4} + \frac{5}{3} \,{\left (c d^{5} e + 2 \, a d^{3} e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{6} + 5 \, a d^{4} e^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.34521, size = 286, normalized size = 7.33 \begin{align*} \frac{1}{7} x^{7} e^{5} d c + \frac{5}{6} x^{6} e^{4} d^{2} c + \frac{1}{6} x^{6} e^{6} a + 2 x^{5} e^{3} d^{3} c + x^{5} e^{5} d a + \frac{5}{2} x^{4} e^{2} d^{4} c + \frac{5}{2} x^{4} e^{4} d^{2} a + \frac{5}{3} x^{3} e d^{5} c + \frac{10}{3} x^{3} e^{3} d^{3} a + \frac{1}{2} x^{2} d^{6} c + \frac{5}{2} x^{2} e^{2} d^{4} a + x e d^{5} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.106722, size = 136, normalized size = 3.49 \begin{align*} a d^{5} e x + \frac{c d e^{5} x^{7}}{7} + x^{6} \left (\frac{a e^{6}}{6} + \frac{5 c d^{2} e^{4}}{6}\right ) + x^{5} \left (a d e^{5} + 2 c d^{3} e^{3}\right ) + x^{4} \left (\frac{5 a d^{2} e^{4}}{2} + \frac{5 c d^{4} e^{2}}{2}\right ) + x^{3} \left (\frac{10 a d^{3} e^{3}}{3} + \frac{5 c d^{5} e}{3}\right ) + x^{2} \left (\frac{5 a d^{4} e^{2}}{2} + \frac{c d^{6}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.26842, size = 162, normalized size = 4.15 \begin{align*} \frac{1}{7} \, c d x^{7} e^{5} + \frac{5}{6} \, c d^{2} x^{6} e^{4} + 2 \, c d^{3} x^{5} e^{3} + \frac{5}{2} \, c d^{4} x^{4} e^{2} + \frac{5}{3} \, c d^{5} x^{3} e + \frac{1}{2} \, c d^{6} x^{2} + \frac{1}{6} \, a x^{6} e^{6} + a d x^{5} e^{5} + \frac{5}{2} \, a d^{2} x^{4} e^{4} + \frac{10}{3} \, a d^{3} x^{3} e^{3} + \frac{5}{2} \, a d^{4} x^{2} e^{2} + a d^{5} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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