### 3.1852 $$\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx$$

Optimal. Leaf size=111 $-\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4}$

[Out]

-((c*d^2 - a*e^2)^3*(d + e*x)^5)/(5*e^4) + (c*d*(c*d^2 - a*e^2)^2*(d + e*x)^6)/(2*e^4) - (3*c^2*d^2*(c*d^2 - a
*e^2)*(d + e*x)^7)/(7*e^4) + (c^3*d^3*(d + e*x)^8)/(8*e^4)

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Rubi [A]  time = 0.160326, antiderivative size = 111, 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} $-\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((c*d^2 - a*e^2)^3*(d + e*x)^5)/(5*e^4) + (c*d*(c*d^2 - a*e^2)^2*(d + e*x)^6)/(2*e^4) - (3*c^2*d^2*(c*d^2 - a
*e^2)*(d + e*x)^7)/(7*e^4) + (c^3*d^3*(d + e*x)^8)/(8*e^4)

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) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^4}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^5}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^6}{e^3}+\frac{c^3 d^3 (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^5}{5 e^4}+\frac{c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{2 e^4}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{7 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0633384, size = 211, normalized size = 1.9 $\frac{1}{280} x \left (28 a^2 c d e^2 x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+56 a^3 e^3 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(56*a^3*e^3*(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) + 28*a^2*c*d*e^2*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) + 8*a*c^2*d^2*e*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^
2 + 70*d*e^3*x^3 + 15*e^4*x^4) + c^3*d^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*
x^4)))/280

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Maple [B]  time = 0.04, size = 531, normalized size = 4.8 \begin{align*}{\frac{{d}^{3}{e}^{4}{c}^{3}{x}^{8}}{8}}+{\frac{ \left ({d}^{4}{e}^{3}{c}^{3}+3\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){e}^{2}{c}^{2}+e \left ( a{d}^{3}{e}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{d}^{3}{e}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( 4\,a{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,a{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( ade \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}ade+{d}^{3}{e}^{3}c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( ade \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}ade+{d}^{3}{e}^{3}c{a}^{2} \right ) +3\,{e}^{3}{a}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{3}{a}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{3}{d}^{3}{e}^{4} \right ){x}^{2}}{2}}+{a}^{3}{d}^{4}{e}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01722, size = 339, normalized size = 3.05 \begin{align*} \frac{1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac{1}{7} \,{\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} +{\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*d^3*e^4*x^8 + a^3*d^4*e^3*x + 1/7*(4*c^3*d^4*e^3 + 3*a*c^2*d^2*e^5)*x^7 + 1/2*(2*c^3*d^5*e^2 + 4*a*c^2
*d^3*e^4 + a^2*c*d*e^6)*x^6 + 1/5*(4*c^3*d^6*e + 18*a*c^2*d^4*e^3 + 12*a^2*c*d^2*e^5 + a^3*e^7)*x^5 + 1/4*(c^3
*d^7 + 12*a*c^2*d^5*e^2 + 18*a^2*c*d^3*e^4 + 4*a^3*d*e^6)*x^4 + (a*c^2*d^6*e + 4*a^2*c*d^4*e^3 + 2*a^3*d^2*e^5
)*x^3 + 1/2*(3*a^2*c*d^5*e^2 + 4*a^3*d^3*e^4)*x^2

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Fricas [B]  time = 1.30077, size = 563, normalized size = 5.07 \begin{align*} \frac{1}{8} x^{8} e^{4} d^{3} c^{3} + \frac{4}{7} x^{7} e^{3} d^{4} c^{3} + \frac{3}{7} x^{7} e^{5} d^{2} c^{2} a + x^{6} e^{2} d^{5} c^{3} + 2 x^{6} e^{4} d^{3} c^{2} a + \frac{1}{2} x^{6} e^{6} d c a^{2} + \frac{4}{5} x^{5} e d^{6} c^{3} + \frac{18}{5} x^{5} e^{3} d^{4} c^{2} a + \frac{12}{5} x^{5} e^{5} d^{2} c a^{2} + \frac{1}{5} x^{5} e^{7} a^{3} + \frac{1}{4} x^{4} d^{7} c^{3} + 3 x^{4} e^{2} d^{5} c^{2} a + \frac{9}{2} x^{4} e^{4} d^{3} c a^{2} + x^{4} e^{6} d a^{3} + x^{3} e d^{6} c^{2} a + 4 x^{3} e^{3} d^{4} c a^{2} + 2 x^{3} e^{5} d^{2} a^{3} + \frac{3}{2} x^{2} e^{2} d^{5} c a^{2} + 2 x^{2} e^{4} d^{3} a^{3} + x e^{3} d^{4} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e^4*d^3*c^3 + 4/7*x^7*e^3*d^4*c^3 + 3/7*x^7*e^5*d^2*c^2*a + x^6*e^2*d^5*c^3 + 2*x^6*e^4*d^3*c^2*a + 1/
2*x^6*e^6*d*c*a^2 + 4/5*x^5*e*d^6*c^3 + 18/5*x^5*e^3*d^4*c^2*a + 12/5*x^5*e^5*d^2*c*a^2 + 1/5*x^5*e^7*a^3 + 1/
4*x^4*d^7*c^3 + 3*x^4*e^2*d^5*c^2*a + 9/2*x^4*e^4*d^3*c*a^2 + x^4*e^6*d*a^3 + x^3*e*d^6*c^2*a + 4*x^3*e^3*d^4*
c*a^2 + 2*x^3*e^5*d^2*a^3 + 3/2*x^2*e^2*d^5*c*a^2 + 2*x^2*e^4*d^3*a^3 + x*e^3*d^4*a^3

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Sympy [B]  time = 0.179508, size = 270, normalized size = 2.43 \begin{align*} a^{3} d^{4} e^{3} x + \frac{c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac{3 a c^{2} d^{2} e^{5}}{7} + \frac{4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac{a^{3} e^{7}}{5} + \frac{12 a^{2} c d^{2} e^{5}}{5} + \frac{18 a c^{2} d^{4} e^{3}}{5} + \frac{4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac{9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac{c^{3} d^{7}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \left (2 a^{3} d^{3} e^{4} + \frac{3 a^{2} c d^{5} e^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**4*e**3*x + c**3*d**3*e**4*x**8/8 + x**7*(3*a*c**2*d**2*e**5/7 + 4*c**3*d**4*e**3/7) + x**6*(a**2*c*d*e
**6/2 + 2*a*c**2*d**3*e**4 + c**3*d**5*e**2) + x**5*(a**3*e**7/5 + 12*a**2*c*d**2*e**5/5 + 18*a*c**2*d**4*e**3
/5 + 4*c**3*d**6*e/5) + x**4*(a**3*d*e**6 + 9*a**2*c*d**3*e**4/2 + 3*a*c**2*d**5*e**2 + c**3*d**7/4) + x**3*(2
*a**3*d**2*e**5 + 4*a**2*c*d**4*e**3 + a*c**2*d**6*e) + x**2*(2*a**3*d**3*e**4 + 3*a**2*c*d**5*e**2/2)

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Giac [B]  time = 1.21402, size = 346, normalized size = 3.12 \begin{align*} \frac{1}{8} \, c^{3} d^{3} x^{8} e^{4} + \frac{4}{7} \, c^{3} d^{4} x^{7} e^{3} + c^{3} d^{5} x^{6} e^{2} + \frac{4}{5} \, c^{3} d^{6} x^{5} e + \frac{1}{4} \, c^{3} d^{7} x^{4} + \frac{3}{7} \, a c^{2} d^{2} x^{7} e^{5} + 2 \, a c^{2} d^{3} x^{6} e^{4} + \frac{18}{5} \, a c^{2} d^{4} x^{5} e^{3} + 3 \, a c^{2} d^{5} x^{4} e^{2} + a c^{2} d^{6} x^{3} e + \frac{1}{2} \, a^{2} c d x^{6} e^{6} + \frac{12}{5} \, a^{2} c d^{2} x^{5} e^{5} + \frac{9}{2} \, a^{2} c d^{3} x^{4} e^{4} + 4 \, a^{2} c d^{4} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{5} x^{2} e^{2} + \frac{1}{5} \, a^{3} x^{5} e^{7} + a^{3} d x^{4} e^{6} + 2 \, a^{3} d^{2} x^{3} e^{5} + 2 \, a^{3} d^{3} x^{2} e^{4} + a^{3} d^{4} x e^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*d^3*x^8*e^4 + 4/7*c^3*d^4*x^7*e^3 + c^3*d^5*x^6*e^2 + 4/5*c^3*d^6*x^5*e + 1/4*c^3*d^7*x^4 + 3/7*a*c^2*
d^2*x^7*e^5 + 2*a*c^2*d^3*x^6*e^4 + 18/5*a*c^2*d^4*x^5*e^3 + 3*a*c^2*d^5*x^4*e^2 + a*c^2*d^6*x^3*e + 1/2*a^2*c
*d*x^6*e^6 + 12/5*a^2*c*d^2*x^5*e^5 + 9/2*a^2*c*d^3*x^4*e^4 + 4*a^2*c*d^4*x^3*e^3 + 3/2*a^2*c*d^5*x^2*e^2 + 1/
5*a^3*x^5*e^7 + a^3*d*x^4*e^6 + 2*a^3*d^2*x^3*e^5 + 2*a^3*d^3*x^2*e^4 + a^3*d^4*x*e^3