### 3.1854 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{d+e x} \, dx$$

Optimal. Leaf size=91 $\frac{e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}$

[Out]

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

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Rubi [A]  time = 0.0992741, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $\frac{e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0391323, size = 123, normalized size = 1.35 $\frac{1}{60} x \left (15 a^2 c d e^2 x \left (6 d^2+8 d e x+3 e^2 x^2\right )+20 a^3 e^3 \left (3 d^2+3 d e x+e^2 x^2\right )+6 a c^2 d^2 e x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+c^3 d^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(20*a^3*e^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*c*d*e^2*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 6*a*c^2*d^2*e*x^
2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + c^3*d^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2)))/60

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Maple [B]  time = 0.038, size = 205, normalized size = 2.3 \begin{align*}{\frac{{c}^{3}{d}^{3}{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( a{e}^{3}{d}^{2}{c}^{2}+2\,{c}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,a{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) dc+cd \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( ae \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,c{d}^{2}ae \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{e}^{2}d \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{3}{a}^{2}{e}^{2} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0788, size = 203, normalized size = 2.23 \begin{align*} \frac{1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac{1}{5} \,{\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52703, size = 309, normalized size = 3.4 \begin{align*} \frac{1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac{1}{5} \,{\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.152798, size = 160, normalized size = 1.76 \begin{align*} a^{3} d^{2} e^{3} x + \frac{c^{3} d^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac{3 a c^{2} d^{2} e^{3}}{5} + \frac{2 c^{3} d^{4} e}{5}\right ) + x^{4} \left (\frac{3 a^{2} c d e^{4}}{4} + \frac{3 a c^{2} d^{3} e^{2}}{2} + \frac{c^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{5}}{3} + 2 a^{2} c d^{2} e^{3} + a c^{2} d^{4} e\right ) + x^{2} \left (a^{3} d e^{4} + \frac{3 a^{2} c d^{3} e^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.24299, size = 213, normalized size = 2.34 \begin{align*} \frac{1}{60} \,{\left (10 \, c^{3} d^{3} x^{6} e^{8} + 24 \, c^{3} d^{4} x^{5} e^{7} + 15 \, c^{3} d^{5} x^{4} e^{6} + 36 \, a c^{2} d^{2} x^{5} e^{9} + 90 \, a c^{2} d^{3} x^{4} e^{8} + 60 \, a c^{2} d^{4} x^{3} e^{7} + 45 \, a^{2} c d x^{4} e^{10} + 120 \, a^{2} c d^{2} x^{3} e^{9} + 90 \, a^{2} c d^{3} x^{2} e^{8} + 20 \, a^{3} x^{3} e^{11} + 60 \, a^{3} d x^{2} e^{10} + 60 \, a^{3} d^{2} x e^{9}\right )} e^{\left (-6\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*c^3*d^3*x^6*e^8 + 24*c^3*d^4*x^5*e^7 + 15*c^3*d^5*x^4*e^6 + 36*a*c^2*d^2*x^5*e^9 + 90*a*c^2*d^3*x^4*e
^8 + 60*a*c^2*d^4*x^3*e^7 + 45*a^2*c*d*x^4*e^10 + 120*a^2*c*d^2*x^3*e^9 + 90*a^2*c*d^3*x^2*e^8 + 20*a^3*x^3*e^
11 + 60*a^3*d*x^2*e^10 + 60*a^3*d^2*x*e^9)*e^(-6)