### 3.1842 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{d+e x} \, dx$$

Optimal. Leaf size=54 $\frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0490563, antiderivative size = 54, 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} $\frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0125745, size = 54, normalized size = 1. $\frac{1}{12} x \left (6 a^2 e^2 (2 d+e x)+4 a c d e x (3 d+2 e x)+c^2 d^2 x^2 (4 d+3 e x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(6*a^2*e^2*(2*d + e*x) + 4*a*c*d*e*x*(3*d + 2*e*x) + c^2*d^2*x^2*(4*d + 3*e*x)))/12

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 77, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{d}^{2}e{x}^{4}}{4}}+{\frac{ \left ( ad{e}^{2}c+cd \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( ae \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{2}ae \right ){x}^{2}}{2}}+{a}^{2}{e}^{2}dx \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)^2/(e*x+d),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10627, size = 86, normalized size = 1.59 \begin{align*} \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\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)^2/(e*x+d),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.52867, size = 136, normalized size = 2.52 \begin{align*} \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\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)^2/(e*x+d),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.168092, size = 66, normalized size = 1.22 \begin{align*} a^{2} d e^{2} x + \frac{c^{2} d^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{a^{2} e^{3}}{2} + a c d^{2} e\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)**2/(e*x+d),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1643, size = 97, normalized size = 1.8 \begin{align*} \frac{1}{12} \,{\left (3 \, c^{2} d^{2} x^{4} e^{5} + 4 \, c^{2} d^{3} x^{3} e^{4} + 8 \, a c d x^{3} e^{6} + 12 \, a c d^{2} x^{2} e^{5} + 6 \, a^{2} x^{2} e^{7} + 12 \, a^{2} d x e^{6}\right )} e^{\left (-4\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)^2/(e*x+d),x, algorithm="giac")

[Out]

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