### 3.1845 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=63 $-\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2}$

[Out]

(c^2*d^2*x)/e^2 - (c*d^2 - a*e^2)^2/(e^3*(d + e*x)) - (2*c*d*(c*d^2 - a*e^2)*Log[d + e*x])/e^3

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Rubi [A]  time = 0.0520952, antiderivative size = 63, 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} $-\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2}$

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)^4,x]

[Out]

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

Mathematica [A]  time = 0.0409001, size = 59, normalized size = 0.94 $\frac{-\frac{\left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (a e^2-c d^2\right ) \log (d+e x)+c^2 d^2 e x}{e^3}$

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)^4,x]

[Out]

(c^2*d^2*e*x - (c*d^2 - a*e^2)^2/(d + e*x) + 2*c*d*(-(c*d^2) + a*e^2)*Log[d + e*x])/e^3

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Maple [A]  time = 0.046, size = 92, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}{d}^{2}x}{{e}^{2}}}+2\,{\frac{cd\ln \left ( ex+d \right ) a}{e}}-2\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{{a}^{2}e}{ex+d}}+2\,{\frac{ac{d}^{2}}{e \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{3} \left ( ex+d \right ) }} \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)^4,x)

[Out]

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

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Maxima [A]  time = 1.01256, size = 107, normalized size = 1.7 \begin{align*} \frac{c^{2} d^{2} x}{e^{2}} - \frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{4} x + d e^{3}} - \frac{2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50685, size = 208, normalized size = 3.3 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + c^{2} d^{3} e x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 2 \,{\left (c^{2} d^{4} - a c d^{2} e^{2} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.647083, size = 71, normalized size = 1.13 \begin{align*} \frac{c^{2} d^{2} x}{e^{2}} + \frac{2 c d \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}}{d e^{3} + e^{4} x} \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)**4,x)

[Out]

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

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Giac [B]  time = 1.2321, size = 181, normalized size = 2.87 \begin{align*} c^{2} d^{2} x e^{\left (-2\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

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