### 3.1834 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^2} \, dx$$

Optimal. Leaf size=26 $\left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e}$

[Out]

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

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Rubi [A]  time = 0.0281776, antiderivative size = 26, 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 = {24, 43} $\left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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

Mathematica [A]  time = 0.0091383, size = 30, normalized size = 1.15 $\frac{\left (a e^2-c d^2\right ) \log (d+e x)}{e^2}+\frac{c d x}{e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 32, normalized size = 1.2 \begin{align*}{\frac{cdx}{e}}+\ln \left ( ex+d \right ) a-{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{{e}^{2}}} \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)/(e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 1.01912, size = 42, normalized size = 1.62 \begin{align*} \frac{c d x}{e} - \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{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)/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56012, size = 62, normalized size = 2.38 \begin{align*} \frac{c d e x -{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{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)/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.392545, size = 26, normalized size = 1. \begin{align*} \frac{c d x}{e} + \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{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)/(e*x+d)**2,x)

[Out]

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

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Giac [B]  time = 1.24215, size = 158, normalized size = 6.08 \begin{align*}{\left (2 \, d e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c d e -{\left (c d^{2} + a e^{2}\right )}{\left (e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-1\right )} - \frac{a d}{x e + d} \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)/(e*x+d)^2,x, algorithm="giac")

[Out]

(2*d*e^(-3)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (x*e + d)*e^(-3) - d^2*e^(-3)/(x*e + d))*c*d*e - (c*d^2 + a
*e^2)*(e^(-1)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - d*e^(-1)/(x*e + d))*e^(-1) - a*d/(x*e + d)