### 3.1868 $$\int \frac{(d+e x)^2}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx$$

Optimal. Leaf size=38 $\frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d}$

[Out]

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

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Rubi [A]  time = 0.0323342, antiderivative size = 38, 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} $\frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0089051, size = 35, normalized size = 0.92 $\frac{\left (c d^2-a e^2\right ) \log (a e+c d x)+c d e x}{c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 45, normalized size = 1.2 \begin{align*}{\frac{ex}{cd}}-{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( cdx+ae \right ) }{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0063, size = 51, normalized size = 1.34 \begin{align*} \frac{e x}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55843, size = 76, normalized size = 2. \begin{align*} \frac{c d e x +{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.376055, size = 32, normalized size = 0.84 \begin{align*} \frac{e x}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.25951, size = 215, normalized size = 5.66 \begin{align*} \frac{x e}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e/(c*d) + 1/2*(c*d^2 - a*e^2)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^2*d^2) + (c^2*d^4 - 2*a*c*d^2*e^
2 + a^2*e^4)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/(sqrt(-c^2*d^4 + 2*a
*c*d^2*e^2 - a^2*e^4)*c^2*d^2)