### 3.1879 $$\int \frac{(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx$$

Optimal. Leaf size=48 $\frac{e \log (a e+c d x)}{c^2 d^2}-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)}$

[Out]

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

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Rubi [A]  time = 0.0363662, antiderivative size = 48, 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} $\frac{e \log (a e+c d x)}{c^2 d^2}-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.012727, size = 47, normalized size = 0.98 $\frac{a e^2-c d^2}{c^2 d^2 (a e+c d x)}+\frac{e \log (a e+c d x)}{c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 55, normalized size = 1.2 \begin{align*}{\frac{a{e}^{2}}{{c}^{2}{d}^{2} \left ( cdx+ae \right ) }}-{\frac{1}{c \left ( cdx+ae \right ) }}+{\frac{e\ln \left ( cdx+ae \right ) }{{c}^{2}{d}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02123, size = 70, normalized size = 1.46 \begin{align*} -\frac{c d^{2} - a e^{2}}{c^{3} d^{3} x + a c^{2} d^{2} e} + \frac{e \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)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.88916, size = 109, normalized size = 2.27 \begin{align*} -\frac{c d^{2} - a e^{2} -{\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3} x + a c^{2} d^{2} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.444943, size = 46, normalized size = 0.96 \begin{align*} \frac{a e^{2} - c d^{2}}{a c^{2} d^{2} e + c^{3} d^{3} x} + \frac{e \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)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

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

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Giac [B]  time = 1.20888, size = 435, normalized size = 9.06 \begin{align*} \frac{{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\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 )}{{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{e \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{c^{3} d^{7} - 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} - a^{3} d e^{6} +{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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