### 3.1878 $$\int \frac{(d+e x)^4}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx$$

Optimal. Leaf size=74 $-\frac{\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}+\frac{e^2 x}{c^2 d^2}$

[Out]

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

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Rubi [A]  time = 0.0591841, antiderivative size = 74, 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} $-\frac{\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}+\frac{e^2 x}{c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0461451, size = 65, normalized size = 0.88 $\frac{-\frac{\left (c d^2-a e^2\right )^2}{a e+c d x}+2 \left (c d^2 e-a e^3\right ) \log (a e+c d x)+c d e^2 x}{c^3 d^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.053, size = 114, normalized size = 1.5 \begin{align*}{\frac{{e}^{2}x}{{c}^{2}{d}^{2}}}-{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}+2\,{\frac{a{e}^{2}}{{c}^{2}d \left ( cdx+ae \right ) }}-{\frac{d}{c \left ( cdx+ae \right ) }}-2\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}{d}^{3}}}+2\,{\frac{e\ln \left ( cdx+ae \right ) }{{c}^{2}d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.02031, size = 120, normalized size = 1.62 \begin{align*} -\frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{c^{4} d^{4} x + a c^{3} d^{3} e} + \frac{e^{2} x}{c^{2} d^{2}} + \frac{2 \,{\left (c d^{2} e - a e^{3}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92026, size = 227, normalized size = 3.07 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + a c d e^{3} x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (c d x + a e\right )}{c^{4} d^{4} x + a c^{3} d^{3} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.68862, size = 85, normalized size = 1.15 \begin{align*} - \frac{a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}}{a c^{3} d^{3} e + c^{4} d^{4} x} + \frac{e^{2} x}{c^{2} d^{2}} - \frac{2 e \left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.29919, size = 525, normalized size = 7.09 \begin{align*} \frac{2 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\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^{5} d^{7} - 2 \, a c^{4} d^{5} e^{2} + a^{2} c^{3} d^{3} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{x e^{2}}{c^{2} d^{2}} + \frac{{\left (c d^{2} e - a e^{3}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{3} d^{3}} - \frac{\frac{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c} + \frac{{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} x}{c d}}{{\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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

2*(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9)*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^5*d^7 - 2*a*c^4*d^5*e^2 + a^2*c^3*d^3*e^4)*sqrt(-c^2*d^4 +
2*a*c*d^2*e^2 - a^2*e^4)) + x*e^2/(c^2*d^2) + (c*d^2*e - a*e^3)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^
3*d^3) - ((c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)/c + (c^4*d^8*e - 4*a*c^3
*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9)*x/(c*d))/((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)