3.1890 \(\int \frac{(d+e x)^4}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=35 \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0131445, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 37} \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

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)^3,x]

[Out]

-(d + e*x)^2/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A]  time = 0.0132388, size = 35, normalized size = 1. \[ -\frac{a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \]

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)^3,x]

[Out]

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

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Maple [A]  time = 0.042, size = 51, normalized size = 1.5 \begin{align*} -{\frac{-a{e}^{2}+c{d}^{2}}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{e}{{c}^{2}{d}^{2} \left ( cdx+ae \right ) }} \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)^3,x)

[Out]

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

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Maxima [A]  time = 1.08134, size = 76, normalized size = 2.17 \begin{align*} -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83694, size = 113, normalized size = 3.23 \begin{align*} -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.61767, size = 60, normalized size = 1.71 \begin{align*} - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{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)**3,x)

[Out]

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

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Giac [B]  time = 3.10052, size = 509, normalized size = 14.54 \begin{align*} -\frac{2 \, c^{5} d^{9} x^{3} e^{3} + 5 \, c^{5} d^{10} x^{2} e^{2} + 4 \, c^{5} d^{11} x e + c^{5} d^{12} - 8 \, a c^{4} d^{7} x^{3} e^{5} - 19 \, a c^{4} d^{8} x^{2} e^{4} - 14 \, a c^{4} d^{9} x e^{3} - 3 \, a c^{4} d^{10} e^{2} + 12 \, a^{2} c^{3} d^{5} x^{3} e^{7} + 26 \, a^{2} c^{3} d^{6} x^{2} e^{6} + 16 \, a^{2} c^{3} d^{7} x e^{5} + 2 \, a^{2} c^{3} d^{8} e^{4} - 8 \, a^{3} c^{2} d^{3} x^{3} e^{9} - 14 \, a^{3} c^{2} d^{4} x^{2} e^{8} - 4 \, a^{3} c^{2} d^{5} x e^{7} + 2 \, a^{3} c^{2} d^{6} e^{6} + 2 \, a^{4} c d x^{3} e^{11} + a^{4} c d^{2} x^{2} e^{10} - 4 \, a^{4} c d^{3} x e^{9} - 3 \, a^{4} c d^{4} e^{8} + a^{5} x^{2} e^{12} + 2 \, a^{5} d x e^{11} + a^{5} d^{2} e^{10}}{2 \,{\left (c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{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)^3,x, algorithm="giac")

[Out]

-1/2*(2*c^5*d^9*x^3*e^3 + 5*c^5*d^10*x^2*e^2 + 4*c^5*d^11*x*e + c^5*d^12 - 8*a*c^4*d^7*x^3*e^5 - 19*a*c^4*d^8*
x^2*e^4 - 14*a*c^4*d^9*x*e^3 - 3*a*c^4*d^10*e^2 + 12*a^2*c^3*d^5*x^3*e^7 + 26*a^2*c^3*d^6*x^2*e^6 + 16*a^2*c^3
*d^7*x*e^5 + 2*a^2*c^3*d^8*e^4 - 8*a^3*c^2*d^3*x^3*e^9 - 14*a^3*c^2*d^4*x^2*e^8 - 4*a^3*c^2*d^5*x*e^7 + 2*a^3*
c^2*d^6*e^6 + 2*a^4*c*d*x^3*e^11 + a^4*c*d^2*x^2*e^10 - 4*a^4*c*d^3*x*e^9 - 3*a^4*c*d^4*e^8 + a^5*x^2*e^12 + 2
*a^5*d*x*e^11 + a^5*d^2*e^10)/((c^6*d^10 - 4*a*c^5*d^8*e^2 + 6*a^2*c^4*d^6*e^4 - 4*a^3*c^3*d^4*e^6 + a^4*c^2*d
^2*e^8)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2)