### 3.1902 $$\int \frac{(d+e x)^4}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx$$

Optimal. Leaf size=20 $-\frac{1}{3 c d (a e+c d x)^3}$

[Out]

-1/(3*c*d*(a*e + c*d*x)^3)

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Rubi [A]  time = 0.0106371, antiderivative size = 20, 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, 32} $-\frac{1}{3 c d (a e+c d x)^3}$

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)^4,x]

[Out]

-1/(3*c*d*(a*e + c*d*x)^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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[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 )^4} \, dx &=\int \frac{1}{(a e+c d x)^4} \, dx\\ &=-\frac{1}{3 c d (a e+c d x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0037334, size = 20, normalized size = 1. $-\frac{1}{3 c d (a e+c d x)^3}$

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)^4,x]

[Out]

-1/(3*c*d*(a*e + c*d*x)^3)

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Maple [A]  time = 0.038, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{3\,cd \left ( cdx+ae \right ) ^{3}}} \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)^4,x)

[Out]

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

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Maxima [B]  time = 1.11815, size = 70, normalized size = 3.5 \begin{align*} -\frac{1}{3 \,{\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.7866, size = 103, normalized size = 5.15 \begin{align*} -\frac{1}{3 \,{\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.672494, size = 58, normalized size = 2.9 \begin{align*} - \frac{1}{3 a^{3} c d e^{3} + 9 a^{2} c^{2} d^{2} e^{2} x + 9 a c^{3} d^{3} e x^{2} + 3 c^{4} d^{4} x^{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)**4,x)

[Out]

-1/(3*a**3*c*d*e**3 + 9*a**2*c**2*d**2*e**2*x + 9*a*c**3*d**3*e*x**2 + 3*c**4*d**4*x**3)

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Giac [B]  time = 3.41515, size = 639, normalized size = 31.95 \begin{align*} -\frac{c^{6} d^{12} x^{3} e^{3} + 3 \, c^{6} d^{13} x^{2} e^{2} + 3 \, c^{6} d^{14} x e + c^{6} d^{15} - 6 \, a c^{5} d^{10} x^{3} e^{5} - 18 \, a c^{5} d^{11} x^{2} e^{4} - 18 \, a c^{5} d^{12} x e^{3} - 6 \, a c^{5} d^{13} e^{2} + 15 \, a^{2} c^{4} d^{8} x^{3} e^{7} + 45 \, a^{2} c^{4} d^{9} x^{2} e^{6} + 45 \, a^{2} c^{4} d^{10} x e^{5} + 15 \, a^{2} c^{4} d^{11} e^{4} - 20 \, a^{3} c^{3} d^{6} x^{3} e^{9} - 60 \, a^{3} c^{3} d^{7} x^{2} e^{8} - 60 \, a^{3} c^{3} d^{8} x e^{7} - 20 \, a^{3} c^{3} d^{9} e^{6} + 15 \, a^{4} c^{2} d^{4} x^{3} e^{11} + 45 \, a^{4} c^{2} d^{5} x^{2} e^{10} + 45 \, a^{4} c^{2} d^{6} x e^{9} + 15 \, a^{4} c^{2} d^{7} e^{8} - 6 \, a^{5} c d^{2} x^{3} e^{13} - 18 \, a^{5} c d^{3} x^{2} e^{12} - 18 \, a^{5} c d^{4} x e^{11} - 6 \, a^{5} c d^{5} e^{10} + a^{6} x^{3} e^{15} + 3 \, a^{6} d x^{2} e^{14} + 3 \, a^{6} d^{2} x e^{13} + a^{6} d^{3} e^{12}}{3 \,{\left (c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{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)^4,x, algorithm="giac")

[Out]

-1/3*(c^6*d^12*x^3*e^3 + 3*c^6*d^13*x^2*e^2 + 3*c^6*d^14*x*e + c^6*d^15 - 6*a*c^5*d^10*x^3*e^5 - 18*a*c^5*d^11
*x^2*e^4 - 18*a*c^5*d^12*x*e^3 - 6*a*c^5*d^13*e^2 + 15*a^2*c^4*d^8*x^3*e^7 + 45*a^2*c^4*d^9*x^2*e^6 + 45*a^2*c
^4*d^10*x*e^5 + 15*a^2*c^4*d^11*e^4 - 20*a^3*c^3*d^6*x^3*e^9 - 60*a^3*c^3*d^7*x^2*e^8 - 60*a^3*c^3*d^8*x*e^7 -
20*a^3*c^3*d^9*e^6 + 15*a^4*c^2*d^4*x^3*e^11 + 45*a^4*c^2*d^5*x^2*e^10 + 45*a^4*c^2*d^6*x*e^9 + 15*a^4*c^2*d^
7*e^8 - 6*a^5*c*d^2*x^3*e^13 - 18*a^5*c*d^3*x^2*e^12 - 18*a^5*c*d^4*x*e^11 - 6*a^5*c*d^5*e^10 + a^6*x^3*e^15 +
3*a^6*d*x^2*e^14 + 3*a^6*d^2*x*e^13 + a^6*d^3*e^12)/((c^7*d^13 - 6*a*c^6*d^11*e^2 + 15*a^2*c^5*d^9*e^4 - 20*a
^3*c^4*d^7*e^6 + 15*a^4*c^3*d^5*e^8 - 6*a^5*c^2*d^3*e^10 + a^6*c*d*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*
e)^3)