### 3.1861 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^8} \, dx$$

Optimal. Leaf size=35 $\frac{(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0152227, 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} $\frac{(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^8,x]

[Out]

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

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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^5} \, dx\\ &=\frac{(a e+c d x)^4}{4 \left (c d^2-a e^2\right ) (d+e x)^4}\\ \end{align*}

Mathematica [B]  time = 0.0397116, size = 100, normalized size = 2.86 $-\frac{a^2 c d e^4 (d+4 e x)+a^3 e^6+a c^2 d^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^3 d^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{4 e^4 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^8,x]

[Out]

-(a^3*e^6 + a^2*c*d*e^4*(d + 4*e*x) + a*c^2*d^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^3*d^3*(d^3 + 4*d^2*e*x + 6
*d*e^2*x^2 + 4*e^3*x^3))/(4*e^4*(d + e*x)^4)

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Maple [B]  time = 0.045, size = 141, normalized size = 4. \begin{align*} -{\frac{3\,{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{c}^{2}{d}^{4}{e}^{2}-{c}^{3}{d}^{6}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x)

[Out]

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

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Maxima [B]  time = 1.13931, size = 213, normalized size = 6.09 \begin{align*} -\frac{4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/4*(4*c^3*d^3*e^3*x^3 + c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + a^3*e^6 + 6*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*
x^2 + 4*(c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^
4*e^4)

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Fricas [B]  time = 1.52173, size = 308, normalized size = 8.8 \begin{align*} -\frac{4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/4*(4*c^3*d^3*e^3*x^3 + c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + a^3*e^6 + 6*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*
x^2 + 4*(c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^
4*e^4)

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Sympy [B]  time = 4.7249, size = 168, normalized size = 4.8 \begin{align*} - \frac{a^{3} e^{6} + a^{2} c d^{2} e^{4} + a c^{2} d^{4} e^{2} + c^{3} d^{6} + 4 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (6 a c^{2} d^{2} e^{4} + 6 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} c d e^{5} + 4 a c^{2} d^{3} e^{3} + 4 c^{3} d^{5} e\right )}{4 d^{4} e^{4} + 16 d^{3} e^{5} x + 24 d^{2} e^{6} x^{2} + 16 d e^{7} x^{3} + 4 e^{8} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**8,x)

[Out]

-(a**3*e**6 + a**2*c*d**2*e**4 + a*c**2*d**4*e**2 + c**3*d**6 + 4*c**3*d**3*e**3*x**3 + x**2*(6*a*c**2*d**2*e*
*4 + 6*c**3*d**4*e**2) + x*(4*a**2*c*d*e**5 + 4*a*c**2*d**3*e**3 + 4*c**3*d**5*e))/(4*d**4*e**4 + 16*d**3*e**5
*x + 24*d**2*e**6*x**2 + 16*d*e**7*x**3 + 4*e**8*x**4)

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Giac [B]  time = 1.18985, size = 373, normalized size = 10.66 \begin{align*} -\frac{{\left (4 \, c^{3} d^{3} x^{6} e^{6} + 18 \, c^{3} d^{4} x^{5} e^{5} + 34 \, c^{3} d^{5} x^{4} e^{4} + 35 \, c^{3} d^{6} x^{3} e^{3} + 21 \, c^{3} d^{7} x^{2} e^{2} + 7 \, c^{3} d^{8} x e + c^{3} d^{9} + 6 \, a c^{2} d^{2} x^{5} e^{7} + 22 \, a c^{2} d^{3} x^{4} e^{6} + 31 \, a c^{2} d^{4} x^{3} e^{5} + 21 \, a c^{2} d^{5} x^{2} e^{4} + 7 \, a c^{2} d^{6} x e^{3} + a c^{2} d^{7} e^{2} + 4 \, a^{2} c d x^{4} e^{8} + 13 \, a^{2} c d^{2} x^{3} e^{7} + 15 \, a^{2} c d^{3} x^{2} e^{6} + 7 \, a^{2} c d^{4} x e^{5} + a^{2} c d^{5} e^{4} + a^{3} x^{3} e^{9} + 3 \, a^{3} d x^{2} e^{8} + 3 \, a^{3} d^{2} x e^{7} + a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{4 \,{\left (x e + d\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/4*(4*c^3*d^3*x^6*e^6 + 18*c^3*d^4*x^5*e^5 + 34*c^3*d^5*x^4*e^4 + 35*c^3*d^6*x^3*e^3 + 21*c^3*d^7*x^2*e^2 +
7*c^3*d^8*x*e + c^3*d^9 + 6*a*c^2*d^2*x^5*e^7 + 22*a*c^2*d^3*x^4*e^6 + 31*a*c^2*d^4*x^3*e^5 + 21*a*c^2*d^5*x^2
*e^4 + 7*a*c^2*d^6*x*e^3 + a*c^2*d^7*e^2 + 4*a^2*c*d*x^4*e^8 + 13*a^2*c*d^2*x^3*e^7 + 15*a^2*c*d^3*x^2*e^6 + 7
*a^2*c*d^4*x*e^5 + a^2*c*d^5*e^4 + a^3*x^3*e^9 + 3*a^3*d*x^2*e^8 + 3*a^3*d^2*x*e^7 + a^3*d^3*e^6)*e^(-4)/(x*e
+ d)^7