### 3.1862 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^9} \, dx$$

Optimal. Leaf size=73 $\frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )}$

[Out]

(a*e + c*d*x)^4/(5*(c*d^2 - a*e^2)*(d + e*x)^5) + (c*d*(a*e + c*d*x)^4)/(20*(c*d^2 - a*e^2)^2*(d + e*x)^4)

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Rubi [A]  time = 0.0262825, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.086, Rules used = {626, 45, 37} $\frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )}$

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)^9,x]

[Out]

(a*e + c*d*x)^4/(5*(c*d^2 - a*e^2)*(d + e*x)^5) + (c*d*(a*e + c*d*x)^4)/(20*(c*d^2 - a*e^2)^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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^9} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^6} \, dx\\ &=\frac{(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{(c d) \int \frac{(a e+c d x)^3}{(d+e x)^5} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac{(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0406616, size = 103, normalized size = 1.41 $-\frac{3 a^2 c d e^4 (d+5 e x)+4 a^3 e^6+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5}$

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)^9,x]

[Out]

-(4*a^3*e^6 + 3*a^2*c*d*e^4*(d + 5*e*x) + 2*a*c^2*d^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + c^3*d^3*(d^3 + 5*d^2*
e*x + 10*d*e^2*x^2 + 10*e^3*x^3))/(20*e^4*(d + e*x)^5)

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Maple [B]  time = 0.046, size = 141, normalized size = 1.9 \begin{align*} -{\frac{{c}^{3}{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{3\,cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\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}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \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)^9,x)

[Out]

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

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Maxima [B]  time = 1.12977, size = 236, normalized size = 3.23 \begin{align*} -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} 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)^9,x, algorithm="maxima")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Fricas [B]  time = 1.50647, size = 352, normalized size = 4.82 \begin{align*} -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} 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)^9,x, algorithm="fricas")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Sympy [B]  time = 18.4928, size = 185, normalized size = 2.53 \begin{align*} - \frac{4 a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 2 a c^{2} d^{4} e^{2} + c^{3} d^{6} + 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (20 a c^{2} d^{2} e^{4} + 10 c^{3} d^{4} e^{2}\right ) + x \left (15 a^{2} c d e^{5} + 10 a c^{2} d^{3} e^{3} + 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \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)**9,x)

[Out]

-(4*a**3*e**6 + 3*a**2*c*d**2*e**4 + 2*a*c**2*d**4*e**2 + c**3*d**6 + 10*c**3*d**3*e**3*x**3 + x**2*(20*a*c**2
*d**2*e**4 + 10*c**3*d**4*e**2) + x*(15*a**2*c*d*e**5 + 10*a*c**2*d**3*e**3 + 5*c**3*d**5*e))/(20*d**5*e**4 +
100*d**4*e**5*x + 200*d**3*e**6*x**2 + 200*d**2*e**7*x**3 + 100*d*e**8*x**4 + 20*e**9*x**5)

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Giac [B]  time = 1.21506, size = 378, normalized size = 5.18 \begin{align*} -\frac{{\left (10 \, c^{3} d^{3} x^{6} e^{6} + 40 \, c^{3} d^{4} x^{5} e^{5} + 65 \, c^{3} d^{5} x^{4} e^{4} + 56 \, c^{3} d^{6} x^{3} e^{3} + 28 \, c^{3} d^{7} x^{2} e^{2} + 8 \, c^{3} d^{8} x e + c^{3} d^{9} + 20 \, a c^{2} d^{2} x^{5} e^{7} + 70 \, a c^{2} d^{3} x^{4} e^{6} + 92 \, a c^{2} d^{4} x^{3} e^{5} + 56 \, a c^{2} d^{5} x^{2} e^{4} + 16 \, a c^{2} d^{6} x e^{3} + 2 \, a c^{2} d^{7} e^{2} + 15 \, a^{2} c d x^{4} e^{8} + 48 \, a^{2} c d^{2} x^{3} e^{7} + 54 \, a^{2} c d^{3} x^{2} e^{6} + 24 \, a^{2} c d^{4} x e^{5} + 3 \, a^{2} c d^{5} e^{4} + 4 \, a^{3} x^{3} e^{9} + 12 \, a^{3} d x^{2} e^{8} + 12 \, a^{3} d^{2} x e^{7} + 4 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{20 \,{\left (x e + d\right )}^{8}} \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)^9,x, algorithm="giac")

[Out]

-1/20*(10*c^3*d^3*x^6*e^6 + 40*c^3*d^4*x^5*e^5 + 65*c^3*d^5*x^4*e^4 + 56*c^3*d^6*x^3*e^3 + 28*c^3*d^7*x^2*e^2
+ 8*c^3*d^8*x*e + c^3*d^9 + 20*a*c^2*d^2*x^5*e^7 + 70*a*c^2*d^3*x^4*e^6 + 92*a*c^2*d^4*x^3*e^5 + 56*a*c^2*d^5*
x^2*e^4 + 16*a*c^2*d^6*x*e^3 + 2*a*c^2*d^7*e^2 + 15*a^2*c*d*x^4*e^8 + 48*a^2*c*d^2*x^3*e^7 + 54*a^2*c*d^3*x^2*
e^6 + 24*a^2*c*d^4*x*e^5 + 3*a^2*c*d^5*e^4 + 4*a^3*x^3*e^9 + 12*a^3*d*x^2*e^8 + 12*a^3*d^2*x*e^7 + 4*a^3*d^3*e
^6)*e^(-4)/(x*e + d)^8