3.1864 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=111 \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac{c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac{\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac{c^3 d^3}{4 e^4 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0682159, antiderivative size = 111, 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} \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac{c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac{\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac{c^3 d^3}{4 e^4 (d+e x)^4} \]

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)^11,x]

[Out]

(c*d^2 - a*e^2)^3/(7*e^4*(d + e*x)^7) - (c*d*(c*d^2 - a*e^2)^2)/(2*e^4*(d + e*x)^6) + (3*c^2*d^2*(c*d^2 - a*e^
2))/(5*e^4*(d + e*x)^5) - (c^3*d^3)/(4*e^4*(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 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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^8} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^8}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^7}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^6}+\frac{c^3 d^3}{e^3 (d+e x)^5}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac{c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac{c^3 d^3}{4 e^4 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.03442, size = 103, normalized size = 0.93 \[ -\frac{10 a^2 c d e^4 (d+7 e x)+20 a^3 e^6+4 a c^2 d^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^3 d^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )}{140 e^4 (d+e x)^7} \]

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)^11,x]

[Out]

-(20*a^3*e^6 + 10*a^2*c*d*e^4*(d + 7*e*x) + 4*a*c^2*d^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^3*d^3*(d^3 + 7*d^
2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3))/(140*e^4*(d + e*x)^7)

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Maple [A]  time = 0.047, size = 141, normalized size = 1.3 \begin{align*} -{\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}}{7\,{e}^{4} \left ( ex+d \right ) ^{7}}}-{\frac{{c}^{3}{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{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)^11,x)

[Out]

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

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Maxima [A]  time = 1.06908, size = 266, normalized size = 2.4 \begin{align*} -\frac{35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \,{\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \,{\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} 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)^11,x, algorithm="maxima")

[Out]

-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a
*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a*c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*
x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

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Fricas [A]  time = 1.52223, size = 406, normalized size = 3.66 \begin{align*} -\frac{35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \,{\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \,{\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} 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)^11,x, algorithm="fricas")

[Out]

-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a
*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a*c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*
x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

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Sympy [B]  time = 100.358, size = 209, normalized size = 1.88 \begin{align*} - \frac{20 a^{3} e^{6} + 10 a^{2} c d^{2} e^{4} + 4 a c^{2} d^{4} e^{2} + c^{3} d^{6} + 35 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (84 a c^{2} d^{2} e^{4} + 21 c^{3} d^{4} e^{2}\right ) + x \left (70 a^{2} c d e^{5} + 28 a c^{2} d^{3} e^{3} + 7 c^{3} d^{5} e\right )}{140 d^{7} e^{4} + 980 d^{6} e^{5} x + 2940 d^{5} e^{6} x^{2} + 4900 d^{4} e^{7} x^{3} + 4900 d^{3} e^{8} x^{4} + 2940 d^{2} e^{9} x^{5} + 980 d e^{10} x^{6} + 140 e^{11} x^{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)**11,x)

[Out]

-(20*a**3*e**6 + 10*a**2*c*d**2*e**4 + 4*a*c**2*d**4*e**2 + c**3*d**6 + 35*c**3*d**3*e**3*x**3 + x**2*(84*a*c*
*2*d**2*e**4 + 21*c**3*d**4*e**2) + x*(70*a**2*c*d*e**5 + 28*a*c**2*d**3*e**3 + 7*c**3*d**5*e))/(140*d**7*e**4
+ 980*d**6*e**5*x + 2940*d**5*e**6*x**2 + 4900*d**4*e**7*x**3 + 4900*d**3*e**8*x**4 + 2940*d**2*e**9*x**5 + 9
80*d*e**10*x**6 + 140*e**11*x**7)

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Giac [B]  time = 1.20712, size = 378, normalized size = 3.41 \begin{align*} -\frac{{\left (35 \, c^{3} d^{3} x^{6} e^{6} + 126 \, c^{3} d^{4} x^{5} e^{5} + 175 \, c^{3} d^{5} x^{4} e^{4} + 120 \, c^{3} d^{6} x^{3} e^{3} + 45 \, c^{3} d^{7} x^{2} e^{2} + 10 \, c^{3} d^{8} x e + c^{3} d^{9} + 84 \, a c^{2} d^{2} x^{5} e^{7} + 280 \, a c^{2} d^{3} x^{4} e^{6} + 340 \, a c^{2} d^{4} x^{3} e^{5} + 180 \, a c^{2} d^{5} x^{2} e^{4} + 40 \, a c^{2} d^{6} x e^{3} + 4 \, a c^{2} d^{7} e^{2} + 70 \, a^{2} c d x^{4} e^{8} + 220 \, a^{2} c d^{2} x^{3} e^{7} + 240 \, a^{2} c d^{3} x^{2} e^{6} + 100 \, a^{2} c d^{4} x e^{5} + 10 \, a^{2} c d^{5} e^{4} + 20 \, a^{3} x^{3} e^{9} + 60 \, a^{3} d x^{2} e^{8} + 60 \, a^{3} d^{2} x e^{7} + 20 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{140 \,{\left (x e + d\right )}^{10}} \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)^11,x, algorithm="giac")

[Out]

-1/140*(35*c^3*d^3*x^6*e^6 + 126*c^3*d^4*x^5*e^5 + 175*c^3*d^5*x^4*e^4 + 120*c^3*d^6*x^3*e^3 + 45*c^3*d^7*x^2*
e^2 + 10*c^3*d^8*x*e + c^3*d^9 + 84*a*c^2*d^2*x^5*e^7 + 280*a*c^2*d^3*x^4*e^6 + 340*a*c^2*d^4*x^3*e^5 + 180*a*
c^2*d^5*x^2*e^4 + 40*a*c^2*d^6*x*e^3 + 4*a*c^2*d^7*e^2 + 70*a^2*c*d*x^4*e^8 + 220*a^2*c*d^2*x^3*e^7 + 240*a^2*
c*d^3*x^2*e^6 + 100*a^2*c*d^4*x*e^5 + 10*a^2*c*d^5*e^4 + 20*a^3*x^3*e^9 + 60*a^3*d*x^2*e^8 + 60*a^3*d^2*x*e^7
+ 20*a^3*d^3*e^6)*e^(-4)/(x*e + d)^10