3.1863 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{10}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.071321, 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} \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{4 e^4 (d+e x)^4}-\frac{3 c d \left (c d^2-a e^2\right )^2}{5 e^4 (d+e x)^5}+\frac{\left (c d^2-a e^2\right )^3}{6 e^4 (d+e x)^6}-\frac{c^3 d^3}{3 e^4 (d+e x)^3} \]

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)^10,x]

[Out]

(c*d^2 - a*e^2)^3/(6*e^4*(d + e*x)^6) - (3*c*d*(c*d^2 - a*e^2)^2)/(5*e^4*(d + e*x)^5) + (3*c^2*d^2*(c*d^2 - a*
e^2))/(4*e^4*(d + e*x)^4) - (c^3*d^3)/(3*e^4*(d + e*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 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)^{10}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^7} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^7}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^6}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^5}+\frac{c^3 d^3}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^3}{6 e^4 (d+e x)^6}-\frac{3 c d \left (c d^2-a e^2\right )^2}{5 e^4 (d+e x)^5}+\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{4 e^4 (d+e x)^4}-\frac{c^3 d^3}{3 e^4 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0349013, size = 103, normalized size = 0.93 \[ -\frac{6 a^2 c d e^4 (d+6 e x)+10 a^3 e^6+3 a c^2 d^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^3 d^3 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \]

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)^10,x]

[Out]

-(10*a^3*e^6 + 6*a^2*c*d*e^4*(d + 6*e*x) + 3*a*c^2*d^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^3*d^3*(d^3 + 6*d^2
*e*x + 15*d*e^2*x^2 + 20*e^3*x^3))/(60*e^4*(d + e*x)^6)

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Maple [A]  time = 0.045, size = 141, normalized size = 1.3 \begin{align*} -{\frac{3\,{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\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}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{3\,cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \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)^10,x)

[Out]

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

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Maxima [A]  time = 1.15409, size = 251, normalized size = 2.26 \begin{align*} -\frac{20 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} + 10 \, a^{3} e^{6} + 15 \,{\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 6 \, a^{2} c d e^{5}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53929, size = 378, normalized size = 3.41 \begin{align*} -\frac{20 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} + 10 \, a^{3} e^{6} + 15 \,{\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 6 \, a^{2} c d e^{5}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 24.9106, size = 197, normalized size = 1.77 \begin{align*} - \frac{10 a^{3} e^{6} + 6 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6} + 20 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (45 a c^{2} d^{2} e^{4} + 15 c^{3} d^{4} e^{2}\right ) + x \left (36 a^{2} c d e^{5} + 18 a c^{2} d^{3} e^{3} + 6 c^{3} d^{5} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \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)**10,x)

[Out]

-(10*a**3*e**6 + 6*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6 + 20*c**3*d**3*e**3*x**3 + x**2*(45*a*c**
2*d**2*e**4 + 15*c**3*d**4*e**2) + x*(36*a**2*c*d*e**5 + 18*a*c**2*d**3*e**3 + 6*c**3*d**5*e))/(60*d**6*e**4 +
360*d**5*e**5*x + 900*d**4*e**6*x**2 + 1200*d**3*e**7*x**3 + 900*d**2*e**8*x**4 + 360*d*e**9*x**5 + 60*e**10*
x**6)

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Giac [B]  time = 1.20137, size = 378, normalized size = 3.41 \begin{align*} -\frac{{\left (20 \, c^{3} d^{3} x^{6} e^{6} + 75 \, c^{3} d^{4} x^{5} e^{5} + 111 \, c^{3} d^{5} x^{4} e^{4} + 84 \, c^{3} d^{6} x^{3} e^{3} + 36 \, c^{3} d^{7} x^{2} e^{2} + 9 \, c^{3} d^{8} x e + c^{3} d^{9} + 45 \, a c^{2} d^{2} x^{5} e^{7} + 153 \, a c^{2} d^{3} x^{4} e^{6} + 192 \, a c^{2} d^{4} x^{3} e^{5} + 108 \, a c^{2} d^{5} x^{2} e^{4} + 27 \, a c^{2} d^{6} x e^{3} + 3 \, a c^{2} d^{7} e^{2} + 36 \, a^{2} c d x^{4} e^{8} + 114 \, a^{2} c d^{2} x^{3} e^{7} + 126 \, a^{2} c d^{3} x^{2} e^{6} + 54 \, a^{2} c d^{4} x e^{5} + 6 \, a^{2} c d^{5} e^{4} + 10 \, a^{3} x^{3} e^{9} + 30 \, a^{3} d x^{2} e^{8} + 30 \, a^{3} d^{2} x e^{7} + 10 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{9}} \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)^10,x, algorithm="giac")

[Out]

-1/60*(20*c^3*d^3*x^6*e^6 + 75*c^3*d^4*x^5*e^5 + 111*c^3*d^5*x^4*e^4 + 84*c^3*d^6*x^3*e^3 + 36*c^3*d^7*x^2*e^2
+ 9*c^3*d^8*x*e + c^3*d^9 + 45*a*c^2*d^2*x^5*e^7 + 153*a*c^2*d^3*x^4*e^6 + 192*a*c^2*d^4*x^3*e^5 + 108*a*c^2*
d^5*x^2*e^4 + 27*a*c^2*d^6*x*e^3 + 3*a*c^2*d^7*e^2 + 36*a^2*c*d*x^4*e^8 + 114*a^2*c*d^2*x^3*e^7 + 126*a^2*c*d^
3*x^2*e^6 + 54*a^2*c*d^4*x*e^5 + 6*a^2*c*d^5*e^4 + 10*a^3*x^3*e^9 + 30*a^3*d*x^2*e^8 + 30*a^3*d^2*x*e^7 + 10*a
^3*d^3*e^6)*e^(-4)/(x*e + d)^9