3.1850 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=77 \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0491455, antiderivative size = 77, 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} \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^9,x]

[Out]

-(c*d^2 - a*e^2)^2/(6*e^3*(d + e*x)^6) + (2*c*d*(c*d^2 - a*e^2))/(5*e^3*(d + e*x)^5) - (c^2*d^2)/(4*e^3*(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 )^2}{(d+e x)^9} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^7} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^7}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^6}+\frac{c^2 d^2}{e^2 (d+e x)^5}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}+\frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{c^2 d^2}{4 e^3 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0250267, size = 61, normalized size = 0.79 \[ -\frac{10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^9,x]

[Out]

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

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Maple [A]  time = 0.045, size = 83, normalized size = 1.1 \begin{align*} -{\frac{{c}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{2\,cd \left ( a{e}^{2}-c{d}^{2} \right ) }{5\,{e}^{3} \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)^2/(e*x+d)^9,x)

[Out]

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

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Maxima [A]  time = 1.05644, size = 176, normalized size = 2.29 \begin{align*} -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\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)^2/(e*x+d)^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58881, size = 267, normalized size = 3.47 \begin{align*} -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\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)^2/(e*x+d)^9,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.18706, size = 138, normalized size = 1.79 \begin{align*} - \frac{10 a^{2} e^{4} + 4 a c d^{2} e^{2} + c^{2} d^{4} + 15 c^{2} d^{2} e^{2} x^{2} + x \left (24 a c d e^{3} + 6 c^{2} d^{3} e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} 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)**2/(e*x+d)**9,x)

[Out]

-(10*a**2*e**4 + 4*a*c*d**2*e**2 + c**2*d**4 + 15*c**2*d**2*e**2*x**2 + x*(24*a*c*d*e**3 + 6*c**2*d**3*e))/(60
*d**6*e**3 + 360*d**5*e**4*x + 900*d**4*e**5*x**2 + 1200*d**3*e**6*x**3 + 900*d**2*e**7*x**4 + 360*d*e**8*x**5
+ 60*e**9*x**6)

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Giac [A]  time = 1.20518, size = 189, normalized size = 2.45 \begin{align*} -\frac{{\left (15 \, c^{2} d^{2} x^{4} e^{4} + 36 \, c^{2} d^{3} x^{3} e^{3} + 28 \, c^{2} d^{4} x^{2} e^{2} + 8 \, c^{2} d^{5} x e + c^{2} d^{6} + 24 \, a c d x^{3} e^{5} + 52 \, a c d^{2} x^{2} e^{4} + 32 \, a c d^{3} x e^{3} + 4 \, a c d^{4} e^{2} + 10 \, a^{2} x^{2} e^{6} + 20 \, a^{2} d x e^{5} + 10 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{60 \,{\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)^2/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/60*(15*c^2*d^2*x^4*e^4 + 36*c^2*d^3*x^3*e^3 + 28*c^2*d^4*x^2*e^2 + 8*c^2*d^5*x*e + c^2*d^6 + 24*a*c*d*x^3*e
^5 + 52*a*c*d^2*x^2*e^4 + 32*a*c*d^3*x*e^3 + 4*a*c*d^4*e^2 + 10*a^2*x^2*e^6 + 20*a^2*d*x*e^5 + 10*a^2*d^2*e^4)
*e^(-3)/(x*e + d)^8