3.1849 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.04866, 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} \[ \frac{c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac{\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}-\frac{c^2 d^2}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0297077, size = 61, normalized size = 0.79 \[ -\frac{6 a^2 e^4+3 a c d e^2 (d+5 e x)+c^2 d^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*a^2*e^4 + 3*a*c*d*e^2*(d + 5*e*x) + c^2*d^2*(d^2 + 5*d*e*x + 10*e^2*x^2))/(30*e^3*(d + e*x)^5)

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Maple [A]  time = 0.044, size = 83, normalized size = 1.1 \begin{align*} -{\frac{cd \left ( a{e}^{2}-c{d}^{2} \right ) }{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{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)^8,x)

[Out]

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

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Maxima [A]  time = 1.058, size = 161, normalized size = 2.09 \begin{align*} -\frac{10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \,{\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} 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)^8,x, algorithm="maxima")

[Out]

-1/30*(10*c^2*d^2*e^2*x^2 + c^2*d^4 + 3*a*c*d^2*e^2 + 6*a^2*e^4 + 5*(c^2*d^3*e + 3*a*c*d*e^3)*x)/(e^8*x^5 + 5*
d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Fricas [A]  time = 1.6301, size = 243, normalized size = 3.16 \begin{align*} -\frac{10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \,{\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} 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)^8,x, algorithm="fricas")

[Out]

-1/30*(10*c^2*d^2*e^2*x^2 + c^2*d^4 + 3*a*c*d^2*e^2 + 6*a^2*e^4 + 5*(c^2*d^3*e + 3*a*c*d*e^3)*x)/(e^8*x^5 + 5*
d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Sympy [A]  time = 2.08551, size = 126, normalized size = 1.64 \begin{align*} - \frac{6 a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4} + 10 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} + 5 c^{2} d^{3} e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} 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)**2/(e*x+d)**8,x)

[Out]

-(6*a**2*e**4 + 3*a*c*d**2*e**2 + c**2*d**4 + 10*c**2*d**2*e**2*x**2 + x*(15*a*c*d*e**3 + 5*c**2*d**3*e))/(30*
d**5*e**3 + 150*d**4*e**4*x + 300*d**3*e**5*x**2 + 300*d**2*e**6*x**3 + 150*d*e**7*x**4 + 30*e**8*x**5)

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Giac [A]  time = 1.22419, size = 189, normalized size = 2.45 \begin{align*} -\frac{{\left (10 \, c^{2} d^{2} x^{4} e^{4} + 25 \, c^{2} d^{3} x^{3} e^{3} + 21 \, c^{2} d^{4} x^{2} e^{2} + 7 \, c^{2} d^{5} x e + c^{2} d^{6} + 15 \, a c d x^{3} e^{5} + 33 \, a c d^{2} x^{2} e^{4} + 21 \, a c d^{3} x e^{3} + 3 \, a c d^{4} e^{2} + 6 \, a^{2} x^{2} e^{6} + 12 \, a^{2} d x e^{5} + 6 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{30 \,{\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)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/30*(10*c^2*d^2*x^4*e^4 + 25*c^2*d^3*x^3*e^3 + 21*c^2*d^4*x^2*e^2 + 7*c^2*d^5*x*e + c^2*d^6 + 15*a*c*d*x^3*e
^5 + 33*a*c*d^2*x^2*e^4 + 21*a*c*d^3*x*e^3 + 3*a*c*d^4*e^2 + 6*a^2*x^2*e^6 + 12*a^2*d*x*e^5 + 6*a^2*d^2*e^4)*e
^(-3)/(x*e + d)^7