### 3.1848 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^7} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0498161, 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} $\frac{2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac{\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac{c^2 d^2}{2 e^3 (d+e x)^2}$

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)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0227091, size = 61, normalized size = 0.79 $-\frac{3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4}$

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)^7,x]

[Out]

-(3*a^2*e^4 + 2*a*c*d*e^2*(d + 4*e*x) + c^2*d^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(12*e^3*(d + e*x)^4)

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Maple [A]  time = 0.043, size = 83, normalized size = 1.1 \begin{align*} -{\frac{{c}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{2\,cd \left ( a{e}^{2}-c{d}^{2} \right ) }{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \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)^7,x)

[Out]

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

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Maxima [A]  time = 1.04598, size = 146, normalized size = 1.9 \begin{align*} -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} 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)^7,x, algorithm="maxima")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d
*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Fricas [A]  time = 1.51108, size = 217, normalized size = 2.82 \begin{align*} -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} 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)^7,x, algorithm="fricas")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d
*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 1.47222, size = 114, normalized size = 1.48 \begin{align*} - \frac{3 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (8 a c d e^{3} + 4 c^{2} d^{3} e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \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)**7,x)

[Out]

-(3*a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*(8*a*c*d*e**3 + 4*c**2*d**3*e))/(12*d*
*4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12*e**7*x**4)

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Giac [A]  time = 1.23884, size = 189, normalized size = 2.45 \begin{align*} -\frac{{\left (6 \, c^{2} d^{2} x^{4} e^{4} + 16 \, c^{2} d^{3} x^{3} e^{3} + 15 \, c^{2} d^{4} x^{2} e^{2} + 6 \, c^{2} d^{5} x e + c^{2} d^{6} + 8 \, a c d x^{3} e^{5} + 18 \, a c d^{2} x^{2} e^{4} + 12 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 3 \, a^{2} x^{2} e^{6} + 6 \, a^{2} d x e^{5} + 3 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{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)^7,x, algorithm="giac")

[Out]

-1/12*(6*c^2*d^2*x^4*e^4 + 16*c^2*d^3*x^3*e^3 + 15*c^2*d^4*x^2*e^2 + 6*c^2*d^5*x*e + c^2*d^6 + 8*a*c*d*x^3*e^5
+ 18*a*c*d^2*x^2*e^4 + 12*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + 3*a^2*x^2*e^6 + 6*a^2*d*x*e^5 + 3*a^2*d^2*e^4)*e^(-
3)/(x*e + d)^6