### 3.1462 $$\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^7} \, dx$$

Optimal. Leaf size=65 $\frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4}$

[Out]

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

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Rubi [A]  time = 0.0365427, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {27, 43} $\frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 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{a^2+2 a b x+b^2 x^2}{(d+e x)^7} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^7} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^7}-\frac{2 b (b d-a e)}{e^2 (d+e x)^6}+\frac{b^2}{e^2 (d+e x)^5}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}+\frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{b^2}{4 e^3 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0211527, size = 55, normalized size = 0.85 $-\frac{10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]

[Out]

-(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^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 = 71, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{ \left ( 2\,ae-2\,bd \right ) b}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^7,x)

[Out]

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

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Maxima [B]  time = 1.20014, size = 162, normalized size = 2.49 \begin{align*} -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\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((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^2)*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 [B]  time = 1.77717, size = 251, normalized size = 3.86 \begin{align*} -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\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((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^2)*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 = 1.57847, size = 128, normalized size = 1.97 \begin{align*} - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d 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((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**7,x)

[Out]

-(10*a**2*e**2 + 4*a*b*d*e + b**2*d**2 + 15*b**2*e**2*x**2 + x*(24*a*b*e**2 + 6*b**2*d*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 = 1.1518, size = 81, normalized size = 1.25 \begin{align*} -\frac{{\left (15 \, b^{2} x^{2} e^{2} + 6 \, b^{2} d x e + b^{2} d^{2} + 24 \, a b x e^{2} + 4 \, a b d e + 10 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(15*b^2*x^2*e^2 + 6*b^2*d*x*e + b^2*d^2 + 24*a*b*x*e^2 + 4*a*b*d*e + 10*a^2*e^2)*e^(-3)/(x*e + d)^6