3.1704 \(\int \frac{(A+B x) (d+e x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{-2 a B e+A b e+b B d}{2 b^3 (a+b x)^2}-\frac{(A b-a B) (b d-a e)}{3 b^3 (a+b x)^3}-\frac{B e}{b^3 (a+b x)} \]

[Out]

-((A*b - a*B)*(b*d - a*e))/(3*b^3*(a + b*x)^3) - (b*B*d + A*b*e - 2*a*B*e)/(2*b^3*(a + b*x)^2) - (B*e)/(b^3*(a
+ b*x))

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Rubi [A]  time = 0.0588126, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ -\frac{-2 a B e+A b e+b B d}{2 b^3 (a+b x)^2}-\frac{(A b-a B) (b d-a e)}{3 b^3 (a+b x)^3}-\frac{B e}{b^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b - a*B)*(b*d - a*e))/(3*b^3*(a + b*x)^3) - (b*B*d + A*b*e - 2*a*B*e)/(2*b^3*(a + b*x)^2) - (B*e)/(b^3*(a
+ b*x))

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(A+B x) (d+e x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac{(A b-a B) (b d-a e)}{b^2 (a+b x)^4}+\frac{b B d+A b e-2 a B e}{b^2 (a+b x)^3}+\frac{B e}{b^2 (a+b x)^2}\right ) \, dx\\ &=-\frac{(A b-a B) (b d-a e)}{3 b^3 (a+b x)^3}-\frac{b B d+A b e-2 a B e}{2 b^3 (a+b x)^2}-\frac{B e}{b^3 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0334332, size = 61, normalized size = 0.84 \[ -\frac{B \left (2 a^2 e+a b (d+6 e x)+3 b^2 x (d+2 e x)\right )+A b (a e+2 b d+3 b e x)}{6 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*b*(2*b*d + a*e + 3*b*e*x) + B*(2*a^2*e + 3*b^2*x*(d + 2*e*x) + a*b*(d + 6*e*x)))/(6*b^3*(a + b*x)^3)

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Maple [A]  time = 0.004, size = 79, normalized size = 1.1 \begin{align*} -{\frac{Be}{{b}^{3} \left ( bx+a \right ) }}-{\frac{Abe-2\,aBe+Bbd}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{-aAeb+Ad{b}^{2}+Be{a}^{2}-Bdab}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*e/b^3/(b*x+a)-1/2*(A*b*e-2*B*a*e+B*b*d)/b^3/(b*x+a)^2-1/3*(-A*a*b*e+A*b^2*d+B*a^2*e-B*a*b*d)/b^3/(b*x+a)^3

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Maxima [A]  time = 1.01855, size = 131, normalized size = 1.79 \begin{align*} -\frac{6 \, B b^{2} e x^{2} +{\left (B a b + 2 \, A b^{2}\right )} d +{\left (2 \, B a^{2} + A a b\right )} e + 3 \,{\left (B b^{2} d +{\left (2 \, B a b + A b^{2}\right )} e\right )} x}{6 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53485, size = 205, normalized size = 2.81 \begin{align*} -\frac{6 \, B b^{2} e x^{2} +{\left (B a b + 2 \, A b^{2}\right )} d +{\left (2 \, B a^{2} + A a b\right )} e + 3 \,{\left (B b^{2} d +{\left (2 \, B a b + A b^{2}\right )} e\right )} x}{6 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.84714, size = 107, normalized size = 1.47 \begin{align*} - \frac{A a b e + 2 A b^{2} d + 2 B a^{2} e + B a b d + 6 B b^{2} e x^{2} + x \left (3 A b^{2} e + 6 B a b e + 3 B b^{2} d\right )}{6 a^{3} b^{3} + 18 a^{2} b^{4} x + 18 a b^{5} x^{2} + 6 b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-(A*a*b*e + 2*A*b**2*d + 2*B*a**2*e + B*a*b*d + 6*B*b**2*e*x**2 + x*(3*A*b**2*e + 6*B*a*b*e + 3*B*b**2*d))/(6*
a**3*b**3 + 18*a**2*b**4*x + 18*a*b**5*x**2 + 6*b**6*x**3)

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Giac [A]  time = 1.13196, size = 101, normalized size = 1.38 \begin{align*} -\frac{6 \, B b^{2} x^{2} e + 3 \, B b^{2} d x + 6 \, B a b x e + 3 \, A b^{2} x e + B a b d + 2 \, A b^{2} d + 2 \, B a^{2} e + A a b e}{6 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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