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]

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

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Rubi [A]  time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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]

-1/6*(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)))/(b^3*(a + b*x)^3)

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fricas [A]  time = 1.01, size = 97, normalized size = 1.33 \[ -\frac {6 \, B b^{2} e x^{2} + {\left (B a b + 2 \, A b^{2}\right )} d + {\left (2 \, B a^{2} + A a b\right )} e + 3 \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

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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giac [A]  time = 0.16, size = 75, normalized size = 1.03 \[ -\frac {6 \, B b^{2} x^{2} e + 3 \, B b^{2} d x + 6 \, B a b x e + 3 \, A b^{2} x e + B a b d + 2 \, A b^{2} d + 2 \, B a^{2} e + A a b e}{6 \, {\left (b x + a\right )}^{3} b^{3}} \]

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)

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maple [A]  time = 0.05, size = 79, normalized size = 1.08 \[ -\frac {B e}{\left (b x +a \right ) b^{3}}-\frac {-a A e b +A d \,b^{2}+B e \,a^{2}-a B d b}{3 \left (b x +a \right )^{3} b^{3}}-\frac {A b e -2 a B e +B b d}{2 \left (b x +a \right )^{2} b^{3}} \]

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]

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

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maxima [A]  time = 0.56, size = 97, normalized size = 1.33 \[ -\frac {6 \, B b^{2} e x^{2} + {\left (B a b + 2 \, A b^{2}\right )} d + {\left (2 \, B a^{2} + A a b\right )} e + 3 \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

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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mupad [B]  time = 0.05, size = 91, normalized size = 1.25 \[ -\frac {\frac {2\,A\,b^2\,d+2\,B\,a^2\,e+A\,a\,b\,e+B\,a\,b\,d}{6\,b^3}+\frac {x\,\left (A\,b\,e+2\,B\,a\,e+B\,b\,d\right )}{2\,b^2}+\frac {B\,e\,x^2}{b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.49, size = 107, normalized size = 1.47 \[ \frac {- A a b e - 2 A b^{2} d - 2 B a^{2} e - B a b d - 6 B b^{2} e x^{2} + x \left (- 3 A b^{2} e - 6 B a b e - 3 B b^{2} d\right )}{6 a^{3} b^{3} + 18 a^{2} b^{4} x + 18 a b^{5} x^{2} + 6 b^{6} x^{3}} \]

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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