3.1948 \(\int \frac{(a+b x) (d+e x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0241647, antiderivative size = 38, 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, 43} \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0089424, size = 27, normalized size = 0.71 \[ -\frac{a e+3 b d+4 b e x}{12 b^2 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 35, normalized size = 0.9 \begin{align*} -{\frac{e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{-ae+bd}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \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)^3,x)

[Out]

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

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Maxima [A]  time = 0.963735, size = 82, normalized size = 2.16 \begin{align*} -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53512, size = 128, normalized size = 3.37 \begin{align*} -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.59745, size = 65, normalized size = 1.71 \begin{align*} - \frac{a e + 3 b d + 4 b e x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \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)**3,x)

[Out]

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

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Giac [A]  time = 1.13264, size = 36, normalized size = 0.95 \begin{align*} -\frac{4 \, b x e + 3 \, b d + a e}{12 \,{\left (b x + a\right )}^{4} b^{2}} \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)^3,x, algorithm="giac")

[Out]

-1/12*(4*b*x*e + 3*b*d + a*e)/((b*x + a)^4*b^2)