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

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

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Rubi [A]  time = 0.02, antiderivative size = 38, 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, 43} \begin {gather*} -\frac {b d-a e}{4 b^2 (a+b x)^4}-\frac {e}{3 b^2 (a+b x)^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a e+3 b d+4 b e x}{12 b^2 (a+b x)^4} \end {gather*}

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]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 61, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.16, size = 27, normalized size = 0.71 \begin {gather*} -\frac {4 \, b x e + 3 \, b d + a e}{12 \, {\left (b x + a\right )}^{4} b^{2}} \end {gather*}

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)

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maple [A]  time = 0.09, size = 35, normalized size = 0.92 \begin {gather*} -\frac {e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {-a e +b d}{4 \left (b x +a \right )^{4} b^{2}} \end {gather*}

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 = 0.53, size = 61, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 2.01, size = 63, normalized size = 1.66 \begin {gather*} -\frac {\frac {a\,e+3\,b\,d}{12\,b^2}+\frac {e\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}

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)^3,x)

[Out]

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

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sympy [B]  time = 0.46, size = 65, normalized size = 1.71 \begin {gather*} \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 {gather*}

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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