3.1519 \(\int \frac {(d+e x)^2}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 37} \[ -\frac {(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d + e*x)^3/(3*(b*d - a*e)*(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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.89 \[ -\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.59, size = 84, normalized size = 3.00 \[ -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b^2*e^2*x^2 + b^2*d^2 + a*b*d*e + a^2*e^2 + 3*(b^2*d*e + a*b*e^2)*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 [B]  time = 0.16, size = 58, normalized size = 2.07 \[ -\frac {3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 70, normalized size = 2.50 \[ -\frac {e^{2}}{\left (b x +a \right ) b^{3}}+\frac {\left (a e -b d \right ) e}{\left (b x +a \right )^{2} b^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 \left (b x +a \right )^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.41, size = 84, normalized size = 3.00 \[ -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b^2*e^2*x^2 + b^2*d^2 + a*b*d*e + a^2*e^2 + 3*(b^2*d*e + a*b*e^2)*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 = 0.04, size = 80, normalized size = 2.86 \[ -\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,b^3}+\frac {e^2\,x^2}{b}+\frac {e\,x\,\left (a\,e+b\,d\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.61, size = 88, normalized size = 3.14 \[ \frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**2*e**2 - a*b*d*e - b**2*d**2 - 3*b**2*e**2*x**2 + x*(-3*a*b*e**2 - 3*b**2*d*e))/(3*a**3*b**3 + 9*a**2*b**
4*x + 9*a*b**5*x**2 + 3*b**6*x**3)

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