3.1530 \(\int \frac {(d+e x)^3}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=58 \[ \frac {e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac {(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]

[Out]

-1/5*(e*x+d)^4/(-a*e+b*d)/(b*x+a)^5+1/20*e*(e*x+d)^4/(-a*e+b*d)^2/(b*x+a)^4

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac {e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac {(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d + e*x)^4/(5*(b*d - a*e)*(a + b*x)^5) + (e*(d + e*x)^4)/(20*(b*d - a*e)^2*(a + b*x)^4)

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]

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^3}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^4}{5 (b d-a e) (a+b x)^5}-\frac {e \int \frac {(d+e x)^3}{(a+b x)^5} \, dx}{5 (b d-a e)}\\ &=-\frac {(d+e x)^4}{5 (b d-a e) (a+b x)^5}+\frac {e (d+e x)^4}{20 (b d-a e)^2 (a+b x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 97, normalized size = 1.67 \[ -\frac {a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (3 d^2+10 d e x+10 e^2 x^2\right )+b^3 \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right )}{20 b^4 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(3*d^2 + 10*d*e*x + 10*e^2*x^2) + b^3*(4*d^3 + 15*d^2*e*x +
20*d*e^2*x^2 + 10*e^3*x^3))/(b^4*(a + b*x)^5)

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fricas [B]  time = 0.89, size = 160, normalized size = 2.76 \[ -\frac {10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \, {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*e^3*x^3 + 4*b^3*d^3 + 3*a*b^2*d^2*e + 2*a^2*b*d*e^2 + a^3*e^3 + 10*(2*b^3*d*e^2 + a*b^2*e^3)*x^2
+ 5*(3*b^3*d^2*e + 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

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giac [B]  time = 0.16, size = 109, normalized size = 1.88 \[ -\frac {10 \, b^{3} x^{3} e^{3} + 20 \, b^{3} d x^{2} e^{2} + 15 \, b^{3} d^{2} x e + 4 \, b^{3} d^{3} + 10 \, a b^{2} x^{2} e^{3} + 10 \, a b^{2} d x e^{2} + 3 \, a b^{2} d^{2} e + 5 \, a^{2} b x e^{3} + 2 \, a^{2} b d e^{2} + a^{3} e^{3}}{20 \, {\left (b x + a\right )}^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^3*e^3 + 20*b^3*d*x^2*e^2 + 15*b^3*d^2*x*e + 4*b^3*d^3 + 10*a*b^2*x^2*e^3 + 10*a*b^2*d*x*e^2 +
3*a*b^2*d^2*e + 5*a^2*b*x*e^3 + 2*a^2*b*d*e^2 + a^3*e^3)/((b*x + a)^5*b^4)

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maple [B]  time = 0.05, size = 121, normalized size = 2.09 \[ -\frac {e^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {\left (a e -b d \right ) e^{2}}{\left (b x +a \right )^{3} b^{4}}-\frac {3 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) e}{4 \left (b x +a \right )^{4} b^{4}}-\frac {-a^{3} e^{3}+3 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e +b^{3} d^{3}}{5 \left (b x +a \right )^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.42, size = 160, normalized size = 2.76 \[ -\frac {10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \, {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*e^3*x^3 + 4*b^3*d^3 + 3*a*b^2*d^2*e + 2*a^2*b*d*e^2 + a^3*e^3 + 10*(2*b^3*d*e^2 + a*b^2*e^3)*x^2
+ 5*(3*b^3*d^2*e + 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

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mupad [B]  time = 0.53, size = 154, normalized size = 2.66 \[ -\frac {\frac {a^3\,e^3+2\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+4\,b^3\,d^3}{20\,b^4}+\frac {e^3\,x^3}{2\,b}+\frac {e\,x\,\left (a^2\,e^2+2\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4\,b^3}+\frac {e^2\,x^2\,\left (a\,e+2\,b\,d\right )}{2\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^3*e^3 + 4*b^3*d^3 + 3*a*b^2*d^2*e + 2*a^2*b*d*e^2)/(20*b^4) + (e^3*x^3)/(2*b) + (e*x*(a^2*e^2 + 3*b^2*d^2
+ 2*a*b*d*e))/(4*b^3) + (e^2*x^2*(a*e + 2*b*d))/(2*b^2))/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a
^2*b^3*x^3 + 5*a^4*b*x)

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sympy [B]  time = 2.03, size = 172, normalized size = 2.97 \[ \frac {- a^{3} e^{3} - 2 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 4 b^{3} d^{3} - 10 b^{3} e^{3} x^{3} + x^{2} \left (- 10 a b^{2} e^{3} - 20 b^{3} d e^{2}\right ) + x \left (- 5 a^{2} b e^{3} - 10 a b^{2} d e^{2} - 15 b^{3} d^{2} e\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(-a**3*e**3 - 2*a**2*b*d*e**2 - 3*a*b**2*d**2*e - 4*b**3*d**3 - 10*b**3*e**3*x**3 + x**2*(-10*a*b**2*e**3 - 20
*b**3*d*e**2) + x*(-5*a**2*b*e**3 - 10*a*b**2*d*e**2 - 15*b**3*d**2*e))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*
a**3*b**6*x**2 + 200*a**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

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