3.1946 \(\int \frac {(a+b x) (d+e x)^3}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]

[Out]

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

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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 = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 37} \[ -\frac {(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d + e*x)^4/(4*(b*d - a*e)*(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]

Rubi steps

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

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Mathematica [B]  time = 0.03, size = 91, normalized size = 3.25 \[ -\frac {a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{4 b^4 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.52, size = 143, normalized size = 5.11 \[ -\frac {4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 106, normalized size = 3.79 \[ -\frac {4 \, b^{3} x^{3} e^{3} + 6 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e + b^{3} d^{3} + 6 \, a b^{2} x^{2} e^{3} + 4 \, a b^{2} d x e^{2} + a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + a^{2} b d e^{2} + a^{3} e^{3}}{4 \, {\left (b x + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 122, normalized size = 4.36 \[ -\frac {e^{3}}{\left (b x +a \right ) b^{4}}+\frac {3 \left (a e -b d \right ) e^{2}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) e}{\left (b x +a \right )^{3} b^{4}}-\frac {-e^{3} a^{3}+3 a^{2} b d \,e^{2}-3 a \,d^{2} e \,b^{2}+d^{3} b^{3}}{4 \left (b x +a \right )^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.52, size = 143, normalized size = 5.11 \[ -\frac {4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 135, normalized size = 4.82 \[ -\frac {\frac {a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3}{4\,b^4}+\frac {e^3\,x^3}{b}+\frac {e\,x\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{b^3}+\frac {3\,e^2\,x^2\,\left (a\,e+b\,d\right )}{2\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2)/(4*b^4) + (e^3*x^3)/b + (e*x*(a^2*e^2 + b^2*d^2 + a*b*d*e))/
b^3 + (3*e^2*x^2*(a*e + b*d))/(2*b^2))/(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 = 1.83, size = 155, normalized size = 5.54 \[ \frac {- a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e - b^{3} d^{3} - 4 b^{3} e^{3} x^{3} + x^{2} \left (- 6 a b^{2} e^{3} - 6 b^{3} d e^{2}\right ) + x \left (- 4 a^{2} b e^{3} - 4 a b^{2} d e^{2} - 4 b^{3} d^{2} e\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**3*e**3 - a**2*b*d*e**2 - a*b**2*d**2*e - b**3*d**3 - 4*b**3*e**3*x**3 + x**2*(-6*a*b**2*e**3 - 6*b**3*d*e
**2) + x*(-4*a**2*b*e**3 - 4*a*b**2*d*e**2 - 4*b**3*d**2*e))/(4*a**4*b**4 + 16*a**3*b**5*x + 24*a**2*b**6*x**2
+ 16*a*b**7*x**3 + 4*b**8*x**4)

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