3.1937 \(\int \frac{(a+b x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac{3 e (b d-a e)^2}{b^4 (a+b x)}-\frac{(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac{e^3 x}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.12876, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac{3 e (b d-a e)^2}{b^4 (a+b x)}-\frac{(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac{e^3 x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{3} \int \frac{1}{b^{3}}\, dx - \frac{3 e^{2} \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{3 e \left (a e - b d\right )^{2}}{b^{4} \left (a + b x\right )} + \frac{\left (a e - b d\right )^{3}}{2 b^{4} \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

e**3*Integral(b**(-3), x) - 3*e**2*(a*e - b*d)*log(a + b*x)/b**4 - 3*e*(a*e - b*
d)**2/(b**4*(a + b*x)) + (a*e - b*d)**3/(2*b**4*(a + b*x)**2)

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Mathematica [A]  time = 0.081187, size = 114, normalized size = 1.46 \[ \frac{-5 a^3 e^3+a^2 b e^2 (9 d-4 e x)+a b^2 e \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 (a+b x)^2 (a e-b d) \log (a+b x)+b^3 \left (-\left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )}{2 b^4 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*a^3*e^3 + a^2*b*e^2*(9*d - 4*e*x) + a*b^2*e*(-3*d^2 + 12*d*e*x + 4*e^2*x^2)
- b^3*(d^3 + 6*d^2*e*x - 2*e^3*x^3) - 6*e^2*(-(b*d) + a*e)*(a + b*x)^2*Log[a + b
*x])/(2*b^4*(a + b*x)^2)

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Maple [B]  time = 0.012, size = 160, normalized size = 2.1 \[{\frac{{e}^{3}x}{{b}^{3}}}+{\frac{{a}^{3}{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}d{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,a{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{d}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{e}^{3}\ln \left ( bx+a \right ) a}{{b}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) d}{{b}^{3}}}-3\,{\frac{{a}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{ad{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{e{d}^{2}}{{b}^{2} \left ( bx+a \right ) }} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.717802, size = 169, normalized size = 2.17 \[ \frac{e^{3} x}{b^{3}} - \frac{b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{3 \,{\left (b d e^{2} - a e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]

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)^2,x, algorithm="maxima")

[Out]

e^3*x/b^3 - 1/2*(b^3*d^3 + 3*a*b^2*d^2*e - 9*a^2*b*d*e^2 + 5*a^3*e^3 + 6*(b^3*d^
2*e - 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^6*x^2 + 2*a*b^5*x + a^2*b^4) + 3*(b*d*e^2
- a*e^3)*log(b*x + a)/b^4

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Fricas [A]  time = 0.287986, size = 254, normalized size = 3.26 \[ \frac{2 \, b^{3} e^{3} x^{3} + 4 \, a b^{2} e^{3} x^{2} - b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - 2 \,{\left (3 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a^{2} b d e^{2} - a^{3} e^{3} +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/2*(2*b^3*e^3*x^3 + 4*a*b^2*e^3*x^2 - b^3*d^3 - 3*a*b^2*d^2*e + 9*a^2*b*d*e^2 -
5*a^3*e^3 - 2*(3*b^3*d^2*e - 6*a*b^2*d*e^2 + 2*a^2*b*e^3)*x + 6*(a^2*b*d*e^2 -
a^3*e^3 + (b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(a*b^2*d*e^2 - a^2*b*e^3)*x)*log(b*x +
a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)

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Sympy [A]  time = 4.23304, size = 128, normalized size = 1.64 \[ - \frac{5 a^{3} e^{3} - 9 a^{2} b d e^{2} + 3 a b^{2} d^{2} e + b^{3} d^{3} + x \left (6 a^{2} b e^{3} - 12 a b^{2} d e^{2} + 6 b^{3} d^{2} e\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac{e^{3} x}{b^{3}} - \frac{3 e^{2} \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{4}} \]

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.290574, size = 144, normalized size = 1.85 \[ \frac{x e^{3}}{b^{3}} + \frac{3 \,{\left (b d e^{2} - a e^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \]

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)^2,x, algorithm="giac")

[Out]

x*e^3/b^3 + 3*(b*d*e^2 - a*e^3)*ln(abs(b*x + a))/b^4 - 1/2*(b^3*d^3 + 3*a*b^2*d^
2*e - 9*a^2*b*d*e^2 + 5*a^3*e^3 + 6*(b^3*d^2*e - 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(
(b*x + a)^2*b^4)