3.1496 \(\int \frac{(d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e^2 x (3 b d-2 a e)}{b^3}+\frac{e^3 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.149105, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e^2 x (3 b d-2 a e)}{b^3}+\frac{e^3 x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*Integral(x, x)/b**2 + 4*e**2*(2*a*e - 3*b*d)*Integral(-1/4, x)/b**3 + 3*e*(
a*e - b*d)**2*log(a + b*x)/b**4 + (a*e - b*d)**3/(b**4*(a + b*x))

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Mathematica [A]  time = 0.102404, size = 72, normalized size = 0.96 \[ \frac{2 b e^2 x (3 b d-2 a e)-\frac{2 (b d-a e)^3}{a+b x}+6 e (b d-a e)^2 \log (a+b x)+b^2 e^3 x^2}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 149, normalized size = 2. \[{\frac{{e}^{3}{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{{e}^{3}xa}{{b}^{3}}}+3\,{\frac{{e}^{2}xd}{{b}^{2}}}+3\,{\frac{{e}^{3}\ln \left ( bx+a \right ){a}^{2}}{{b}^{4}}}-6\,{\frac{{e}^{2}\ln \left ( bx+a \right ) ad}{{b}^{3}}}+3\,{\frac{e\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}+{\frac{{a}^{3}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{a{d}^{2}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{3}}{b \left ( bx+a \right ) }} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.682934, size = 159, normalized size = 2.12 \[ -\frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{b^{5} x + a b^{4}} + \frac{b e^{3} x^{2} + 2 \,{\left (3 \, b d e^{2} - 2 \, a e^{3}\right )} x}{2 \, b^{3}} + \frac{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215903, size = 234, normalized size = 3.12 \[ \frac{b^{3} e^{3} x^{3} - 2 \, b^{3} d^{3} + 6 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} + 3 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.73626, size = 100, normalized size = 1.33 \[ \frac{a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{a b^{4} + b^{5} x} + \frac{e^{3} x^{2}}{2 b^{2}} - \frac{x \left (2 a e^{3} - 3 b d e^{2}\right )}{b^{3}} + \frac{3 e \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{4}} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212218, size = 154, normalized size = 2.05 \[ \frac{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{b^{2} x^{2} e^{3} + 6 \, b^{2} d x e^{2} - 4 \, a b x e^{3}}{2 \, b^{4}} - \frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{{\left (b x + a\right )} b^{4}} \]

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),x, algorithm="giac")

[Out]

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