3.1941 \(\int \frac{a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=82 \[ \frac{e^2 \log (a+b x)}{(b d-a e)^3}-\frac{e^2 \log (d+e x)}{(b d-a e)^3}+\frac{e}{(a+b x) (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.107804, antiderivative size = 82, 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 \[ \frac{e^2 \log (a+b x)}{(b d-a e)^3}-\frac{e^2 \log (d+e x)}{(b d-a e)^3}+\frac{e}{(a+b x) (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.2689, size = 68, normalized size = 0.83 \[ - \frac{e^{2} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{3}} + \frac{e^{2} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{3}} + \frac{e}{\left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115769, size = 67, normalized size = 0.82 \[ \frac{\frac{(b d-a e) (3 a e-b d+2 b e x)}{(a+b x)^2}+2 e^2 \log (a+b x)-2 e^2 \log (d+e x)}{2 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 81, normalized size = 1. \[{\frac{1}{ \left ( 2\,ae-2\,bd \right ) \left ( bx+a \right ) ^{2}}}+{\frac{e}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{3}}}+{\frac{{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.717402, size = 273, normalized size = 3.33 \[ \frac{e^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{e^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac{2 \, b e x - b d + 3 \, a e}{2 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.281986, size = 327, normalized size = 3.99 \[ -\frac{b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - a b e^{2}\right )} x - 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.84898, size = 381, normalized size = 4.65 \[ \frac{e^{2} \log{\left (x + \frac{- \frac{a^{4} e^{6}}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac{4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} - \frac{b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} - \frac{e^{2} \log{\left (x + \frac{\frac{a^{4} e^{6}}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac{4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} + \frac{b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} + \frac{3 a e - b d + 2 b e x}{2 a^{4} e^{2} - 4 a^{3} b d e + 2 a^{2} b^{2} d^{2} + x^{2} \left (2 a^{2} b^{2} e^{2} - 4 a b^{3} d e + 2 b^{4} d^{2}\right ) + x \left (4 a^{3} b e^{2} - 8 a^{2} b^{2} d e + 4 a b^{3} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28028, size = 219, normalized size = 2.67 \[ \frac{b e^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac{e^{3}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac{b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - a b e^{2}\right )} x}{2 \,{\left (b d - a e\right )}^{3}{\left (b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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