3.1933 \(\int \frac{a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{(d+e x) (b d-a e)}+\frac{b \log (a+b x)}{(b d-a e)^2}-\frac{b \log (d+e x)}{(b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.0741045, antiderivative size = 56, 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 \[ \frac{1}{(d+e x) (b d-a e)}+\frac{b \log (a+b x)}{(b d-a e)^2}-\frac{b \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.8979, size = 46, normalized size = 0.82 \[ \frac{b \log{\left (a + b x \right )}}{\left (a e - b d\right )^{2}} - \frac{b \log{\left (d + e x \right )}}{\left (a e - b d\right )^{2}} - \frac{1}{\left (d + e x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0447896, size = 53, normalized size = 0.95 \[ \frac{b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d}{(d+e x) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 58, normalized size = 1. \[{\frac{b\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{2}}}-{\frac{1}{ \left ( ae-bd \right ) \left ( ex+d \right ) }}-{\frac{b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.708437, size = 122, normalized size = 2.18 \[ \frac{b \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{b \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{1}{b d^{2} - a d e +{\left (b d e - a e^{2}\right )} x} \]

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)*(e*x + d)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.290692, size = 124, normalized size = 2.21 \[ \frac{b d - a e +{\left (b e x + b d\right )} \log \left (b x + a\right ) -{\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \]

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)*(e*x + d)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.17403, size = 233, normalized size = 4.16 \[ - \frac{b \log{\left (x + \frac{- \frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac{1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280992, size = 111, normalized size = 1.98 \[ \frac{b e{\rm ln}\left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac{e}{{\left (b d e - a e^{2}\right )}{\left (x e + d\right )}} \]

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)*(e*x + d)^2),x, algorithm="giac")

[Out]

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