3.1930 \(\int \frac{(a+b x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0440028, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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),x)

[Out]

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

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Mathematica [A]  time = 0.0122822, size = 25, normalized size = 1. \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 1.3 \[{\frac{ex}{b}}-{\frac{\ln \left ( bx+a \right ) ae}{{b}^{2}}}+{\frac{d\ln \left ( bx+a \right ) }{b}} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.713989, size = 34, normalized size = 1.36 \[ \frac{e x}{b} + \frac{{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \]

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),x, algorithm="maxima")

[Out]

e*x/b + (b*d - a*e)*log(b*x + a)/b^2

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Fricas [A]  time = 0.273331, size = 32, normalized size = 1.28 \[ \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.27874, size = 20, normalized size = 0.8 \[ \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \]

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),x)

[Out]

e*x/b - (a*e - b*d)*log(a + b*x)/b**2

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GIAC/XCAS [A]  time = 0.278502, size = 38, normalized size = 1.52 \[ \frac{x e}{b} + \frac{{\left (b d - a e\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]

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),x, algorithm="giac")

[Out]

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