3.1696 \(\int \frac{(A+B x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=60 \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.125692, antiderivative size = 60, 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 \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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),x)

[Out]

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

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Mathematica [A]  time = 0.076946, size = 56, normalized size = 0.93 \[ \frac{-\frac{(A b-a B) (b d-a e)}{a+b x}+\log (a+b x) (-2 a B e+A b e+b B d)+b B e x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 106, normalized size = 1.8 \[{\frac{Bex}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) Ae}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) aBe}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) Bd}{{b}^{2}}}+{\frac{aAe}{{b}^{2} \left ( bx+a \right ) }}-{\frac{Ad}{b \left ( bx+a \right ) }}-{\frac{Be{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Bad}{{b}^{2} \left ( bx+a \right ) }} \]

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]

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

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Maxima [A]  time = 0.688286, size = 104, normalized size = 1.73 \[ \frac{B e x}{b^{2}} + \frac{{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e}{b^{4} x + a b^{3}} + \frac{{\left (B b d -{\left (2 \, B a - A b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]

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]

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

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Fricas [A]  time = 0.26794, size = 147, normalized size = 2.45 \[ \frac{B b^{2} e x^{2} + B a b e x +{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e +{\left (B a b d -{\left (2 \, B a^{2} - A a b\right )} e +{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]

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

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Sympy [A]  time = 2.67038, size = 71, normalized size = 1.18 \[ \frac{B e x}{b^{2}} - \frac{- A a b e + A b^{2} d + B a^{2} e - B a b d}{a b^{3} + b^{4} x} - \frac{\left (- A b e + 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{3}} \]

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]

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

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GIAC/XCAS [A]  time = 0.344184, size = 100, normalized size = 1.67 \[ \frac{B x e}{b^{2}} + \frac{{\left (B b d - 2 \, B a e + A b e\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{B a b d - A b^{2} d - B a^{2} e + A a b e}{{\left (b x + a\right )} b^{3}} \]

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]

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