3.1149 \(\int \frac{(A+B x) (d+e x)}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=86 \[ \frac{\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac{\log (b+c x) (A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)}{b^2 c (b+c x)}-\frac{A d}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.197853, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac{\log (b+c x) (A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)}{b^2 c (b+c x)}-\frac{A d}{b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.2793, size = 82, normalized size = 0.95 \[ - \frac{A d}{b^{2} x} + \frac{\left (A c - B b\right ) \left (b e - c d\right )}{b^{2} c \left (b + c x\right )} + \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (x \right )}}{b^{3}} - \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (b + c x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)/(c*x**2+b*x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.185735, size = 80, normalized size = 0.93 \[ -\frac{\frac{b (b B-A c) (b e-c d)}{c (b+c x)}-\log (x) (A b e-2 A c d+b B d)+\log (b+c x) (A b e-2 A c d+b B d)+\frac{A b d}{x}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 133, normalized size = 1.6 \[{\frac{\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{Ac\ln \left ( x \right ) d}{{b}^{3}}}+{\frac{\ln \left ( x \right ) Bd}{{b}^{2}}}-{\frac{Ad}{{b}^{2}x}}+{\frac{Ae}{b \left ( cx+b \right ) }}-{\frac{Acd}{{b}^{2} \left ( cx+b \right ) }}-{\frac{Be}{c \left ( cx+b \right ) }}+{\frac{Bd}{b \left ( cx+b \right ) }}-{\frac{\ln \left ( cx+b \right ) Ae}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Acd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) Bd}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.692529, size = 143, normalized size = 1.66 \[ -\frac{A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} - \frac{{\left (A b e +{\left (B b - 2 \, A c\right )} d\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (A b e +{\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276527, size = 248, normalized size = 2.88 \[ -\frac{A b^{2} c d -{\left ({\left (B b^{2} c - 2 \, A b c^{2}\right )} d -{\left (B b^{3} - A b^{2} c\right )} e\right )} x +{\left ({\left (A b c^{2} e +{\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} +{\left (A b^{2} c e +{\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \left (c x + b\right ) -{\left ({\left (A b c^{2} e +{\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} +{\left (A b^{2} c e +{\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \left (x\right )}{b^{3} c^{2} x^{2} + b^{4} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.00042, size = 233, normalized size = 2.71 \[ - \frac{A b c d + x \left (- A b c e + 2 A c^{2} d + B b^{2} e - B b c d\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{A b^{2} e - 2 A b c d + B b^{2} d - b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} - \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{A b^{2} e - 2 A b c d + B b^{2} d + b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)/(c*x**2+b*x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.282615, size = 151, normalized size = 1.76 \[ \frac{{\left (B b d - 2 \, A c d + A b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (B b c d - 2 \, A c^{2} d + A b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} + \frac{B b c d x - 2 \, A c^{2} d x - B b^{2} x e + A b c x e - A b c d}{{\left (c x^{2} + b x\right )} b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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