3.1142 \(\int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=68 \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.170699, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.2591, size = 51, normalized size = 0.75 \[ \frac{A \log{\left (x \right )}}{b d} - \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{d \left (b e - c d\right )} + \frac{\left (A c - B b\right ) \log{\left (b + c x \right )}}{b \left (b e - c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0694552, size = 63, normalized size = 0.93 \[ \frac{\log (b+c x) (A c d-b B d)+b (B d-A e) \log (d+e x)+A \log (x) (b e-c d)}{b d (b e-c d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 94, normalized size = 1.4 \[{\frac{A\ln \left ( x \right ) }{bd}}+{\frac{\ln \left ( cx+b \right ) Ac}{ \left ( be-cd \right ) b}}-{\frac{\ln \left ( cx+b \right ) B}{be-cd}}-{\frac{\ln \left ( ex+d \right ) Ae}{d \left ( be-cd \right ) }}+{\frac{\ln \left ( ex+d \right ) B}{be-cd}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.722578, size = 92, normalized size = 1.35 \[ \frac{{\left (B b - A c\right )} \log \left (c x + b\right )}{b c d - b^{2} e} - \frac{{\left (B d - A e\right )} \log \left (e x + d\right )}{c d^{2} - b d e} + \frac{A \log \left (x\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.49535, size = 88, normalized size = 1.29 \[ \frac{{\left (B b - A c\right )} d \log \left (c x + b\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right ) +{\left (A c d - A b e\right )} \log \left (x\right )}{b c d^{2} - b^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295542, size = 184, normalized size = 2.71 \[ -\frac{A{\rm ln}\left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b d} + \frac{{\left (2 \, B b d - A c d - A b e\right )}{\rm ln}\left (\frac{{\left | 2 \, c x e + c d + b e -{\left | c d - b e \right |} \right |}}{{\left | 2 \, c x e + c d + b e +{\left | c d - b e \right |} \right |}}\right )}{2 \, b d{\left | c d - b e \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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