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

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 = 0.0727457, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]

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))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )} \, dx &=\int \left (\frac{A}{b d x}-\frac{c (b B-A c)}{b (-c d+b e) (b+c x)}-\frac{e (B d-A e)}{d (c d-b e) (d+e x)}\right ) \, dx\\ &=\frac{A \log (x)}{b d}+\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)}\\ \end{align*}

Mathematica [A]  time = 0.0445822, 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 = 0.007, size = 94, normalized size = 1.4 \begin{align*}{\frac{A\ln \left ( x \right ) }{bd}}-{\frac{\ln \left ( ex+d \right ) Ae}{d \left ( be-cd \right ) }}+{\frac{\ln \left ( ex+d \right ) B}{be-cd}}+{\frac{\ln \left ( cx+b \right ) Ac}{b \left ( be-cd \right ) }}-{\frac{\ln \left ( cx+b \right ) B}{be-cd}} \end{align*}

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

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Maxima [A]  time = 1.14539, size = 92, normalized size = 1.35 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x),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 = 8.18057, size = 143, normalized size = 2.1 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x),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. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.33476, size = 184, normalized size = 2.71 \begin{align*} -\frac{A \log \left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{A \log \left ({\left | x \right |}\right )}{b d} + \frac{{\left (2 \, B b d - A c d - A b e\right )} \log \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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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