### 3.264 $$\int \frac{1}{(d+e x) (b x+c x^2)} \, dx$$

Optimal. Leaf size=53 $-\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}+\frac{\log (x)}{b d}$

[Out]

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

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Rubi [A]  time = 0.0462377, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {698} $-\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}+\frac{\log (x)}{b d}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)),x]

[Out]

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

Rule 698

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

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) \left (b x+c x^2\right )} \, dx &=\int \left (\frac{1}{b d x}+\frac{c^2}{b (-c d+b e) (b+c x)}+\frac{e^2}{d (c d-b e) (d+e x)}\right ) \, dx\\ &=\frac{\log (x)}{b d}-\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}\\ \end{align*}

Mathematica [A]  time = 0.0233908, size = 48, normalized size = 0.91 $\frac{-c d \log (b+c x)+b e \log (d+e x)-b e \log (x)+c d \log (x)}{b c d^2-b^2 d e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.052, size = 54, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{bd}}+{\frac{c\ln \left ( cx+b \right ) }{b \left ( be-cd \right ) }}-{\frac{e\ln \left ( ex+d \right ) }{d \left ( be-cd \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.10489, size = 72, normalized size = 1.36 \begin{align*} -\frac{c \log \left (c x + b\right )}{b c d - b^{2} e} + \frac{e \log \left (e x + d\right )}{c d^{2} - b d e} + \frac{\log \left (x\right )}{b d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.23065, size = 109, normalized size = 2.06 \begin{align*} -\frac{c d \log \left (c x + b\right ) - b e \log \left (e x + d\right ) -{\left (c d - 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(1/(e*x+d)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 78.8291, size = 583, normalized size = 11. \begin{align*} - \frac{e \log{\left (x + \frac{- \frac{2 b^{6} e^{6}}{\left (b e - c d\right )^{2}} + \frac{6 b^{5} c d e^{5}}{\left (b e - c d\right )^{2}} - \frac{8 b^{4} c^{2} d^{2} e^{4}}{\left (b e - c d\right )^{2}} + \frac{3 b^{4} c d e^{4}}{b e - c d} + 2 b^{4} e^{4} + \frac{6 b^{3} c^{3} d^{3} e^{3}}{\left (b e - c d\right )^{2}} - \frac{6 b^{3} c^{2} d^{2} e^{3}}{b e - c d} - 3 b^{3} c d e^{3} - \frac{2 b^{2} c^{4} d^{4} e^{2}}{\left (b e - c d\right )^{2}} + \frac{3 b^{2} c^{3} d^{3} e^{2}}{b e - c d} + 2 b^{2} c^{2} d^{2} e^{2} - 3 b c^{3} d^{3} e + 2 c^{4} d^{4}}{2 b^{3} c e^{4} - 3 b^{2} c^{2} d e^{3} - 3 b c^{3} d^{2} e^{2} + 2 c^{4} d^{3} e} \right )}}{d \left (b e - c d\right )} + \frac{c \log{\left (x + \frac{- \frac{2 b^{4} c^{2} d^{2} e^{4}}{\left (b e - c d\right )^{2}} + 2 b^{4} e^{4} + \frac{6 b^{3} c^{3} d^{3} e^{3}}{\left (b e - c d\right )^{2}} - \frac{3 b^{3} c^{2} d^{2} e^{3}}{b e - c d} - 3 b^{3} c d e^{3} - \frac{8 b^{2} c^{4} d^{4} e^{2}}{\left (b e - c d\right )^{2}} + \frac{6 b^{2} c^{3} d^{3} e^{2}}{b e - c d} + 2 b^{2} c^{2} d^{2} e^{2} + \frac{6 b c^{5} d^{5} e}{\left (b e - c d\right )^{2}} - \frac{3 b c^{4} d^{4} e}{b e - c d} - 3 b c^{3} d^{3} e - \frac{2 c^{6} d^{6}}{\left (b e - c d\right )^{2}} + 2 c^{4} d^{4}}{2 b^{3} c e^{4} - 3 b^{2} c^{2} d e^{3} - 3 b c^{3} d^{2} e^{2} + 2 c^{4} d^{3} e} \right )}}{b \left (b e - c d\right )} + \frac{\log{\left (x \right )}}{b d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x**2+b*x),x)

[Out]

-e*log(x + (-2*b**6*e**6/(b*e - c*d)**2 + 6*b**5*c*d*e**5/(b*e - c*d)**2 - 8*b**4*c**2*d**2*e**4/(b*e - c*d)**
2 + 3*b**4*c*d*e**4/(b*e - c*d) + 2*b**4*e**4 + 6*b**3*c**3*d**3*e**3/(b*e - c*d)**2 - 6*b**3*c**2*d**2*e**3/(
b*e - c*d) - 3*b**3*c*d*e**3 - 2*b**2*c**4*d**4*e**2/(b*e - c*d)**2 + 3*b**2*c**3*d**3*e**2/(b*e - c*d) + 2*b*
*2*c**2*d**2*e**2 - 3*b*c**3*d**3*e + 2*c**4*d**4)/(2*b**3*c*e**4 - 3*b**2*c**2*d*e**3 - 3*b*c**3*d**2*e**2 +
2*c**4*d**3*e))/(d*(b*e - c*d)) + c*log(x + (-2*b**4*c**2*d**2*e**4/(b*e - c*d)**2 + 2*b**4*e**4 + 6*b**3*c**3
*d**3*e**3/(b*e - c*d)**2 - 3*b**3*c**2*d**2*e**3/(b*e - c*d) - 3*b**3*c*d*e**3 - 8*b**2*c**4*d**4*e**2/(b*e -
c*d)**2 + 6*b**2*c**3*d**3*e**2/(b*e - c*d) + 2*b**2*c**2*d**2*e**2 + 6*b*c**5*d**5*e/(b*e - c*d)**2 - 3*b*c*
*4*d**4*e/(b*e - c*d) - 3*b*c**3*d**3*e - 2*c**6*d**6/(b*e - c*d)**2 + 2*c**4*d**4)/(2*b**3*c*e**4 - 3*b**2*c*
*2*d*e**3 - 3*b*c**3*d**2*e**2 + 2*c**4*d**3*e))/(b*(b*e - c*d)) + log(x)/(b*d)

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Giac [B]  time = 1.36874, size = 169, normalized size = 3.19 \begin{align*} -\frac{{\left (c d + 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 |}} - \frac{\log \left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{\log \left ({\left | x \right |}\right )}{b d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x),x, algorithm="giac")

[Out]

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