3.18 \(\int \frac{1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac{d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac{f \log (e+f x)}{(b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.0733537, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {2058} \[ \frac{b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac{d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac{f \log (e+f x)}{(b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Int[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^(-1),x]

[Out]

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

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx &=\int \left (\frac{b^2}{(b c-a d) (b e-a f) (a+b x)}+\frac{d^2}{(b c-a d) (-d e+c f) (c+d x)}+\frac{f^2}{(b e-a f) (d e-c f) (e+f x)}\right ) \, dx\\ &=\frac{b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac{d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac{f \log (e+f x)}{(b e-a f) (d e-c f)}\\ \end{align*}

Mathematica [A]  time = 0.0476963, size = 80, normalized size = 0.93 \[ \frac{b \log (a+b x) (c f-d e)+d (b e-a f) \log (c+d x)+f (a d-b c) \log (e+f x)}{(b c-a d) (b e-a f) (c f-d e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^(-1),x]

[Out]

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

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Maple [A]  time = 0.008, size = 87, normalized size = 1. \begin{align*}{\frac{b\ln \left ( bx+a \right ) }{ \left ( af-be \right ) \left ( ad-bc \right ) }}-{\frac{d\ln \left ( dx+c \right ) }{ \left ( cf-de \right ) \left ( ad-bc \right ) }}+{\frac{f\ln \left ( fx+e \right ) }{ \left ( cf-de \right ) \left ( af-be \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3),x)

[Out]

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

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Maxima [A]  time = 1.04476, size = 151, normalized size = 1.76 \begin{align*} \frac{b \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f} - \frac{d \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} e -{\left (b c^{2} - a c d\right )} f} + \frac{f \log \left (f x + e\right )}{b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 28.6856, size = 230, normalized size = 2.67 \begin{align*} \frac{{\left (b c - a d\right )} f \log \left (f x + e\right ) +{\left (b d e - b c f\right )} \log \left (b x + a\right ) -{\left (b d e - a d f\right )} \log \left (d x + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} e^{2} -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f +{\left (a b c^{2} - a^{2} c d\right )} f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3),x, algorithm="fricas")

[Out]

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

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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(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b d f x^{3} + a c e +{\left (b d e + b c f + a d f\right )} x^{2} +{\left (b c e + a d e + a c f\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3),x, algorithm="giac")

[Out]

integrate(1/(b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d*e + a*c*f)*x), x)