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

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Rubi [A]  time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 10.24, size = 112, normalized size = 1.30 \[ \frac {{\left (b c - a d\right )} f \log \left (f x + e\right ) + {\left (b d e - b c f\right )} \log \left (b x + a\right ) - {\left (b d e - a d f\right )} \log \left (d x + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} e^{2} - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f + {\left (a b c^{2} - a^{2} c d\right )} f^{2}} \]

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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giac [A]  time = 0.31, size = 137, normalized size = 1.59 \[ -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a b^{2} c f - a^{2} b d f - b^{3} c e + a b^{2} d e} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d f - a c d^{2} f - b c d^{2} e + a d^{3} e} + \frac {f^{2} \log \left ({\left | f x + e \right |}\right )}{a c f^{3} - b c f^{2} e - a d f^{2} e + b d f e^{2}} \]

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]

-b^2*log(abs(b*x + a))/(a*b^2*c*f - a^2*b*d*f - b^3*c*e + a*b^2*d*e) + d^2*log(abs(d*x + c))/(b*c^2*d*f - a*c*
d^2*f - b*c*d^2*e + a*d^3*e) + f^2*log(abs(f*x + e))/(a*c*f^3 - b*c*f^2*e - a*d*f^2*e + b*d*f*e^2)

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maple [A]  time = 0.01, size = 87, normalized size = 1.01 \[ \frac {b \ln \left (b x +a \right )}{\left (a f -b e \right ) \left (a d -b c \right )}-\frac {d \ln \left (d x +c \right )}{\left (c f -d e \right ) \left (a d -b c \right )}+\frac {f \ln \left (f x +e \right )}{\left (c f -d e \right ) \left (a f -b e \right )} \]

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]

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

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maxima [A]  time = 0.65, size = 112, normalized size = 1.30 \[ \frac {b \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f} - \frac {d \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} e - {\left (b c^{2} - a c d\right )} f} + \frac {f \log \left (f x + e\right )}{b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f} \]

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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mupad [B]  time = 2.33, size = 106, normalized size = 1.23 \[ \frac {b\,\ln \left (a+b\,x\right )}{b^2\,c\,e+a^2\,d\,f-a\,b\,c\,f-a\,b\,d\,e}+\frac {d\,\ln \left (c+d\,x\right )}{a\,d^2\,e+b\,c^2\,f-a\,c\,d\,f-b\,c\,d\,e}+\frac {f\,\ln \left (e+f\,x\right )}{a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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