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

Optimal. Leaf size=234 \[ -\frac{b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac{d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac{2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac{f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) +
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(d*e - c*f)^3) + (2*f^3*(2*b*d*e - b*c*f - a*d*f
)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

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Rubi [A]  time = 0.405178, antiderivative size = 234, 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} \[ -\frac{b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac{d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac{2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac{f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]

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)^(-2),x]

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) +
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(d*e - c*f)^3) + (2*f^3*(2*b*d*e - b*c*f - a*d*f
)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

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}{\left (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\right )^2} \, dx &=\int \left (\frac{b^4}{(b c-a d)^2 (b e-a f)^2 (a+b x)^2}-\frac{2 b^4 (b d e+b c f-2 a d f)}{(b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac{d^4}{(b c-a d)^2 (-d e+c f)^2 (c+d x)^2}-\frac{2 d^4 (b d e-2 b c f+a d f)}{(b c-a d)^3 (-d e+c f)^3 (c+d x)}+\frac{f^4}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac{2 f^4 (-2 b d e+b c f+a d f)}{(b e-a f)^3 (d e-c f)^3 (e+f x)}\right ) \, dx\\ &=-\frac{b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac{d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac{f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac{2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac{2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac{2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3}\\ \end{align*}

Mathematica [A]  time = 0.652643, size = 232, normalized size = 0.99 \[ -\frac{b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac{d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}-\frac{2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (c f-d e)^3}-\frac{f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}-\frac{2 f^3 \log (e+f x) (a d f+b c f-2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]

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)^(-2),x]

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) -
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(-(d*e) + c*f)^3) - (2*f^3*(-2*b*d*e + b*c*f + a
*d*f)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

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Maple [A]  time = 0.05, size = 398, normalized size = 1.7 \begin{align*} -{\frac{{b}^{3}}{ \left ( af-be \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) adf}{ \left ( af-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) cf}{ \left ( af-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) de}{ \left ( af-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{{d}^{3}}{ \left ( cf-de \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) af}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) bcf}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) be}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{{f}^{3}}{ \left ( cf-de \right ) ^{2} \left ( af-be \right ) ^{2} \left ( fx+e \right ) }}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) ad}{ \left ( cf-de \right ) ^{3} \left ( af-be \right ) ^{3}}}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) bc}{ \left ( cf-de \right ) ^{3} \left ( af-be \right ) ^{3}}}+4\,{\frac{{f}^{3}\ln \left ( fx+e \right ) bde}{ \left ( cf-de \right ) ^{3} \left ( af-be \right ) ^{3}}} \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)^2,x)

[Out]

-b^3/(a*f-b*e)^2/(a*d-b*c)^2/(b*x+a)+4*b^3/(a*f-b*e)^3/(a*d-b*c)^3*ln(b*x+a)*a*d*f-2*b^4/(a*f-b*e)^3/(a*d-b*c)
^3*ln(b*x+a)*c*f-2*b^4/(a*f-b*e)^3/(a*d-b*c)^3*ln(b*x+a)*d*e-d^3/(c*f-d*e)^2/(a*d-b*c)^2/(d*x+c)+2*d^4/(c*f-d*
e)^3/(a*d-b*c)^3*ln(d*x+c)*a*f-4*d^3/(c*f-d*e)^3/(a*d-b*c)^3*ln(d*x+c)*b*c*f+2*d^4/(c*f-d*e)^3/(a*d-b*c)^3*ln(
d*x+c)*b*e-f^3/(c*f-d*e)^2/(a*f-b*e)^2/(f*x+e)-2*f^4/(c*f-d*e)^3/(a*f-b*e)^3*ln(f*x+e)*a*d-2*f^4/(c*f-d*e)^3/(
a*f-b*e)^3*ln(f*x+e)*b*c+4*f^3/(c*f-d*e)^3/(a*f-b*e)^3*ln(f*x+e)*b*d*e

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Maxima [B]  time = 1.85039, size = 2830, normalized size = 12.09 \begin{align*} \text{result too large to display} \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)^2,x, algorithm="maxima")

[Out]

-2*(b^4*d*e + (b^4*c - 2*a*b^3*d)*f)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*e
^3 - 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*e^2*f + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d
+ 3*a^4*b^2*c*d^2 - a^5*b*d^3)*e*f^2 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*f^3) + 2*(b*d
^4*e - (2*b*c*d^3 - a*d^4)*f)*log(d*x + c)/((b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)*e^3 - 3*
(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*e^2*f + 3*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b
*c^3*d^3 - a^3*c^2*d^4)*e*f^2 - (b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*f^3) + 2*(2*b*d*e*f^
3 - (b*c + a*d)*f^4)*log(f*x + e)/(b^3*d^3*e^6 + a^3*c^3*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f + 3*(b^3*c^2*d
+ 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^3 + 3*(a*b^2*
c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^5) - ((b^3*c*d^2 + a*b^2*d^3)*e^3 - 2
*(b^3*c^2*d + a^2*b*d^3)*e^2*f + (b^3*c^3 + a^3*d^3)*e*f^2 + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^3 + 2*(
b^3*d^3*e^2*f - (b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3)*f^3)*x^2 + (2*b^3*d^3*e^
3 - (b^3*c*d^2 + a*b^2*d^3)*e^2*f - (b^3*c^2*d + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 2
*a^3*d^3)*f^3)*x)/((a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*e^5 - 2*(a*b^4*c^4*d - a^2*b^3*c^3*d^2
- a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*e^4*f + (a*b^4*c^5 + 2*a^2*b^3*c^4*d - 6*a^3*b^2*c^3*d^2 + 2*a^4*b*c^2*d^3 +
a^5*c*d^4)*e^3*f^2 - 2*(a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*e^2*f^3 + (a^3*b^2*c^5 - 2*
a^4*b*c^4*d + a^5*c^3*d^2)*e*f^4 + ((b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*e^4*f - 2*(b^5*c^3*d^2 - a*b^4
*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^3*f^2 + (b^5*c^4*d + 2*a*b^4*c^3*d^2 - 6*a^2*b^3*c^2*d^3 + 2*a^3*b^2
*c*d^4 + a^4*b*d^5)*e^2*f^3 - 2*(a*b^4*c^4*d - a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*e*f^4 + (a^2*b
^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*f^5)*x^3 + ((b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*e^5 - (b
^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^4*f - (b^5*c^4*d - 2*a*b^4*c^3*d^2 + 2*a^2*b^3*c^2
*d^3 - 2*a^3*b^2*c*d^4 + a^4*b*d^5)*e^3*f^2 + (b^5*c^5 + a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 +
a^4*b*c*d^4 + a^5*d^5)*e^2*f^3 - (2*a*b^4*c^5 - a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - a^4*b*c^2*d^3 + 2*a^5*c*d
^4)*e*f^4 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*f^5)*x^2 + ((b^5*c^3*d^2 - a*b^4*c^2*d
^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^5 - (2*b^5*c^4*d - a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - a^3*b^2*c*d^4 + 2*a
^4*b*d^5)*e^4*f + (b^5*c^5 + a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + a^5*d^5)*e^3*
f^2 - (a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^3*b^2*c^3*d^2 - 2*a^4*b*c^2*d^3 + a^5*c*d^4)*e^2*f^3 - (a^2*b^3*c^5 -
a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*e*f^4 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*f^5)*x)

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Fricas [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)^2,x, algorithm="fricas")

[Out]

Timed out

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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)**2,x)

[Out]

Timed out

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Giac [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)^2,x, algorithm="giac")

[Out]

Timed out