3.1698 \(\int \frac{1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx\)

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 = 0.892468, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac{d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac{2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac{f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^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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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**2/(d*x+c)**2/(f*x+e)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.45155, 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[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^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 = 0.058, size = 398, normalized size = 1.7 \[ -{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{2} \left ( cf-de \right ) ^{2} \left ( dx+c \right ) }}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) af}{ \left ( ad-bc \right ) ^{3} \left ( cf-de \right ) ^{3}}}-4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) bcf}{ \left ( ad-bc \right ) ^{3} \left ( cf-de \right ) ^{3}}}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) be}{ \left ( ad-bc \right ) ^{3} \left ( cf-de \right ) ^{3}}}-{\frac{{b}^{3}}{ \left ( ad-bc \right ) ^{2} \left ( af-be \right ) ^{2} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) adf}{ \left ( ad-bc \right ) ^{3} \left ( af-be \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) cf}{ \left ( ad-bc \right ) ^{3} \left ( af-be \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) de}{ \left ( ad-bc \right ) ^{3} \left ( af-be \right ) ^{3}}}-{\frac{{f}^{3}}{ \left ( af-be \right ) ^{2} \left ( cf-de \right ) ^{2} \left ( fx+e \right ) }}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) ad}{ \left ( af-be \right ) ^{3} \left ( cf-de \right ) ^{3}}}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) bc}{ \left ( af-be \right ) ^{3} \left ( cf-de \right ) ^{3}}}+4\,{\frac{{f}^{3}\ln \left ( fx+e \right ) bde}{ \left ( af-be \right ) ^{3} \left ( cf-de \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52638, size = 2830, normalized size = 12.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^2*(f*x + e)^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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**2/(d*x+c)**2/(f*x+e)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.453101, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done