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]
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Rubi [A] time = 0.888164, 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 \[ -\frac{b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac{d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac{2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac{f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]
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]
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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/(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]
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Mathematica [A] time = 1.33016, 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]
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Maple [A] time = 0.066, size = 398, normalized size = 1.7 \[ -{\frac{{f}^{3}}{ \left ( cf-de \right ) ^{2} \left ( fa-be \right ) ^{2} \left ( fx+e \right ) }}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) ad}{ \left ( cf-de \right ) ^{3} \left ( fa-be \right ) ^{3}}}-2\,{\frac{{f}^{4}\ln \left ( fx+e \right ) bc}{ \left ( cf-de \right ) ^{3} \left ( fa-be \right ) ^{3}}}+4\,{\frac{{f}^{3}\ln \left ( fx+e \right ) bde}{ \left ( cf-de \right ) ^{3} \left ( fa-be \right ) ^{3}}}-{\frac{{d}^{3}}{ \left ( cf-de \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) af}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) bcf}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) be}{ \left ( cf-de \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{3}}{ \left ( fa-be \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) adf}{ \left ( fa-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) cf}{ \left ( fa-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) de}{ \left ( fa-be \right ) ^{3} \left ( ad-bc \right ) ^{3}}} \]
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]
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Maxima [A] time = 0.86502, size = 2830, normalized size = 12.09 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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)^(-2),x, algorithm="maxima")
[Out]
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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((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)^(-2),x, algorithm="fricas")
[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/(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]
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GIAC/XCAS [A] time = 0.280415, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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)^(-2),x, algorithm="giac")
[Out]