3.436 \(\int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^3 x^5} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 a+b x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{3 b (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{6 b c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

(2*a + b*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*b*(b + 2*c*x))/(2*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)) + (6*b*c*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^
2 - 4*a*c)^(5/2)

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Rubi [A]  time = 0.0985487, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{2 a+b x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{3 b (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{6 b c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c + a/x^2 + b/x)^3*x^5),x]

[Out]

(2*a + b*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*b*(b + 2*c*x))/(2*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)) + (6*b*c*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^
2 - 4*a*c)^(5/2)

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Rubi in Sympy [A]  time = 18.1801, size = 97, normalized size = 0.94 \[ \frac{6 b c \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 b \left (b + 2 c x\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} + \frac{2 a + b x}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c+a/x**2+b/x)**3/x**5,x)

[Out]

6*b*c*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(5/2) - 3*b*(b + 2
*c*x)/(2*(-4*a*c + b**2)**2*(a + b*x + c*x**2)) + (2*a + b*x)/(2*(-4*a*c + b**2)
*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.169536, size = 102, normalized size = 0.99 \[ \frac{\frac{\left (b^2-4 a c\right ) (2 a+b x)}{(a+x (b+c x))^2}-\frac{12 b c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{3 b (b+2 c x)}{a+x (b+c x)}}{2 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + a/x^2 + b/x)^3*x^5),x]

[Out]

(((b^2 - 4*a*c)*(2*a + b*x))/(a + x*(b + c*x))^2 - (3*b*(b + 2*c*x))/(a + x*(b +
c*x)) - (12*b*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*
(b^2 - 4*a*c)^2)

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Maple [A]  time = 0.005, size = 130, normalized size = 1.3 \[{\frac{-bx-2\,a}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{bcx}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{3\,{b}^{2}}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-6\,{\frac{bc}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^3/x^5,x)

[Out]

1/2*(-b*x-2*a)/(4*a*c-b^2)/(c*x^2+b*x+a)^2-3*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)*c*x-3
/2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)-6*b/(4*a*c-b^2)^(5/2)*c*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26734, size = 1, normalized size = 0.01 \[ \left [\frac{6 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} + 2 \, a b^{2} c x + a^{2} b c +{\left (b^{3} c + 2 \, a b c^{2}\right )} x^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (6 \, b c^{2} x^{3} + 9 \, b^{2} c x^{2} + a b^{2} + 8 \, a^{2} c + 2 \,{\left (b^{3} + 5 \, a b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{12 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} + 2 \, a b^{2} c x + a^{2} b c +{\left (b^{3} c + 2 \, a b c^{2}\right )} x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (6 \, b c^{2} x^{3} + 9 \, b^{2} c x^{2} + a b^{2} + 8 \, a^{2} c + 2 \,{\left (b^{3} + 5 \, a b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^5),x, algorithm="fricas")

[Out]

[1/2*(6*(b*c^3*x^4 + 2*b^2*c^2*x^3 + 2*a*b^2*c*x + a^2*b*c + (b^3*c + 2*a*b*c^2)
*x^2)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 -
2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) - (6*b*c^2*x^3 + 9*b^2*c*x^2 + a*b^
2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c +
16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*x)*sqrt(b^2 - 4*a*c)), -1/2*(12*(b*c^3*x^4 + 2*b^2*c^2*x^3 + 2*
a*b^2*c*x + a^2*b*c + (b^3*c + 2*a*b*c^2)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x
+ b)/(b^2 - 4*a*c)) + (6*b*c^2*x^3 + 9*b^2*c*x^2 + a*b^2 + 8*a^2*c + 2*(b^3 + 5
*a*b*c)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 -
8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b
^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqr
t(-b^2 + 4*a*c))]

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Sympy [A]  time = 5.11295, size = 479, normalized size = 4.65 \[ 3 b c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 192 a^{3} b c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{5} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{7} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c}{6 b c^{2}} \right )} - 3 b c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{192 a^{3} b c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{5} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{7} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c}{6 b c^{2}} \right )} - \frac{8 a^{2} c + a b^{2} + 9 b^{2} c x^{2} + 6 b c^{2} x^{3} + x \left (10 a b c + 2 b^{3}\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+a/x**2+b/x)**3/x**5,x)

[Out]

3*b*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-192*a**3*b*c**4*sqrt(-1/(4*a*c - b**2
)**5) + 144*a**2*b**3*c**3*sqrt(-1/(4*a*c - b**2)**5) - 36*a*b**5*c**2*sqrt(-1/(
4*a*c - b**2)**5) + 3*b**7*c*sqrt(-1/(4*a*c - b**2)**5) + 3*b**2*c)/(6*b*c**2))
- 3*b*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (192*a**3*b*c**4*sqrt(-1/(4*a*c - b**
2)**5) - 144*a**2*b**3*c**3*sqrt(-1/(4*a*c - b**2)**5) + 36*a*b**5*c**2*sqrt(-1/
(4*a*c - b**2)**5) - 3*b**7*c*sqrt(-1/(4*a*c - b**2)**5) + 3*b**2*c)/(6*b*c**2))
- (8*a**2*c + a*b**2 + 9*b**2*c*x**2 + 6*b*c**2*x**3 + x*(10*a*b*c + 2*b**3))/(
32*a**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x**4*(32*a**2*c**4 - 16*a*b**2*c**
3 + 2*b**4*c**2) + x**3*(64*a**2*b*c**3 - 32*a*b**3*c**2 + 4*b**5*c) + x**2*(64*
a**3*c**3 - 12*a*b**4*c + 2*b**6) + x*(64*a**3*b*c**2 - 32*a**2*b**3*c + 4*a*b**
5))

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GIAC/XCAS [A]  time = 0.283169, size = 182, normalized size = 1.77 \[ -\frac{6 \, b c \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{6 \, b c^{2} x^{3} + 9 \, b^{2} c x^{2} + 2 \, b^{3} x + 10 \, a b c x + a b^{2} + 8 \, a^{2} c}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^5),x, algorithm="giac")

[Out]

-6*b*c*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sq
rt(-b^2 + 4*a*c)) - 1/2*(6*b*c^2*x^3 + 9*b^2*c*x^2 + 2*b^3*x + 10*a*b*c*x + a*b^
2 + 8*a^2*c)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(c*x^2 + b*x + a)^2)