3.2207 \(\int \frac{1}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=136 \[ \frac{40 c^3 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{10 c^2 (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{5 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{b+2 c x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

[Out]

-(b + 2*c*x)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*c*(b + 2*c*x))/(3*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)^2) - (10*c^2*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x
+ c*x^2)) + (40*c^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi [A]  time = 0.127699, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{40 c^3 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{10 c^2 (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{5 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{b+2 c x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-4),x]

[Out]

-(b + 2*c*x)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*c*(b + 2*c*x))/(3*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)^2) - (10*c^2*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x
+ c*x^2)) + (40*c^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi in Sympy [A]  time = 20.9494, size = 131, normalized size = 0.96 \[ \frac{40 c^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} - \frac{10 c^{2} \left (b + 2 c x\right )}{\left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} + \frac{5 c \left (b + 2 c x\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} - \frac{b + 2 c x}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**4,x)

[Out]

40*c**3*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(7/2) - 10*c**2*
(b + 2*c*x)/((-4*a*c + b**2)**3*(a + b*x + c*x**2)) + 5*c*(b + 2*c*x)/(3*(-4*a*c
+ b**2)**2*(a + b*x + c*x**2)**2) - (b + 2*c*x)/(3*(-4*a*c + b**2)*(a + b*x + c
*x**2)**3)

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Mathematica [A]  time = 0.286266, size = 134, normalized size = 0.99 \[ -\frac{\frac{120 c^3 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{5 c \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{\left (b^2-4 a c\right )^2 (b+2 c x)}{(a+x (b+c x))^3}+\frac{30 c^2 (b+2 c x)}{a+x (b+c x)}}{3 \left (b^2-4 a c\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-4),x]

[Out]

-(((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x))^3 - (5*c*(b^2 - 4*a*c)*(b + 2*
c*x))/(a + x*(b + c*x))^2 + (30*c^2*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^3*Ar
cTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(3*(b^2 - 4*a*c)^3)

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Maple [A]  time = 0.007, size = 189, normalized size = 1.4 \[{\frac{2\,cx+b}{ \left ( 12\,ac-3\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{10\,{c}^{2}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{5\,bc}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+20\,{\frac{{c}^{3}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+10\,{\frac{b{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+40\,{\frac{{c}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) ^{7/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^4,x)

[Out]

1/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^3+10/3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^2
*x+5/3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^2*b+20*c^3/(4*a*c-b^2)^3/(c*x^2+b*x+a)*x+10
*c^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)*b+40*c^3/(4*a*c-b^2)^(7/2)*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((c*x^2 + b*x + a)^(-4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218505, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-4),x, algorithm="fricas")

[Out]

[-1/3*(60*(c^6*x^6 + 3*b*c^5*x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^5)
*x^4 + (b^3*c^3 + 6*a*b*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*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)) + (60*c^5*x^5 + 150*b*c^4*x^4 + b^5 - 13*a*b^3*c + 66*a^
2*b*c^2 + 10*(11*b^2*c^3 + 16*a*c^4)*x^3 + 15*(b^3*c^2 + 16*a*b*c^3)*x^2 - 3*(b^
4*c - 18*a*b^2*c^2 - 44*a^2*c^3)*x)*sqrt(b^2 - 4*a*c))/((a^3*b^6 - 12*a^4*b^4*c
+ 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^
3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*
(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9
- 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8
- 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7
- 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*sqrt(b^2 - 4*a*c)), -1/3*(120
*(c^6*x^6 + 3*b*c^5*x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^5)*x^4 + (b
^3*c^3 + 6*a*b*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*x^2)*arctan(-sqrt(-b^2 + 4*a*c
)*(2*c*x + b)/(b^2 - 4*a*c)) + (60*c^5*x^5 + 150*b*c^4*x^4 + b^5 - 13*a*b^3*c +
66*a^2*b*c^2 + 10*(11*b^2*c^3 + 16*a*c^4)*x^3 + 15*(b^3*c^2 + 16*a*b*c^3)*x^2 -
3*(b^4*c - 18*a*b^2*c^2 - 44*a^2*c^3)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6 - 12*a^4*
b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -
64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^
5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4
+ (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(
a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^
2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 9.23172, size = 777, normalized size = 5.71 \[ - 20 c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{- 5120 a^{4} c^{7} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 5120 a^{3} b^{2} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 1920 a^{2} b^{4} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 320 a b^{6} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 20 b^{8} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b c^{3}}{40 c^{4}} \right )} + 20 c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} \log{\left (x + \frac{5120 a^{4} c^{7} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 5120 a^{3} b^{2} c^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 1920 a^{2} b^{4} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} - 320 a b^{6} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b^{8} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b c^{3}}{40 c^{4}} \right )} + \frac{66 a^{2} b c^{2} - 13 a b^{3} c + b^{5} + 150 b c^{4} x^{4} + 60 c^{5} x^{5} + x^{3} \left (160 a c^{4} + 110 b^{2} c^{3}\right ) + x^{2} \left (240 a b c^{3} + 15 b^{3} c^{2}\right ) + x \left (132 a^{2} c^{3} + 54 a b^{2} c^{2} - 3 b^{4} c\right )}{192 a^{6} c^{3} - 144 a^{5} b^{2} c^{2} + 36 a^{4} b^{4} c - 3 a^{3} b^{6} + x^{6} \left (192 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 36 a b^{4} c^{4} - 3 b^{6} c^{3}\right ) + x^{5} \left (576 a^{3} b c^{5} - 432 a^{2} b^{3} c^{4} + 108 a b^{5} c^{3} - 9 b^{7} c^{2}\right ) + x^{4} \left (576 a^{4} c^{5} + 144 a^{3} b^{2} c^{4} - 324 a^{2} b^{4} c^{3} + 99 a b^{6} c^{2} - 9 b^{8} c\right ) + x^{3} \left (1152 a^{4} b c^{4} - 672 a^{3} b^{3} c^{3} + 72 a^{2} b^{5} c^{2} + 18 a b^{7} c - 3 b^{9}\right ) + x^{2} \left (576 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 324 a^{3} b^{4} c^{2} + 99 a^{2} b^{6} c - 9 a b^{8}\right ) + x \left (576 a^{5} b c^{3} - 432 a^{4} b^{3} c^{2} + 108 a^{3} b^{5} c - 9 a^{2} b^{7}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**4,x)

[Out]

-20*c**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-5120*a**4*c**7*sqrt(-1/(4*a*c - b*
*2)**7) + 5120*a**3*b**2*c**6*sqrt(-1/(4*a*c - b**2)**7) - 1920*a**2*b**4*c**5*s
qrt(-1/(4*a*c - b**2)**7) + 320*a*b**6*c**4*sqrt(-1/(4*a*c - b**2)**7) - 20*b**8
*c**3*sqrt(-1/(4*a*c - b**2)**7) + 20*b*c**3)/(40*c**4)) + 20*c**3*sqrt(-1/(4*a*
c - b**2)**7)*log(x + (5120*a**4*c**7*sqrt(-1/(4*a*c - b**2)**7) - 5120*a**3*b**
2*c**6*sqrt(-1/(4*a*c - b**2)**7) + 1920*a**2*b**4*c**5*sqrt(-1/(4*a*c - b**2)**
7) - 320*a*b**6*c**4*sqrt(-1/(4*a*c - b**2)**7) + 20*b**8*c**3*sqrt(-1/(4*a*c -
b**2)**7) + 20*b*c**3)/(40*c**4)) + (66*a**2*b*c**2 - 13*a*b**3*c + b**5 + 150*b
*c**4*x**4 + 60*c**5*x**5 + x**3*(160*a*c**4 + 110*b**2*c**3) + x**2*(240*a*b*c*
*3 + 15*b**3*c**2) + x*(132*a**2*c**3 + 54*a*b**2*c**2 - 3*b**4*c))/(192*a**6*c*
*3 - 144*a**5*b**2*c**2 + 36*a**4*b**4*c - 3*a**3*b**6 + x**6*(192*a**3*c**6 - 1
44*a**2*b**2*c**5 + 36*a*b**4*c**4 - 3*b**6*c**3) + x**5*(576*a**3*b*c**5 - 432*
a**2*b**3*c**4 + 108*a*b**5*c**3 - 9*b**7*c**2) + x**4*(576*a**4*c**5 + 144*a**3
*b**2*c**4 - 324*a**2*b**4*c**3 + 99*a*b**6*c**2 - 9*b**8*c) + x**3*(1152*a**4*b
*c**4 - 672*a**3*b**3*c**3 + 72*a**2*b**5*c**2 + 18*a*b**7*c - 3*b**9) + x**2*(5
76*a**5*c**4 + 144*a**4*b**2*c**3 - 324*a**3*b**4*c**2 + 99*a**2*b**6*c - 9*a*b*
*8) + x*(576*a**5*b*c**3 - 432*a**4*b**3*c**2 + 108*a**3*b**5*c - 9*a**2*b**7))

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GIAC/XCAS [A]  time = 0.202839, size = 297, normalized size = 2.18 \[ -\frac{40 \, c^{3} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{60 \, c^{5} x^{5} + 150 \, b c^{4} x^{4} + 110 \, b^{2} c^{3} x^{3} + 160 \, a c^{4} x^{3} + 15 \, b^{3} c^{2} x^{2} + 240 \, a b c^{3} x^{2} - 3 \, b^{4} c x + 54 \, a b^{2} c^{2} x + 132 \, a^{2} c^{3} x + b^{5} - 13 \, a b^{3} c + 66 \, a^{2} b c^{2}}{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-4),x, algorithm="giac")

[Out]

-40*c^3*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c
^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(60*c^5*x^5 + 150*b*c^4*x^4 + 110*b^2
*c^3*x^3 + 160*a*c^4*x^3 + 15*b^3*c^2*x^2 + 240*a*b*c^3*x^2 - 3*b^4*c*x + 54*a*b
^2*c^2*x + 132*a^2*c^3*x + b^5 - 13*a*b^3*c + 66*a^2*b*c^2)/((b^6 - 12*a*b^4*c +
48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)