3.2202 \(\int \frac{x^5}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=145 \[ -\frac{20 a^2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(x^5*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*b*x^3*(2*a + b*x))
/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (5*a*b*x*(2*a + b*x))/((b^2 - 4*a*c)^
3*(a + b*x + c*x^2)) - (20*a^2*b*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 = 0.158768, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{20 a^2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x + c*x^2)^4,x]

[Out]

-(x^5*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*b*x^3*(2*a + b*x))
/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (5*a*b*x*(2*a + b*x))/((b^2 - 4*a*c)^
3*(a + b*x + c*x^2)) - (20*a^2*b*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 = 31.6362, size = 141, normalized size = 0.97 \[ - \frac{20 a^{2} b \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} - \frac{5 a b x \left (2 a + b x\right )}{\left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} + \frac{5 b x^{3} \left (2 a + b x\right )}{6 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} - \frac{x^{5} \left (b + 2 c x\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**2+b*x+a)**4,x)

[Out]

-20*a**2*b*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(7/2) - 5*a*b
*x*(2*a + b*x)/((-4*a*c + b**2)**3*(a + b*x + c*x**2)) + 5*b*x**3*(2*a + b*x)/(6
*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**2) - x**5*(b + 2*c*x)/(3*(-4*a*c + b**2)
*(a + b*x + c*x**2)**3)

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Mathematica [A]  time = 0.530982, size = 266, normalized size = 1.83 \[ \frac{1}{6} \left (-\frac{120 a^2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+\frac{3 \left (-64 a^3 c^3+38 a^2 b^2 c^2-20 a^2 b c^3 x-12 a b^4 c+b^6\right )}{c^3 \left (4 a c-b^2\right )^3 (a+x (b+c x))}-\frac{2 \left (2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x\right )}{c^4 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{48 a^3 c^3-61 a^2 b^2 c^2+70 a^2 b c^3 x+19 a b^4 c-40 a b^3 c^2 x-2 b^6+5 b^5 c x}{c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x + c*x^2)^4,x]

[Out]

((-2*b^6 + 19*a*b^4*c - 61*a^2*b^2*c^2 + 48*a^3*c^3 + 5*b^5*c*x - 40*a*b^3*c^2*x
+ 70*a^2*b*c^3*x)/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (3*(b^6 - 12*a*b^
4*c + 38*a^2*b^2*c^2 - 64*a^3*c^3 - 20*a^2*b*c^3*x))/(c^3*(-b^2 + 4*a*c)^3*(a +
x*(b + c*x))) - (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*
x)))/(c^4*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (120*a^2*b*ArcTan[(b + 2*c*x)/Sq
rt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/6

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Maple [B]  time = 0.022, size = 486, normalized size = 3.4 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{3}} \left ( -10\,{\frac{{a}^{2}b{c}^{2}{x}^{5}}{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}-{\frac{ \left ( 64\,{a}^{3}{c}^{3}+2\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{4}}{2\,c \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{b \left ( 224\,{a}^{3}{c}^{3}+62\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{3}}{6\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{a \left ( 64\,{a}^{3}{c}^{3}+32\,{a}^{2}{b}^{2}{c}^{2}+17\,a{b}^{4}c-{b}^{6} \right ){x}^{2}}{2\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{{a}^{2}b \left ( 44\,{a}^{2}{c}^{2}+18\,ac{b}^{2}-{b}^{4} \right ) x}{2\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{{a}^{3} \left ( 64\,{a}^{2}{c}^{2}+18\,ac{b}^{2}-{b}^{4} \right ) }{6\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }} \right ) }-20\,{\frac{{a}^{2}b}{ \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^2+b*x+a)^4,x)

[Out]

(-10*a^2*b/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*x^5-1/2*(64*a^3*c^3+2*
a^2*b^2*c^2+12*a*b^4*c-b^6)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^4-1/6
*b*(224*a^3*c^3+62*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12
*a*b^4*c-b^6)*x^3-1/2*a*(64*a^3*c^3+32*a^2*b^2*c^2+17*a*b^4*c-b^6)/c^2/(64*a^3*c
^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*a^2*b*(44*a^2*c^2+18*a*b^2*c-b^4)/c^2/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-1/6*a^3*(64*a^2*c^2+18*a*b^2*c-b^4)
/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3-20*a^2*b/(64*a^
3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*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(x^5/(c*x^2 + b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224105, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^2 + b*x + a)^4,x, algorithm="fricas")

[Out]

[-1/6*(60*(a^2*b*c^5*x^6 + 3*a^2*b^2*c^4*x^5 + 3*a^4*b^2*c^2*x + a^5*b*c^2 + 3*(
a^2*b^3*c^3 + a^3*b*c^4)*x^4 + (a^2*b^4*c^2 + 6*a^3*b^2*c^3)*x^3 + 3*(a^3*b^3*c^
2 + a^4*b*c^3)*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*a^2*b*c^4*x^5 -
a^3*b^4 + 18*a^4*b^2*c + 64*a^5*c^2 - 3*(b^6*c - 12*a*b^4*c^2 - 2*a^2*b^2*c^3 -
64*a^3*c^4)*x^4 - (b^7 - 12*a*b^5*c - 62*a^2*b^3*c^2 - 224*a^3*b*c^3)*x^3 - 3*(
a*b^6 - 17*a^2*b^4*c - 32*a^3*b^2*c^2 - 64*a^4*c^3)*x^2 - 3*(a^2*b^5 - 18*a^3*b^
3*c - 44*a^4*b*c^2)*x)*sqrt(b^2 - 4*a*c))/((a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^
5*b^2*c^4 - 64*a^6*c^5 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*
x^6 + 3*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*x^5 + 3*(b^8*c^
3 - 11*a*b^6*c^4 + 36*a^2*b^4*c^5 - 16*a^3*b^2*c^6 - 64*a^4*c^7)*x^4 + (b^9*c^2
- 6*a*b^7*c^3 - 24*a^2*b^5*c^4 + 224*a^3*b^3*c^5 - 384*a^4*b*c^6)*x^3 + 3*(a*b^8
*c^2 - 11*a^2*b^6*c^3 + 36*a^3*b^4*c^4 - 16*a^4*b^2*c^5 - 64*a^5*c^6)*x^2 + 3*(a
^2*b^7*c^2 - 12*a^3*b^5*c^3 + 48*a^4*b^3*c^4 - 64*a^5*b*c^5)*x)*sqrt(b^2 - 4*a*c
)), 1/6*(120*(a^2*b*c^5*x^6 + 3*a^2*b^2*c^4*x^5 + 3*a^4*b^2*c^2*x + a^5*b*c^2 +
3*(a^2*b^3*c^3 + a^3*b*c^4)*x^4 + (a^2*b^4*c^2 + 6*a^3*b^2*c^3)*x^3 + 3*(a^3*b^3
*c^2 + a^4*b*c^3)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (
60*a^2*b*c^4*x^5 - a^3*b^4 + 18*a^4*b^2*c + 64*a^5*c^2 - 3*(b^6*c - 12*a*b^4*c^2
- 2*a^2*b^2*c^3 - 64*a^3*c^4)*x^4 - (b^7 - 12*a*b^5*c - 62*a^2*b^3*c^2 - 224*a^
3*b*c^3)*x^3 - 3*(a*b^6 - 17*a^2*b^4*c - 32*a^3*b^2*c^2 - 64*a^4*c^3)*x^2 - 3*(a
^2*b^5 - 18*a^3*b^3*c - 44*a^4*b*c^2)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6*c^2 - 12*
a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2
*c^7 - 64*a^3*c^8)*x^6 + 3*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c
^7)*x^5 + 3*(b^8*c^3 - 11*a*b^6*c^4 + 36*a^2*b^4*c^5 - 16*a^3*b^2*c^6 - 64*a^4*c
^7)*x^4 + (b^9*c^2 - 6*a*b^7*c^3 - 24*a^2*b^5*c^4 + 224*a^3*b^3*c^5 - 384*a^4*b*
c^6)*x^3 + 3*(a*b^8*c^2 - 11*a^2*b^6*c^3 + 36*a^3*b^4*c^4 - 16*a^4*b^2*c^5 - 64*
a^5*c^6)*x^2 + 3*(a^2*b^7*c^2 - 12*a^3*b^5*c^3 + 48*a^4*b^3*c^4 - 64*a^5*b*c^5)*
x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 13.4001, size = 898, normalized size = 6.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**2+b*x+a)**4,x)

[Out]

10*a**2*b*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-2560*a**6*b*c**4*sqrt(-1/(4*a*c -
b**2)**7) + 2560*a**5*b**3*c**3*sqrt(-1/(4*a*c - b**2)**7) - 960*a**4*b**5*c**2
*sqrt(-1/(4*a*c - b**2)**7) + 160*a**3*b**7*c*sqrt(-1/(4*a*c - b**2)**7) - 10*a*
*2*b**9*sqrt(-1/(4*a*c - b**2)**7) + 10*a**2*b**2)/(20*a**2*b*c)) - 10*a**2*b*sq
rt(-1/(4*a*c - b**2)**7)*log(x + (2560*a**6*b*c**4*sqrt(-1/(4*a*c - b**2)**7) -
2560*a**5*b**3*c**3*sqrt(-1/(4*a*c - b**2)**7) + 960*a**4*b**5*c**2*sqrt(-1/(4*a
*c - b**2)**7) - 160*a**3*b**7*c*sqrt(-1/(4*a*c - b**2)**7) + 10*a**2*b**9*sqrt(
-1/(4*a*c - b**2)**7) + 10*a**2*b**2)/(20*a**2*b*c)) - (64*a**5*c**2 + 18*a**4*b
**2*c - a**3*b**4 + 60*a**2*b*c**4*x**5 + x**4*(192*a**3*c**4 + 6*a**2*b**2*c**3
+ 36*a*b**4*c**2 - 3*b**6*c) + x**3*(224*a**3*b*c**3 + 62*a**2*b**3*c**2 + 12*a
*b**5*c - b**7) + x**2*(192*a**4*c**3 + 96*a**3*b**2*c**2 + 51*a**2*b**4*c - 3*a
*b**6) + x*(132*a**4*b*c**2 + 54*a**3*b**3*c - 3*a**2*b**5))/(384*a**6*c**5 - 28
8*a**5*b**2*c**4 + 72*a**4*b**4*c**3 - 6*a**3*b**6*c**2 + x**6*(384*a**3*c**8 -
288*a**2*b**2*c**7 + 72*a*b**4*c**6 - 6*b**6*c**5) + x**5*(1152*a**3*b*c**7 - 86
4*a**2*b**3*c**6 + 216*a*b**5*c**5 - 18*b**7*c**4) + x**4*(1152*a**4*c**7 + 288*
a**3*b**2*c**6 - 648*a**2*b**4*c**5 + 198*a*b**6*c**4 - 18*b**8*c**3) + x**3*(23
04*a**4*b*c**6 - 1344*a**3*b**3*c**5 + 144*a**2*b**5*c**4 + 36*a*b**7*c**3 - 6*b
**9*c**2) + x**2*(1152*a**5*c**6 + 288*a**4*b**2*c**5 - 648*a**3*b**4*c**4 + 198
*a**2*b**6*c**3 - 18*a*b**8*c**2) + x*(1152*a**5*b*c**5 - 864*a**4*b**3*c**4 + 2
16*a**3*b**5*c**3 - 18*a**2*b**7*c**2))

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GIAC/XCAS [A]  time = 0.209631, size = 440, normalized size = 3.03 \[ \frac{20 \, a^{2} b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{60 \, a^{2} b c^{4} x^{5} - 3 \, b^{6} c x^{4} + 36 \, a b^{4} c^{2} x^{4} + 6 \, a^{2} b^{2} c^{3} x^{4} + 192 \, a^{3} c^{4} x^{4} - b^{7} x^{3} + 12 \, a b^{5} c x^{3} + 62 \, a^{2} b^{3} c^{2} x^{3} + 224 \, a^{3} b c^{3} x^{3} - 3 \, a b^{6} x^{2} + 51 \, a^{2} b^{4} c x^{2} + 96 \, a^{3} b^{2} c^{2} x^{2} + 192 \, a^{4} c^{3} x^{2} - 3 \, a^{2} b^{5} x + 54 \, a^{3} b^{3} c x + 132 \, a^{4} b c^{2} x - a^{3} b^{4} + 18 \, a^{4} b^{2} c + 64 \, a^{5} c^{2}}{6 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^2 + b*x + a)^4,x, algorithm="giac")

[Out]

20*a^2*b*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)) + 1/6*(60*a^2*b*c^4*x^5 - 3*b^6*c*x^4 + 36
*a*b^4*c^2*x^4 + 6*a^2*b^2*c^3*x^4 + 192*a^3*c^4*x^4 - b^7*x^3 + 12*a*b^5*c*x^3
+ 62*a^2*b^3*c^2*x^3 + 224*a^3*b*c^3*x^3 - 3*a*b^6*x^2 + 51*a^2*b^4*c*x^2 + 96*a
^3*b^2*c^2*x^2 + 192*a^4*c^3*x^2 - 3*a^2*b^5*x + 54*a^3*b^3*c*x + 132*a^4*b*c^2*
x - a^3*b^4 + 18*a^4*b^2*c + 64*a^5*c^2)/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5)*(c*x^2 + b*x + a)^3)