3.2215 \(\int \frac{1}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=171 \[ -\frac{140 c^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[Out]

-(b + 2*c*x)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*c*(b + 2*c*x))/(6*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)^3) - (35*c^2*(b + 2*c*x))/(6*(b^2 - 4*a*c)^3*(a + b*
x + c*x^2)^2) + (35*c^3*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (140*
c^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi [A]  time = 0.163805, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{140 c^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-5),x]

[Out]

-(b + 2*c*x)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*c*(b + 2*c*x))/(6*(b^2 -
4*a*c)^2*(a + b*x + c*x^2)^3) - (35*c^2*(b + 2*c*x))/(6*(b^2 - 4*a*c)^3*(a + b*
x + c*x^2)^2) + (35*c^3*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (140*
c^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi in Sympy [A]  time = 30.1343, size = 167, normalized size = 0.98 \[ - \frac{140 c^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{2}}} + \frac{35 c^{3} \left (b + 2 c x\right )}{\left (- 4 a c + b^{2}\right )^{4} \left (a + b x + c x^{2}\right )} - \frac{35 c^{2} \left (b + 2 c x\right )}{6 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )^{2}} + \frac{7 c \left (b + 2 c x\right )}{6 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{3}} - \frac{b + 2 c x}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-140*c**4*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(9/2) + 35*c**
3*(b + 2*c*x)/((-4*a*c + b**2)**4*(a + b*x + c*x**2)) - 35*c**2*(b + 2*c*x)/(6*(
-4*a*c + b**2)**3*(a + b*x + c*x**2)**2) + 7*c*(b + 2*c*x)/(6*(-4*a*c + b**2)**2
*(a + b*x + c*x**2)**3) - (b + 2*c*x)/(4*(-4*a*c + b**2)*(a + b*x + c*x**2)**4)

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Mathematica [A]  time = 0.324726, size = 167, normalized size = 0.98 \[ \frac{\frac{1680 c^4 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{70 c^2 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{14 c \left (b^2-4 a c\right )^2 (b+2 c x)}{(a+x (b+c x))^3}-\frac{3 \left (b^2-4 a c\right )^3 (b+2 c x)}{(a+x (b+c x))^4}+\frac{420 c^3 (b+2 c x)}{a+x (b+c x)}}{12 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

((-3*(b^2 - 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x))^4 + (14*c*(b^2 - 4*a*c)^2*(b
+ 2*c*x))/(a + x*(b + c*x))^3 - (70*c^2*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b +
c*x))^2 + (420*c^3*(b + 2*c*x))/(a + x*(b + c*x)) + (1680*c^4*ArcTan[(b + 2*c*x)
/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)

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Maple [A]  time = 0.01, size = 249, normalized size = 1.5 \[{\frac{2\,cx+b}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{4}}}+{\frac{7\,{c}^{2}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,bc}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{35\,{c}^{3}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,b{c}^{2}}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+70\,{\frac{{c}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+35\,{\frac{b{c}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+140\,{\frac{{c}^{4}}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^5,x)

[Out]

1/4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^4+7/3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*
x+7/6*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*b+35/3*c^3/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*x
+35/6*c^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*b+70*c^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)*x+3
5*c^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)*b+140*c^4/(4*a*c-b^2)^(9/2)*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)^(-5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232209, 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)^(-5),x, algorithm="fricas")

[Out]

[1/12*(840*(c^8*x^8 + 4*b*c^7*x^7 + 4*a^3*b*c^4*x + a^4*c^4 + 2*(3*b^2*c^6 + 2*a
*c^7)*x^6 + 4*(b^3*c^5 + 3*a*b*c^6)*x^5 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*x
^4 + 4*(a*b^3*c^4 + 3*a^2*b*c^5)*x^3 + 2*(3*a^2*b^2*c^4 + 2*a^3*c^5)*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)) + (840*c^7*x^7 + 2940*b*c^6*x^6 - 3*b^7 + 50*a
*b^5*c - 326*a^2*b^3*c^2 + 1116*a^3*b*c^3 + 280*(13*b^2*c^5 + 11*a*c^6)*x^5 + 35
0*(5*b^3*c^4 + 22*a*b*c^5)*x^4 + 56*(3*b^4*c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*x^3
- 28*(b^5*c^2 - 28*a*b^3*c^3 - 219*a^2*b*c^4)*x^2 + 8*(b^6*c - 19*a*b^4*c^2 + 1
74*a^2*b^2*c^3 + 279*a^3*c^4)*x)*sqrt(b^2 - 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 9
6*a^6*b^4*c^2 - 256*a^7*b^2*c^3 + 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2
*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a
^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3
+ 256*a^2*b^6*c^4 - 576*a^3*b^4*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b
^11*c - 13*a*b^9*c^2 + 48*a^2*b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a
^5*b*c^6)*x^5 + (b^12 - 4*a*b^10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4
*b^4*c^4 + 1536*a^5*b^2*c^5 + 1536*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*
a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b
^10 - 46*a^3*b^8*c + 256*a^4*b^6*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a
^7*c^5)*x^2 + 4*(a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256
*a^7*b*c^4)*x)*sqrt(b^2 - 4*a*c)), 1/12*(1680*(c^8*x^8 + 4*b*c^7*x^7 + 4*a^3*b*c
^4*x + a^4*c^4 + 2*(3*b^2*c^6 + 2*a*c^7)*x^6 + 4*(b^3*c^5 + 3*a*b*c^6)*x^5 + (b^
4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*x^4 + 4*(a*b^3*c^4 + 3*a^2*b*c^5)*x^3 + 2*(3*a
^2*b^2*c^4 + 2*a^3*c^5)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c
)) + (840*c^7*x^7 + 2940*b*c^6*x^6 - 3*b^7 + 50*a*b^5*c - 326*a^2*b^3*c^2 + 1116
*a^3*b*c^3 + 280*(13*b^2*c^5 + 11*a*c^6)*x^5 + 350*(5*b^3*c^4 + 22*a*b*c^5)*x^4
+ 56*(3*b^4*c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*x^3 - 28*(b^5*c^2 - 28*a*b^3*c^3 -
219*a^2*b*c^4)*x^2 + 8*(b^6*c - 19*a*b^4*c^2 + 174*a^2*b^2*c^3 + 279*a^3*c^4)*x
)*sqrt(-b^2 + 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^
3 + 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 2
56*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 +
256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576*a^3*b
^4*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^2 + 48*a^2*
b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 + (b^12 - 4*a*b^
10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 +
1536*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 -
512*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b
^6*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 + 4*(a^3*b^9 - 16*
a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*sqrt(-b^2 + 4*a
*c))]

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Sympy [A]  time = 18.3209, size = 1153, normalized size = 6.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-70*c**4*sqrt(-1/(4*a*c - b**2)**9)*log(x + (-71680*a**5*c**9*sqrt(-1/(4*a*c - b
**2)**9) + 89600*a**4*b**2*c**8*sqrt(-1/(4*a*c - b**2)**9) - 44800*a**3*b**4*c**
7*sqrt(-1/(4*a*c - b**2)**9) + 11200*a**2*b**6*c**6*sqrt(-1/(4*a*c - b**2)**9) -
1400*a*b**8*c**5*sqrt(-1/(4*a*c - b**2)**9) + 70*b**10*c**4*sqrt(-1/(4*a*c - b*
*2)**9) + 70*b*c**4)/(140*c**5)) + 70*c**4*sqrt(-1/(4*a*c - b**2)**9)*log(x + (7
1680*a**5*c**9*sqrt(-1/(4*a*c - b**2)**9) - 89600*a**4*b**2*c**8*sqrt(-1/(4*a*c
- b**2)**9) + 44800*a**3*b**4*c**7*sqrt(-1/(4*a*c - b**2)**9) - 11200*a**2*b**6*
c**6*sqrt(-1/(4*a*c - b**2)**9) + 1400*a*b**8*c**5*sqrt(-1/(4*a*c - b**2)**9) -
70*b**10*c**4*sqrt(-1/(4*a*c - b**2)**9) + 70*b*c**4)/(140*c**5)) + (1116*a**3*b
*c**3 - 326*a**2*b**3*c**2 + 50*a*b**5*c - 3*b**7 + 2940*b*c**6*x**6 + 840*c**7*
x**7 + x**5*(3080*a*c**6 + 3640*b**2*c**5) + x**4*(7700*a*b*c**5 + 1750*b**3*c**
4) + x**3*(4088*a**2*c**5 + 5656*a*b**2*c**4 + 168*b**4*c**3) + x**2*(6132*a**2*
b*c**4 + 784*a*b**3*c**3 - 28*b**5*c**2) + x*(2232*a**3*c**4 + 1392*a**2*b**2*c*
*3 - 152*a*b**4*c**2 + 8*b**6*c))/(3072*a**8*c**4 - 3072*a**7*b**2*c**3 + 1152*a
**6*b**4*c**2 - 192*a**5*b**6*c + 12*a**4*b**8 + x**8*(3072*a**4*c**8 - 3072*a**
3*b**2*c**7 + 1152*a**2*b**4*c**6 - 192*a*b**6*c**5 + 12*b**8*c**4) + x**7*(1228
8*a**4*b*c**7 - 12288*a**3*b**3*c**6 + 4608*a**2*b**5*c**5 - 768*a*b**7*c**4 + 4
8*b**9*c**3) + x**6*(12288*a**5*c**7 + 6144*a**4*b**2*c**6 - 13824*a**3*b**4*c**
5 + 6144*a**2*b**6*c**4 - 1104*a*b**8*c**3 + 72*b**10*c**2) + x**5*(36864*a**5*b
*c**6 - 24576*a**4*b**3*c**5 + 1536*a**3*b**5*c**4 + 2304*a**2*b**7*c**3 - 624*a
*b**9*c**2 + 48*b**11*c) + x**4*(18432*a**6*c**6 + 18432*a**5*b**2*c**5 - 26880*
a**4*b**4*c**4 + 9600*a**3*b**6*c**3 - 1080*a**2*b**8*c**2 - 48*a*b**10*c + 12*b
**12) + x**3*(36864*a**6*b*c**5 - 24576*a**5*b**3*c**4 + 1536*a**4*b**5*c**3 + 2
304*a**3*b**7*c**2 - 624*a**2*b**9*c + 48*a*b**11) + x**2*(12288*a**7*c**5 + 614
4*a**6*b**2*c**4 - 13824*a**5*b**4*c**3 + 6144*a**4*b**6*c**2 - 1104*a**3*b**8*c
+ 72*a**2*b**10) + x*(12288*a**7*b*c**4 - 12288*a**6*b**3*c**3 + 4608*a**5*b**5
*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

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GIAC/XCAS [A]  time = 0.203752, size = 454, normalized size = 2.65 \[ \frac{140 \, c^{4} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{840 \, c^{7} x^{7} + 2940 \, b c^{6} x^{6} + 3640 \, b^{2} c^{5} x^{5} + 3080 \, a c^{6} x^{5} + 1750 \, b^{3} c^{4} x^{4} + 7700 \, a b c^{5} x^{4} + 168 \, b^{4} c^{3} x^{3} + 5656 \, a b^{2} c^{4} x^{3} + 4088 \, a^{2} c^{5} x^{3} - 28 \, b^{5} c^{2} x^{2} + 784 \, a b^{3} c^{3} x^{2} + 6132 \, a^{2} b c^{4} x^{2} + 8 \, b^{6} c x - 152 \, a b^{4} c^{2} x + 1392 \, a^{2} b^{2} c^{3} x + 2232 \, a^{3} c^{4} x - 3 \, b^{7} + 50 \, a b^{5} c - 326 \, a^{2} b^{3} c^{2} + 1116 \, a^{3} b c^{3}}{12 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

140*c^4*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c
^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*c^7*x^7 + 29
40*b*c^6*x^6 + 3640*b^2*c^5*x^5 + 3080*a*c^6*x^5 + 1750*b^3*c^4*x^4 + 7700*a*b*c
^5*x^4 + 168*b^4*c^3*x^3 + 5656*a*b^2*c^4*x^3 + 4088*a^2*c^5*x^3 - 28*b^5*c^2*x^
2 + 784*a*b^3*c^3*x^2 + 6132*a^2*b*c^4*x^2 + 8*b^6*c*x - 152*a*b^4*c^2*x + 1392*
a^2*b^2*c^3*x + 2232*a^3*c^4*x - 3*b^7 + 50*a*b^5*c - 326*a^2*b^3*c^2 + 1116*a^3
*b*c^3)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*(c*
x^2 + b*x + a)^4)