3.2213 \(\int \frac {x^5}{(a+b x+c x^2)^4} \, dx\)

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]

-1/3*x^5*(2*c*x+b)/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+5/6*b*x^3*(b*x+2*a)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2-5*a*b*x*(b*
x+2*a)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)-20*a^2*b*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)

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Rubi [A]  time = 0.06, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {728, 722, 618, 206} \[ -\frac {20 a^2 b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )} \]

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)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 722

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[(m*(2*c*d - b*e))/((p + 1)*(b^2 - 4*a*c)),
Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b x+c x^2\right )^4} \, dx &=-\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) \int \frac {x^4}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\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}-\frac {(5 a b) \int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\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}-\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 {\left (10 a^2 b\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\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}-\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 {\left (20 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^3}\\ &=-\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}-\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 {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}}\\ \end {align*}

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Mathematica [A]  time = 0.27, 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)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))
/6

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fricas [B]  time = 0.65, size = 1594, normalized size = 10.99 \[ \text {result too large to display} \]

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*(a^3*b^6 - 22*a^4*b^4*c + 8*a^5*b^2*c^2 + 256*a^6*c^3 - 60*(a^2*b^3*c^4 - 4*a^3*b*c^5)*x^5 + 3*(b^8*c -
16*a*b^6*c^2 + 46*a^2*b^4*c^3 - 56*a^3*b^2*c^4 + 256*a^4*c^5)*x^4 + (b^9 - 16*a*b^7*c - 14*a^2*b^5*c^2 + 24*a^
3*b^3*c^3 + 896*a^4*b*c^4)*x^3 + 3*(a*b^8 - 21*a^2*b^6*c + 36*a^3*b^4*c^2 + 64*a^4*b^2*c^3 + 256*a^5*c^4)*x^2
+ 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)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x +
b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 3*(a^2*b^7 - 22*a^3*b^5*c + 28*a^4*b^3*c^2 +
176*a^5*b*c^3)*x)/(a^3*b^8*c^2 - 16*a^4*b^6*c^3 + 96*a^5*b^4*c^4 - 256*a^6*b^2*c^5 + 256*a^7*c^6 + (b^8*c^5 -
16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^6 + 3*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c
^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^5 + 3*(b^10*c^3 - 15*a*b^8*c^4 + 80*a^2*b^6*c^5 - 160*a^3*b^4*c^6 + 25
6*a^5*c^8)*x^4 + (b^11*c^2 - 10*a*b^9*c^3 + 320*a^3*b^5*c^5 - 1280*a^4*b^3*c^6 + 1536*a^5*b*c^7)*x^3 + 3*(a*b^
10*c^2 - 15*a^2*b^8*c^3 + 80*a^3*b^6*c^4 - 160*a^4*b^4*c^5 + 256*a^6*c^7)*x^2 + 3*(a^2*b^9*c^2 - 16*a^3*b^7*c^
3 + 96*a^4*b^5*c^4 - 256*a^5*b^3*c^5 + 256*a^6*b*c^6)*x), -1/6*(a^3*b^6 - 22*a^4*b^4*c + 8*a^5*b^2*c^2 + 256*a
^6*c^3 - 60*(a^2*b^3*c^4 - 4*a^3*b*c^5)*x^5 + 3*(b^8*c - 16*a*b^6*c^2 + 46*a^2*b^4*c^3 - 56*a^3*b^2*c^4 + 256*
a^4*c^5)*x^4 + (b^9 - 16*a*b^7*c - 14*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 896*a^4*b*c^4)*x^3 + 3*(a*b^8 - 21*a^2*b^
6*c + 36*a^3*b^4*c^2 + 64*a^4*b^2*c^3 + 256*a^5*c^4)*x^2 + 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)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 3*(a^2*b^7 - 22*a^3*b^
5*c + 28*a^4*b^3*c^2 + 176*a^5*b*c^3)*x)/(a^3*b^8*c^2 - 16*a^4*b^6*c^3 + 96*a^5*b^4*c^4 - 256*a^6*b^2*c^5 + 25
6*a^7*c^6 + (b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^6 + 3*(b^9*c^4 - 16*a*
b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^5 + 3*(b^10*c^3 - 15*a*b^8*c^4 + 80*a^2*b^6*c^5
- 160*a^3*b^4*c^6 + 256*a^5*c^8)*x^4 + (b^11*c^2 - 10*a*b^9*c^3 + 320*a^3*b^5*c^5 - 1280*a^4*b^3*c^6 + 1536*a^
5*b*c^7)*x^3 + 3*(a*b^10*c^2 - 15*a^2*b^8*c^3 + 80*a^3*b^6*c^4 - 160*a^4*b^4*c^5 + 256*a^6*c^7)*x^2 + 3*(a^2*b
^9*c^2 - 16*a^3*b^7*c^3 + 96*a^4*b^5*c^4 - 256*a^5*b^3*c^5 + 256*a^6*b*c^6)*x)]

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giac [B]  time = 0.25, size = 326, normalized size = 2.25 \[ \frac {20 \, a^{2} b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {60 \, a^{2} b c^{4} x^{5} - 3 \, b^{6} c x^{4} + 36 \, a b^{4} c^{2} x^{4} + 6 \, a^{2} b^{2} c^{3} x^{4} + 192 \, a^{3} c^{4} x^{4} - b^{7} x^{3} + 12 \, a b^{5} c x^{3} + 62 \, a^{2} b^{3} c^{2} x^{3} + 224 \, a^{3} b c^{3} x^{3} - 3 \, a b^{6} x^{2} + 51 \, a^{2} b^{4} c x^{2} + 96 \, a^{3} b^{2} c^{2} x^{2} + 192 \, a^{4} c^{3} x^{2} - 3 \, a^{2} b^{5} x + 54 \, a^{3} b^{3} c x + 132 \, a^{4} b c^{2} x - a^{3} b^{4} + 18 \, a^{4} b^{2} c + 64 \, a^{5} c^{2}}{6 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \]

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)

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maple [B]  time = 0.07, size = 486, normalized size = 3.35 \[ -\frac {20 a^{2} b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}+\frac {-\frac {10 a^{2} b \,c^{2} x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (64 a^{3} c^{3}+2 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c}-\frac {\left (44 a^{2} c^{2}+18 a \,b^{2} c -b^{4}\right ) a^{2} b x}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (224 a^{3} c^{3}+62 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) b \,x^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (64 a^{2} c^{2}+18 a \,b^{2} c -b^{4}\right ) a^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (64 a^{3} c^{3}+32 a^{2} b^{2} c^{2}+17 a \,b^{4} c -b^{6}\right ) a \,x^{2}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}}{\left (c \,x^{2}+b x +a \right )^{3}} \]

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)/(6
4*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c*x^4-1/6*b*(224*a^3*c^3+62*a^2*b^2*c^2+12*a*b^4*c-b^6)/(64*a^3*c^3-4
8*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2*x^3-1/2*a*(64*a^3*c^3+32*a^2*b^2*c^2+17*a*b^4*c-b^6)/(64*a^3*c^3-48*a^2*b^2*
c^2+12*a*b^4*c-b^6)/c^2*x^2-1/2*a^2*b*(44*a^2*c^2+18*a*b^2*c-b^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c
^2*x-1/6*a^3*(64*a^2*c^2+18*a*b^2*c-b^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2)/(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(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 0.93, size = 563, normalized size = 3.88 \[ \frac {\frac {a^3\,\left (64\,a^2\,c^2+18\,a\,b^2\,c-b^4\right )}{6\,c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^4\,\left (64\,a^3\,c^3+2\,a^2\,b^2\,c^2+12\,a\,b^4\,c-b^6\right )}{2\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {10\,a^2\,b\,c^2\,x^5}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {a\,x^2\,\left (64\,a^3\,c^3+32\,a^2\,b^2\,c^2+17\,a\,b^4\,c-b^6\right )}{2\,c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {b\,x^3\,\left (224\,a^3\,c^3+62\,a^2\,b^2\,c^2+12\,a\,b^4\,c-b^6\right )}{6\,c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a^2\,b\,x\,\left (44\,a^2\,c^2+18\,a\,b^2\,c-b^4\right )}{2\,c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}+\frac {20\,a^2\,b\,\mathrm {atan}\left (\frac {\left (\frac {10\,a^2\,b^2}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {20\,a^2\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^{7/2}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{10\,a^2\,b}\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x + c*x^2)^4,x)

[Out]

((a^3*(64*a^2*c^2 - b^4 + 18*a*b^2*c))/(6*c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^4*(64*a^3
*c^3 - b^6 + 2*a^2*b^2*c^2 + 12*a*b^4*c))/(2*c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (10*a^2*b*c
^2*x^5)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (a*x^2*(64*a^3*c^3 - b^6 + 32*a^2*b^2*c^2 + 17*a*b^
4*c))/(2*c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (b*x^3*(224*a^3*c^3 - b^6 + 62*a^2*b^2*c^2 +
12*a*b^4*c))/(6*c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (a^2*b*x*(44*a^2*c^2 - b^4 + 18*a*b^2*
c))/(2*c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3*b^2*
c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) + (20*a^2*b*atan((((10*a^2*b^2)/(4*a*c - b
^2)^(7/2) + (20*a^2*b*c*x)/(4*a*c - b^2)^(7/2))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(10*a^2*b)))
/(4*a*c - b^2)^(7/2)

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sympy [B]  time = 2.63, size = 898, normalized size = 6.19 \[ 10 a^{2} b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 2560 a^{6} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 2560 a^{5} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 960 a^{4} b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 160 a^{3} b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 10 a^{2} b^{9} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 10 a^{2} b^{2}}{20 a^{2} b c} \right )} - 10 a^{2} b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {2560 a^{6} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 2560 a^{5} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 960 a^{4} b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 160 a^{3} b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 10 a^{2} b^{9} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 10 a^{2} b^{2}}{20 a^{2} b c} \right )} + \frac {- 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} \left (- 192 a^{3} c^{4} - 6 a^{2} b^{2} c^{3} - 36 a b^{4} c^{2} + 3 b^{6} c\right ) + x^{3} \left (- 224 a^{3} b c^{3} - 62 a^{2} b^{3} c^{2} - 12 a b^{5} c + b^{7}\right ) + x^{2} \left (- 192 a^{4} c^{3} - 96 a^{3} b^{2} c^{2} - 51 a^{2} b^{4} c + 3 a b^{6}\right ) + x \left (- 132 a^{4} b c^{2} - 54 a^{3} b^{3} c + 3 a^{2} b^{5}\right )}{384 a^{6} c^{5} - 288 a^{5} b^{2} c^{4} + 72 a^{4} b^{4} c^{3} - 6 a^{3} b^{6} c^{2} + x^{6} \left (384 a^{3} c^{8} - 288 a^{2} b^{2} c^{7} + 72 a b^{4} c^{6} - 6 b^{6} c^{5}\right ) + x^{5} \left (1152 a^{3} b c^{7} - 864 a^{2} b^{3} c^{6} + 216 a b^{5} c^{5} - 18 b^{7} c^{4}\right ) + x^{4} \left (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}\right ) + x^{3} \left (2304 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}\right ) + x^{2} \left (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}\right ) + x \left (1152 a^{5} b c^{5} - 864 a^{4} b^{3} c^{4} + 216 a^{3} b^{5} c^{3} - 18 a^{2} b^{7} c^{2}\right )} \]

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*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)) + (-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 - 288*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 - 864*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*(2304*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 + 216*a**3*b**5*c*
*3 - 18*a**2*b**7*c**2))

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