3.19.81 \(\int \frac {x^6}{(a+b x+c x^2)^4} \, dx\)

Optimal. Leaf size=146 \[ \frac {40 a^3 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {722, 618, 206} \begin {gather*} \frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 a^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 {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x + c*x^2)^4,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx &=\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {(10 a) \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 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {\left (10 a^2\right ) \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 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 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^3\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (40 a^3\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 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 a^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}}\\ \end {align*}

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Mathematica [B]  time = 0.20, size = 314, normalized size = 2.15 \begin {gather*} \frac {40 a^3 \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 {a^3 c^2 (5 b-2 c x)+a^2 b^2 c (9 c x-5 b)+a b^4 (b-6 c x)+b^6 x}{3 c^5 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac {74 a^3 b c^3-44 a^3 c^4 x-48 a^2 b^3 c^2+48 a^2 b^2 c^3 x+12 a b^5 c-12 a b^4 c^2 x-b^7+b^6 c x}{c^4 \left (4 a c-b^2\right )^3 (a+x (b+c x))}+\frac {-59 a^3 b c^3+26 a^3 c^4 x+48 a^2 b^3 c^2-72 a^2 b^2 c^3 x-12 a b^5 c+33 a b^4 c^2 x+b^7-4 b^6 c x}{3 c^5 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x + c*x^2)^4,x]

[Out]

(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 59*a^3*b*c^3 - 4*b^6*c*x + 33*a*b^4*c^2*x - 72*a^2*b^2*c^3*x + 26*a^3*c^4
*x)/(3*c^5*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (-b^7 + 12*a*b^5*c - 48*a^2*b^3*c^2 + 74*a^3*b*c^3 + b^6*c*x
- 12*a*b^4*c^2*x + 48*a^2*b^2*c^3*x - 44*a^3*c^4*x)/(c^4*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (b^6*x + a*b^4
*(b - 6*c*x) + a^3*c^2*(5*b - 2*c*x) + a^2*b^2*c*(-5*b + 9*c*x))/(3*c^5*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) +
(40*a^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^6/(a + b*x + c*x^2)^4,x]

[Out]

IntegrateAlgebraic[x^6/(a + b*x + c*x^2)^4, x]

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fricas [B]  time = 0.44, size = 1675, normalized size = 11.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2+b*x+a)^4,x, algorithm="fricas")

[Out]

[-1/3*(a^3*b^7 - 17*a^4*b^5*c + 118*a^5*b^3*c^2 - 264*a^6*b*c^3 + 3*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 -
236*a^3*b^2*c^5 + 176*a^4*c^6)*x^5 + 3*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 206*a^3*b^3*c^4 + 56*a^4*b*c^
5)*x^4 + (b^10 - 11*a*b^8*c + 16*a^2*b^6*c^2 + 334*a^3*b^4*c^3 - 1304*a^4*b^2*c^4 + 640*a^5*c^5)*x^3 + 3*(a*b^
9 - 16*a^2*b^7*c + 101*a^3*b^5*c^2 - 196*a^4*b^3*c^3 - 64*a^5*b*c^4)*x^2 + 60*(a^3*c^6*x^6 + 3*a^3*b*c^5*x^5 +
3*a^5*b*c^3*x + a^6*c^3 + 3*(a^3*b^2*c^4 + a^4*c^5)*x^4 + (a^3*b^3*c^3 + 6*a^4*b*c^4)*x^3 + 3*(a^4*b^2*c^3 +
a^5*c^4)*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^8 - 17*a^3*b^6*c + 118*a^4*b^4*c^2 - 284*a^5*b^2*c^3 + 80*a^6*c^4)*x)/(a^3*b^8*c^3 - 1
6*a^4*b^6*c^4 + 96*a^5*b^4*c^5 - 256*a^6*b^2*c^6 + 256*a^7*c^7 + (b^8*c^6 - 16*a*b^6*c^7 + 96*a^2*b^4*c^8 - 25
6*a^3*b^2*c^9 + 256*a^4*c^10)*x^6 + 3*(b^9*c^5 - 16*a*b^7*c^6 + 96*a^2*b^5*c^7 - 256*a^3*b^3*c^8 + 256*a^4*b*c
^9)*x^5 + 3*(b^10*c^4 - 15*a*b^8*c^5 + 80*a^2*b^6*c^6 - 160*a^3*b^4*c^7 + 256*a^5*c^9)*x^4 + (b^11*c^3 - 10*a*
b^9*c^4 + 320*a^3*b^5*c^6 - 1280*a^4*b^3*c^7 + 1536*a^5*b*c^8)*x^3 + 3*(a*b^10*c^3 - 15*a^2*b^8*c^4 + 80*a^3*b
^6*c^5 - 160*a^4*b^4*c^6 + 256*a^6*c^8)*x^2 + 3*(a^2*b^9*c^3 - 16*a^3*b^7*c^4 + 96*a^4*b^5*c^5 - 256*a^5*b^3*c
^6 + 256*a^6*b*c^7)*x), -1/3*(a^3*b^7 - 17*a^4*b^5*c + 118*a^5*b^3*c^2 - 264*a^6*b*c^3 + 3*(b^8*c^2 - 16*a*b^6
*c^3 + 96*a^2*b^4*c^4 - 236*a^3*b^2*c^5 + 176*a^4*c^6)*x^5 + 3*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 206*a^
3*b^3*c^4 + 56*a^4*b*c^5)*x^4 + (b^10 - 11*a*b^8*c + 16*a^2*b^6*c^2 + 334*a^3*b^4*c^3 - 1304*a^4*b^2*c^4 + 640
*a^5*c^5)*x^3 + 3*(a*b^9 - 16*a^2*b^7*c + 101*a^3*b^5*c^2 - 196*a^4*b^3*c^3 - 64*a^5*b*c^4)*x^2 - 120*(a^3*c^6
*x^6 + 3*a^3*b*c^5*x^5 + 3*a^5*b*c^3*x + a^6*c^3 + 3*(a^3*b^2*c^4 + a^4*c^5)*x^4 + (a^3*b^3*c^3 + 6*a^4*b*c^4)
*x^3 + 3*(a^4*b^2*c^3 + a^5*c^4)*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^8 - 17*a^3*b^6*c + 118*a^4*b^4*c^2 - 284*a^5*b^2*c^3 + 80*a^6*c^4)*x)/(a^3*b^8*c^3 - 16*a^4*b^6*c^
4 + 96*a^5*b^4*c^5 - 256*a^6*b^2*c^6 + 256*a^7*c^7 + (b^8*c^6 - 16*a*b^6*c^7 + 96*a^2*b^4*c^8 - 256*a^3*b^2*c^
9 + 256*a^4*c^10)*x^6 + 3*(b^9*c^5 - 16*a*b^7*c^6 + 96*a^2*b^5*c^7 - 256*a^3*b^3*c^8 + 256*a^4*b*c^9)*x^5 + 3*
(b^10*c^4 - 15*a*b^8*c^5 + 80*a^2*b^6*c^6 - 160*a^3*b^4*c^7 + 256*a^5*c^9)*x^4 + (b^11*c^3 - 10*a*b^9*c^4 + 32
0*a^3*b^5*c^6 - 1280*a^4*b^3*c^7 + 1536*a^5*b*c^8)*x^3 + 3*(a*b^10*c^3 - 15*a^2*b^8*c^4 + 80*a^3*b^6*c^5 - 160
*a^4*b^4*c^6 + 256*a^6*c^8)*x^2 + 3*(a^2*b^9*c^3 - 16*a^3*b^7*c^4 + 96*a^4*b^5*c^5 - 256*a^5*b^3*c^6 + 256*a^6
*b*c^7)*x)]

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giac [B]  time = 0.20, size = 386, normalized size = 2.64 \begin {gather*} -\frac {40 \, a^{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 {3 \, b^{6} c^{2} x^{5} - 36 \, a b^{4} c^{3} x^{5} + 144 \, a^{2} b^{2} c^{4} x^{5} - 132 \, a^{3} c^{5} x^{5} + 3 \, b^{7} c x^{4} - 36 \, a b^{5} c^{2} x^{4} + 144 \, a^{2} b^{3} c^{3} x^{4} - 42 \, a^{3} b c^{4} x^{4} + b^{8} x^{3} - 7 \, a b^{6} c x^{3} - 12 \, a^{2} b^{4} c^{2} x^{3} + 286 \, a^{3} b^{2} c^{3} x^{3} - 160 \, a^{4} c^{4} x^{3} + 3 \, a b^{7} x^{2} - 36 \, a^{2} b^{5} c x^{2} + 159 \, a^{3} b^{3} c^{2} x^{2} + 48 \, a^{4} b c^{3} x^{2} + 3 \, a^{2} b^{6} x - 39 \, a^{3} b^{4} c x + 198 \, a^{4} b^{2} c^{2} x - 60 \, a^{5} c^{3} x + a^{3} b^{5} - 13 \, a^{4} b^{3} c + 66 \, a^{5} b c^{2}}{3 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

-40*a^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*(3*b^6*c^2*x^5 - 36*a*b^4*c^3*x^5 + 144*a^2*b^2*c^4*x^5 - 132*a^3*c^5*x^5 + 3*b^7*c*x^4 - 36*a*b^
5*c^2*x^4 + 144*a^2*b^3*c^3*x^4 - 42*a^3*b*c^4*x^4 + b^8*x^3 - 7*a*b^6*c*x^3 - 12*a^2*b^4*c^2*x^3 + 286*a^3*b^
2*c^3*x^3 - 160*a^4*c^4*x^3 + 3*a*b^7*x^2 - 36*a^2*b^5*c*x^2 + 159*a^3*b^3*c^2*x^2 + 48*a^4*b*c^3*x^2 + 3*a^2*
b^6*x - 39*a^3*b^4*c*x + 198*a^4*b^2*c^2*x - 60*a^5*c^3*x + a^3*b^5 - 13*a^4*b^3*c + 66*a^5*b*c^2)/((b^6*c^3 -
12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*(c*x^2 + b*x + a)^3)

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maple [B]  time = 0.06, size = 531, normalized size = 3.64 \begin {gather*} \frac {40 a^{3} \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 {\left (44 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{5}}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c}-\frac {\left (14 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) b \,x^{4}}{\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 (66 a^{2} c^{2}-13 a \,b^{2} c +b^{4}\right ) a^{3} b}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {\left (16 a^{3} c^{3}+53 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) a 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^{3}}-\frac {\left (20 a^{3} c^{3}-66 a^{2} b^{2} c^{2}+13 a \,b^{4} c -b^{6}\right ) a^{2} x}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}-\frac {\left (160 a^{4} c^{4}-286 a^{3} b^{2} c^{3}+12 a^{2} b^{4} c^{2}+7 a \,b^{6} c -b^{8}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}}{\left (c \,x^{2}+b x +a \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(c*x^2+b*x+a)^4,x)

[Out]

(-(44*a^3*c^3-48*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^5-b*(14*a^3*c^3-48
*a^2*b^2*c^2+12*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^4-1/3/c^3*(160*a^4*c^4-286*a^3*b
^2*c^3+12*a^2*b^4*c^2+7*a*b^6*c-b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b/c^3*a*(16*a^3*c^3+53*a^2
*b^2*c^2-12*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)*x^2-a^2*(20*a^3*c^3-66*a^2*b^2*c^2+13*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^3*x+1/3*(66*a^2*c^2-13*a*b^2*c+b^4)*a^3*b/c^3/(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+40*a^3/(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(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 = 0.98, size = 656, normalized size = 4.49 \begin {gather*} -\frac {\frac {a^3\,\left (66\,a^2\,b\,c^2-13\,a\,b^3\,c+b^5\right )}{3\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^5\,\left (-44\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {x^3\,\left (160\,a^4\,c^4-286\,a^3\,b^2\,c^3+12\,a^2\,b^4\,c^2+7\,a\,b^6\,c-b^8\right )}{3\,c^3\,\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 (-14\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{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\,x^2\,\left (16\,a^3\,b\,c^3+53\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a^2\,x\,\left (-20\,a^3\,c^3+66\,a^2\,b^2\,c^2-13\,a\,b^4\,c+b^6\right )}{c^3\,\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 {40\,a^3\,\mathrm {atan}\left (\frac {\left (\frac {20\,a^3\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {40\,a^3\,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 )}{20\,a^3}\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x + c*x^2)^4,x)

[Out]

- ((a^3*(b^5 + 66*a^2*b*c^2 - 13*a*b^3*c))/(3*c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^5*(b^
6 - 44*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^3*(16
0*a^4*c^4 - b^8 + 12*a^2*b^4*c^2 - 286*a^3*b^2*c^3 + 7*a*b^6*c))/(3*c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 1
2*a*b^4*c)) + (x^4*(b^7 - 14*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/(c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2
- 12*a*b^4*c)) + (a*x^2*(b^7 + 16*a^3*b*c^3 + 53*a^2*b^3*c^2 - 12*a*b^5*c))/(c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b
^2*c^2 - 12*a*b^4*c)) + (a^2*x*(b^6 - 20*a^3*c^3 + 66*a^2*b^2*c^2 - 13*a*b^4*c))/(c^3*(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) - (40*a^3*atan((((20*a^3*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/
((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)) + (40*a^3*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))/(20*a^3)))/(4*a*c - b^2)^(7/2)

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sympy [B]  time = 3.48, size = 938, normalized size = 6.42 \begin {gather*} - 20 a^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 5120 a^{7} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 5120 a^{6} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 1920 a^{5} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 320 a^{4} b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 20 a^{3} b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b}{40 a^{3} c} \right )} + 20 a^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {5120 a^{7} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 5120 a^{6} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 1920 a^{5} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 320 a^{4} b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b}{40 a^{3} c} \right )} + \frac {66 a^{5} b c^{2} - 13 a^{4} b^{3} c + a^{3} b^{5} + x^{5} \left (- 132 a^{3} c^{5} + 144 a^{2} b^{2} c^{4} - 36 a b^{4} c^{3} + 3 b^{6} c^{2}\right ) + x^{4} \left (- 42 a^{3} b c^{4} + 144 a^{2} b^{3} c^{3} - 36 a b^{5} c^{2} + 3 b^{7} c\right ) + x^{3} \left (- 160 a^{4} c^{4} + 286 a^{3} b^{2} c^{3} - 12 a^{2} b^{4} c^{2} - 7 a b^{6} c + b^{8}\right ) + x^{2} \left (48 a^{4} b c^{3} + 159 a^{3} b^{3} c^{2} - 36 a^{2} b^{5} c + 3 a b^{7}\right ) + x \left (- 60 a^{5} c^{3} + 198 a^{4} b^{2} c^{2} - 39 a^{3} b^{4} c + 3 a^{2} b^{6}\right )}{192 a^{6} c^{6} - 144 a^{5} b^{2} c^{5} + 36 a^{4} b^{4} c^{4} - 3 a^{3} b^{6} c^{3} + x^{6} \left (192 a^{3} c^{9} - 144 a^{2} b^{2} c^{8} + 36 a b^{4} c^{7} - 3 b^{6} c^{6}\right ) + x^{5} \left (576 a^{3} b c^{8} - 432 a^{2} b^{3} c^{7} + 108 a b^{5} c^{6} - 9 b^{7} c^{5}\right ) + x^{4} \left (576 a^{4} c^{8} + 144 a^{3} b^{2} c^{7} - 324 a^{2} b^{4} c^{6} + 99 a b^{6} c^{5} - 9 b^{8} c^{4}\right ) + x^{3} \left (1152 a^{4} b c^{7} - 672 a^{3} b^{3} c^{6} + 72 a^{2} b^{5} c^{5} + 18 a b^{7} c^{4} - 3 b^{9} c^{3}\right ) + x^{2} \left (576 a^{5} c^{7} + 144 a^{4} b^{2} c^{6} - 324 a^{3} b^{4} c^{5} + 99 a^{2} b^{6} c^{4} - 9 a b^{8} c^{3}\right ) + x \left (576 a^{5} b c^{6} - 432 a^{4} b^{3} c^{5} + 108 a^{3} b^{5} c^{4} - 9 a^{2} b^{7} c^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**2+b*x+a)**4,x)

[Out]

-20*a**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-5120*a**7*c**4*sqrt(-1/(4*a*c - b**2)**7) + 5120*a**6*b**2*c**3*
sqrt(-1/(4*a*c - b**2)**7) - 1920*a**5*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7) + 320*a**4*b**6*c*sqrt(-1/(4*a*c -
b**2)**7) - 20*a**3*b**8*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b)/(40*a**3*c)) + 20*a**3*sqrt(-1/(4*a*c - b**2
)**7)*log(x + (5120*a**7*c**4*sqrt(-1/(4*a*c - b**2)**7) - 5120*a**6*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7) + 19
20*a**5*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7) - 320*a**4*b**6*c*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b**8*sqrt(
-1/(4*a*c - b**2)**7) + 20*a**3*b)/(40*a**3*c)) + (66*a**5*b*c**2 - 13*a**4*b**3*c + a**3*b**5 + x**5*(-132*a*
*3*c**5 + 144*a**2*b**2*c**4 - 36*a*b**4*c**3 + 3*b**6*c**2) + x**4*(-42*a**3*b*c**4 + 144*a**2*b**3*c**3 - 36
*a*b**5*c**2 + 3*b**7*c) + x**3*(-160*a**4*c**4 + 286*a**3*b**2*c**3 - 12*a**2*b**4*c**2 - 7*a*b**6*c + b**8)
+ x**2*(48*a**4*b*c**3 + 159*a**3*b**3*c**2 - 36*a**2*b**5*c + 3*a*b**7) + x*(-60*a**5*c**3 + 198*a**4*b**2*c*
*2 - 39*a**3*b**4*c + 3*a**2*b**6))/(192*a**6*c**6 - 144*a**5*b**2*c**5 + 36*a**4*b**4*c**4 - 3*a**3*b**6*c**3
+ x**6*(192*a**3*c**9 - 144*a**2*b**2*c**8 + 36*a*b**4*c**7 - 3*b**6*c**6) + x**5*(576*a**3*b*c**8 - 432*a**2
*b**3*c**7 + 108*a*b**5*c**6 - 9*b**7*c**5) + x**4*(576*a**4*c**8 + 144*a**3*b**2*c**7 - 324*a**2*b**4*c**6 +
99*a*b**6*c**5 - 9*b**8*c**4) + x**3*(1152*a**4*b*c**7 - 672*a**3*b**3*c**6 + 72*a**2*b**5*c**5 + 18*a*b**7*c*
*4 - 3*b**9*c**3) + x**2*(576*a**5*c**7 + 144*a**4*b**2*c**6 - 324*a**3*b**4*c**5 + 99*a**2*b**6*c**4 - 9*a*b*
*8*c**3) + x*(576*a**5*b*c**6 - 432*a**4*b**3*c**5 + 108*a**3*b**5*c**4 - 9*a**2*b**7*c**3))

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