3.5.38 \(\int \frac {1}{(c+\frac {a}{x^2}+\frac {b}{x})^3 x^7} \, dx\) [438]

Optimal. Leaf size=185 \[ \frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3} \]

[Out]

1/2*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1/2*(2*b^4-15*a*b^2*c+16*a^2*c^2+2*b*c*(-7*a*c+b^2)*x)/a^
2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+b*(30*a^2*c^2-10*a*b^2*c+b^4)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c
+b^2)^(5/2)+ln(x)/a^3-1/2*ln(c*x^2+b*x+a)/a^3

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Rubi [A]
time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1368, 754, 836, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x)}{a^3}+\frac {16 a^2 c^2+2 b c x \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)^3*x^7),x]

[Out]

(b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2
- 7*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)
/Sqrt[b^2 - 4*a*c]])/(a^3*(b^2 - 4*a*c)^(5/2)) + Log[x]/a^3 - Log[a + b*x + c*x^2]/(2*a^3)

Rule 212

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

Rule 632

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 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 754

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

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

Rule 1368

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + 2*n*p)*(c + b/x^n +
a/x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]

Rubi steps

\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^7} \, dx &=\int \frac {1}{x \left (a+b x+c x^2\right )^3} \, dx\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {-2 \left (b^2-4 a c\right )-3 b c x}{x \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}+\frac {\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}-\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^3}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 178, normalized size = 0.96 \begin {gather*} \frac {\frac {a^2 \left (b^2-2 a c+b c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {a \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c x-14 a b c^2 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac {2 b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+2 \log (x)-\log (a+x (b+c x))}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a/x^2 + b/x)^3*x^7),x]

[Out]

((a^2*(b^2 - 2*a*c + b*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3
*c*x - 14*a*b*c^2*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (2*b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2
*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + 2*Log[x] - Log[a + x*(b + c*x)])/(2*a^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(351\) vs. \(2(175)=350\).
time = 0.06, size = 352, normalized size = 1.90

method	result	size
default	\(-\frac {\frac {\frac {a b \,c^{2} \left (7 a c -b^{2}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {c a \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {3 a^{2} \left (8 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (23 a^{2} b \,c^{2}-9 a \,b^{3} c +b^{5}-\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{a^{3}}+\frac {\ln \left (x \right )}{a^{3}}\)	\(352\)
risch	\(\frac {-\frac {b \,c^{2} \left (7 a c -b^{2}\right ) x^{3}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {12 a^{2} c^{2}-\frac {21}{2} a \,b^{2} c +\frac {3}{2} b^{4}}{a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\ln \left (x \right )}{a^{3}}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (1024 a^{8} c^{5}-1280 a^{7} b^{2} c^{4}+640 a^{6} b^{4} c^{3}-160 a^{5} b^{6} c^{2}+20 a^{4} b^{8} c -a^{3} b^{10}\right ) \textit {\_Z}^{2}+\left (1024 a^{5} c^{5}-1280 a^{4} b^{2} c^{4}+640 a^{3} b^{4} c^{3}-160 a^{2} b^{6} c^{2}+20 a \,b^{8} c -b^{10}\right ) \textit {\_Z} +256 a^{2} c^{5}-95 a \,b^{2} c^{4}+10 b^{4} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (1536 a^{9} c^{5}-2048 a^{8} b^{2} c^{4}+1088 a^{7} b^{4} c^{3}-288 a^{6} b^{6} c^{2}+38 a^{5} b^{8} c -2 a^{4} b^{10}\right ) \textit {\_R}^{2}+\left (768 a^{6} c^{5}-656 a^{5} b^{2} c^{4}+216 a^{4} b^{4} c^{3}-33 a^{3} b^{6} c^{2}+2 a^{2} b^{8} c \right ) \textit {\_R} +49 a^{2} b^{2} c^{4}-14 a \,b^{4} c^{3}+c^{2} b^{6}\right ) x +\left (-256 a^{9} b \,c^{4}+256 a^{8} b^{3} c^{3}-96 a^{7} b^{5} c^{2}+16 a^{6} b^{7} c -a^{5} b^{9}\right ) \textit {\_R}^{2}+\left (368 a^{6} b \,c^{4}-328 a^{5} b^{3} c^{3}+111 a^{4} b^{5} c^{2}-17 a^{3} b^{7} c +a^{2} b^{9}\right ) \textit {\_R} -112 a^{3} b \,c^{4}+72 a^{2} b^{3} c^{3}-15 a \,b^{5} c^{2}+b^{7} c \right )\right )\)	\(645\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+a/x^2+b/x)^3/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/a^3*((a*b*c^2*(7*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*c*a*(16*a^2*c^2-29*a*b^2*c+4*b^4)/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^2+a*b*(a^2*c^2+6*a*b^2*c-b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-3/2*a^2*(8*a^2*c^2-7*a*b^2*c+b^4)/
(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^3-8*a*b^2*c^2+b^4*c)/c
*ln(c*x^2+b*x+a)+2*(23*a^2*b*c^2-9*a*b^3*c+b^5-1/2*(16*a^2*c^3-8*a*b^2*c^2+b^4*c)*b/c)/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))))+ln(x)/a^3

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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(1/(c+a/x^2+b/x)^3/x^7,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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (175) = 350\).
time = 0.61, size = 1985, normalized size = 10.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^7,x, algorithm="fricas")

[Out]

[1/2*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*
x^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*x^2 + (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2
+ (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^4 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^3 + (b^7 - 8*a*b^
5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^2 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x)*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)) + 2*(a*b^7 - 10*a^2*b^5
*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*x - (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2 - 12*a
*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^3 + (
b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c
^2 - 64*a^4*b*c^3)*x)*log(c*x^2 + b*x + a) + 2*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^
2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)
*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a
^3*b^3*c^2 - 64*a^4*b*c^3)*x)*log(x))/(a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 + (a^3*b^6*c^2 - 1
2*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*x^4 + 2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c
^4)*x^3 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*x^2 + 2*(a^4*b^7 - 12*a^5*b
^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*x), 1/2*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*
b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*x^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*x^2
+ 2*(a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^4 + 2*(b^6*c - 10*a*b^
4*c^2 + 30*a^2*b^2*c^3)*x^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^2 + 2*(a*b^6 - 10*a^2*b^4*c
+ 30*a^3*b^2*c^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(a*b^7 - 10*
a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*x - (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2
- 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*
x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^
3*b^3*c^2 - 64*a^4*b*c^3)*x)*log(c*x^2 + b*x + a) + 2*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 +
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3
*b*c^4)*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c
+ 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x)*log(x))/(a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 + (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)*x^4 + 2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*
a^6*b*c^4)*x^3 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*x^2 + 2*(a^4*b^7 - 1
2*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*x)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x**2+b/x)**3/x**7,x)

[Out]

Timed out

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Giac [A]
time = 2.94, size = 239, normalized size = 1.29 \begin {gather*} -\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, a^{2} b^{4} - 21 \, a^{3} b^{2} c + 24 \, a^{4} c^{2} + 2 \, {\left (a b^{3} c^{2} - 7 \, a^{2} b c^{3}\right )} x^{3} + {\left (4 \, a b^{4} c - 29 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 6 \, a^{2} b^{3} c - a^{3} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^7,x, algorithm="giac")

[Out]

-(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2
)*sqrt(-b^2 + 4*a*c)) - 1/2*log(c*x^2 + b*x + a)/a^3 + log(abs(x))/a^3 + 1/2*(3*a^2*b^4 - 21*a^3*b^2*c + 24*a^
4*c^2 + 2*(a*b^3*c^2 - 7*a^2*b*c^3)*x^3 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*x^2 + 2*(a*b^5 - 6*a^2*b^3
*c - a^3*b*c^2)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*a^3)

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Mupad [B]
time = 2.46, size = 1089, normalized size = 5.89 \begin {gather*} \frac {\ln \left (x\right )}{a^3}+\frac {\frac {3\,\left (8\,a^2\,c^2-7\,a\,b^2\,c+b^4\right )}{2\,a\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (16\,a^2\,c^3-29\,a\,b^2\,c^2+4\,b^4\,c\right )}{2\,a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {b\,x\,\left (a^2\,c^2+6\,a\,b^2\,c-b^4\right )}{a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {b\,c^2\,x^3\,\left (7\,a\,c-b^2\right )}{a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-\frac {\ln \left (1536\,a^6\,c^5-2\,b^{11}\,x-2\,a\,b^{10}+2\,a\,b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^2\,b^8\,c+2\,b^6\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-303\,a^3\,b^6\,c^2+1160\,a^4\,b^4\,c^3-2160\,a^5\,b^2\,c^4-17\,a^2\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^3\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-321\,a^2\,b^7\,c^2\,x+1286\,a^3\,b^5\,c^3\,x-2560\,a^4\,b^3\,c^4\,x-48\,a^3\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+40\,a\,b^9\,c\,x+2016\,a^5\,b\,c^5\,x-20\,a\,b^4\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (1024\,a^5\,c^5-b^{10}+b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-160\,a^2\,b^6\,c^2+640\,a^3\,b^4\,c^3-1280\,a^4\,b^2\,c^4+20\,a\,b^8\,c+30\,a^2\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^3\,{\left (4\,a\,c-b^2\right )}^5}+\frac {\ln \left (2\,a\,b^{10}+2\,b^{11}\,x-1536\,a^6\,c^5+2\,a\,b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^8\,c+2\,b^6\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^6\,c^2-1160\,a^4\,b^4\,c^3+2160\,a^5\,b^2\,c^4-17\,a^2\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^3\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^7\,c^2\,x-1286\,a^3\,b^5\,c^3\,x+2560\,a^4\,b^3\,c^4\,x-48\,a^3\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-40\,a\,b^9\,c\,x-2016\,a^5\,b\,c^5\,x-20\,a\,b^4\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{10}-1024\,a^5\,c^5+b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+160\,a^2\,b^6\,c^2-640\,a^3\,b^4\,c^3+1280\,a^4\,b^2\,c^4-20\,a\,b^8\,c+30\,a^2\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^3\,{\left (4\,a\,c-b^2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(c + a/x^2 + b/x)^3),x)

[Out]

log(x)/a^3 + ((3*(b^4 + 8*a^2*c^2 - 7*a*b^2*c))/(2*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(4*b^4*c + 16*a^2*
c^3 - 29*a*b^2*c^2))/(2*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*x*(a^2*c^2 - b^4 + 6*a*b^2*c))/(a^2*(b^4 + 16
*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^3*(7*a*c - b^2))/(a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c + b^2) +
a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) - (log(1536*a^6*c^5 - 2*b^11*x - 2*a*b^10 + 2*a*b^5*(-(4*a*c - b^2)^5)^(1
/2) + 39*a^2*b^8*c + 2*b^6*x*(-(4*a*c - b^2)^5)^(1/2) - 303*a^3*b^6*c^2 + 1160*a^4*b^4*c^3 - 2160*a^5*b^2*c^4
- 17*a^2*b^3*c*(-(4*a*c - b^2)^5)^(1/2) + 39*a^3*b*c^2*(-(4*a*c - b^2)^5)^(1/2) - 321*a^2*b^7*c^2*x + 1286*a^3
*b^5*c^3*x - 2560*a^4*b^3*c^4*x - 48*a^3*c^3*x*(-(4*a*c - b^2)^5)^(1/2) + 40*a*b^9*c*x + 2016*a^5*b*c^5*x - 20
*a*b^4*c*x*(-(4*a*c - b^2)^5)^(1/2) + 63*a^2*b^2*c^2*x*(-(4*a*c - b^2)^5)^(1/2))*(1024*a^5*c^5 - b^10 + b^5*(-
(4*a*c - b^2)^5)^(1/2) - 160*a^2*b^6*c^2 + 640*a^3*b^4*c^3 - 1280*a^4*b^2*c^4 + 20*a*b^8*c + 30*a^2*b*c^2*(-(4
*a*c - b^2)^5)^(1/2) - 10*a*b^3*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^3*(4*a*c - b^2)^5) + (log(2*a*b^10 + 2*b^11*
x - 1536*a^6*c^5 + 2*a*b^5*(-(4*a*c - b^2)^5)^(1/2) - 39*a^2*b^8*c + 2*b^6*x*(-(4*a*c - b^2)^5)^(1/2) + 303*a^
3*b^6*c^2 - 1160*a^4*b^4*c^3 + 2160*a^5*b^2*c^4 - 17*a^2*b^3*c*(-(4*a*c - b^2)^5)^(1/2) + 39*a^3*b*c^2*(-(4*a*
c - b^2)^5)^(1/2) + 321*a^2*b^7*c^2*x - 1286*a^3*b^5*c^3*x + 2560*a^4*b^3*c^4*x - 48*a^3*c^3*x*(-(4*a*c - b^2)
^5)^(1/2) - 40*a*b^9*c*x - 2016*a^5*b*c^5*x - 20*a*b^4*c*x*(-(4*a*c - b^2)^5)^(1/2) + 63*a^2*b^2*c^2*x*(-(4*a*
c - b^2)^5)^(1/2))*(b^10 - 1024*a^5*c^5 + b^5*(-(4*a*c - b^2)^5)^(1/2) + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1
280*a^4*b^2*c^4 - 20*a*b^8*c + 30*a^2*b*c^2*(-(4*a*c - b^2)^5)^(1/2) - 10*a*b^3*c*(-(4*a*c - b^2)^5)^(1/2)))/(
2*a^3*(4*a*c - b^2)^5)

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