3.438 \(\int \frac{1}{(c+\frac{a}{x^2}+\frac{b}{x})^3 x^7} \, dx\)

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

[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)

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Rubi [A]  time = 0.220315, antiderivative size = 185, normalized size of antiderivative = 1., 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 = {1354, 740, 822, 800, 634, 618, 206, 628} \[ \frac{16 a^2 c^2+2 b c x \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}-\frac{\log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x)}{a^3}+\frac{-2 a c+b^2+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

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 1354

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]

Rule 740

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 822

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 800

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 634

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 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 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 628

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]

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 ) \operatorname{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*}

Mathematica [A]  time = 0.365448, size = 178, normalized size = 0.96 \[ \frac{\frac{a \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x+2 b^3 c x+2 b^4\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac{2 b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{a^2 \left (-2 a c+b^2+b c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\log (a+x (b+c x))+2 \log (x)}{2 a^3} \]

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]  time = 0.016, size = 781, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)/a^3-7/a/(c*x^2+b*x+a)^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/a^2/(c*x^2+b*x+a)^2*b^3*c^2/(16*a^2*c^2-8
*a*b^2*c+b^4)*x^3+8/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-29/2/a/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^2*b^2+2/a^2/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^4-1/(c*x^2+b*x+a)^2*b/(16*a^2*
c^2-8*a*b^2*c+b^4)*x*c^2-6/a/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c+1/a^2/(c*x^2+b*x+a)^2*b^5/(16*
a^2*c^2-8*a*b^2*c+b^4)*x+12*a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2-21/2/(c*x^2+b*x+a)^2/(16*a^2*c^2-
8*a*b^2*c+b^4)*b^2*c+3/2/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4-8/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*l
n(c*x^2+b*x+a)+4/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^2+b*x+a)*b^2-1/2/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*
x^2+b*x+a)*b^4-30/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2+10/
a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c-1/a^3/(16*a^2*c^2-8
*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 4.5515, size = 4228, normalized size = 22.85 \begin{align*} \text{result too large to display} \end{align*}

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)]

________________________________________________________________________________________

Sympy [B]  time = 21.887, size = 4862, normalized size = 26.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))*log(x + (98304*a**14*c**8*(-b
*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 211968*a**13*b**2*c**7*(-b*
sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3))**2 + 196352*a**12*b**4*c**6*(-b*s
qrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3))**2 - 102528*a**11*b**6*c**5*(-b*sq
rt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 49152*a**11*c**8*(-b*sqrt(-(4*
a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 33120*a**10*b**8*c**4*(-b*sqrt(-(4*a*c -
b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 68544*a**10*b**2*c**7*(-b*sqrt(-(4*a*c - b
**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 6796*a**9*b**10*c**3*(-b*sqrt(-(4*a*c - b**2)**
5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 16
0*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3))**2 - 41296*a**9*b**4*c**6*(-b*sqrt(-(4*a*c - b**2)**5)*
(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 867*a**8*b**12*c**2*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**
2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 14036*a**8*b**6*c**5*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c
**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 49152*a**8*c**8 - 63*a**7*b**14*c*(-b*sqrt(-(4*a*c - b**2)**5)*(30
*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 2935*a**7*b**8*c**4*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**
2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 143424*a**7*b**2*c**7 + 2*a**6*b**16*(-b*sqrt(-(4*a*c - b**2)**
5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 16
0*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3))**2 + 382*a**6*b**10*c**3*(-b*sqrt(-(4*a*c - b**2)**5)*(
30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 155056*a**6*b**4*c**6 - 29*a**5*b**12*c**2*(-b*sqrt(-(4*a
*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 88492*a**5*b**6*c**5 + a**4*b**14*c*(-b*s
qrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3)) - 30185*a**4*b**8*c**4 + 6414*a**3
*b**10*c**3 - 838*a**2*b**12*c**2 + 62*a*b**14*c - 2*b**16)/(69120*a**7*b*c**8 - 102690*a**6*b**3*c**7 + 67554
*a**5*b**5*c**6 - 25155*a**4*b**7*c**5 + 5690*a**3*b**9*c**4 - 780*a**2*b**11*c**3 + 60*a*b**13*c**2 - 2*b**15
*c)) + (b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))*log(x + (98304*a**14*c*
*8*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 211968*a**13*b**2*c**7
*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 196352*a**12*b**4*c**6*(
b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 102528*a**11*b**6*c**5*(b*
sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 1/(2*a**3))**2 - 49152*a**11*c**8*(b*sqrt(-(4
*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 33120*a**10*b**8*c**4*(b*sqrt(-(4*a*c -
b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 68544*a**10*b**2*c**7*(b*sqrt(-(4*a*c - b*
*2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 6796*a**9*b**10*c**3*(b*sqrt(-(4*a*c - b**2)**5)
*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 41296*a**9*b**4*c**6*(b*sqrt(-(4*a*c - b**2)**5)*(30
*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 867*a**8*b**12*c**2*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c*
*2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 14036*a**8*b**6*c**5*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 -
10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 49152*a**8*c**8 - 63*a**7*b**14*c*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*
c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 - 2935*a**7*b**8*c**4*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2
- 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 143424*a**7*b**2*c**7 + 2*a**6*b**16*(b*sqrt(-(4*a*c - b**2)**5)*(30*a
**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3))**2 + 382*a**6*b**10*c**3*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c
**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 155056*a**6*b**4*c**6 - 29*a**5*b**12*c**2*(b*sqrt(-(4*a*c - b**2)
**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) + 88492*a**5*b**6*c**5 + a**4*b**14*c*(b*sqrt(-(4*a*c
- b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(2*a**3*(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)) - 1/(2*a**3)) - 30185*a**4*b**8*c**4 + 6414*a**3*b**10*c**3
- 838*a**2*b**12*c**2 + 62*a*b**14*c - 2*b**16)/(69120*a**7*b*c**8 - 102690*a**6*b**3*c**7 + 67554*a**5*b**5*
c**6 - 25155*a**4*b**7*c**5 + 5690*a**3*b**9*c**4 - 780*a**2*b**11*c**3 + 60*a*b**13*c**2 - 2*b**15*c)) - (-24
*a**3*c**2 + 21*a**2*b**2*c - 3*a*b**4 + x**3*(14*a*b*c**3 - 2*b**3*c**2) + x**2*(-16*a**2*c**3 + 29*a*b**2*c*
*2 - 4*b**4*c) + x*(2*a**2*b*c**2 + 12*a*b**3*c - 2*b**5))/(32*a**6*c**2 - 16*a**5*b**2*c + 2*a**4*b**4 + x**4
*(32*a**4*c**4 - 16*a**3*b**2*c**3 + 2*a**2*b**4*c**2) + x**3*(64*a**4*b*c**3 - 32*a**3*b**3*c**2 + 4*a**2*b**
5*c) + x**2*(64*a**5*c**3 - 12*a**3*b**4*c + 2*a**2*b**6) + x*(64*a**5*b*c**2 - 32*a**4*b**3*c + 4*a**3*b**5))
+ log(x)/a**3

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Giac [A]  time = 1.15702, size = 323, normalized size = 1.75 \begin{align*} -\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{align*}

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)