3.910 \(\int \frac {1}{x^3 (a+b+2 a x^2+a x^4)} \, dx\)

Optimal. Leaf size=89 \[ -\frac {1}{2 x^2 (a+b)}+\frac {\sqrt {a} (a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {b} (a+b)^2}+\frac {a \log \left (a x^4+2 a x^2+a+b\right )}{2 (a+b)^2}-\frac {2 a \log (x)}{(a+b)^2} \]

[Out]

-1/2/(a+b)/x^2-2*a*ln(x)/(a+b)^2+1/2*a*ln(a*x^4+2*a*x^2+a+b)/(a+b)^2+1/2*(a-b)*arctan((x^2+1)*a^(1/2)/b^(1/2))
*a^(1/2)/(a+b)^2/b^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1114, 709, 800, 634, 618, 204, 628} \[ -\frac {1}{2 x^2 (a+b)}+\frac {a \log \left (a x^4+2 a x^2+a+b\right )}{2 (a+b)^2}+\frac {\sqrt {a} (a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {b} (a+b)^2}-\frac {2 a \log (x)}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-1/(2*(a + b)*x^2) + (Sqrt[a]*(a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*Sqrt[b]*(a + b)^2) - (2*a*Log[x]
)/(a + b)^2 + (a*Log[a + b + 2*a*x^2 + a*x^4])/(2*(a + b)^2)

Rule 204

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

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 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), 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[m, -1]

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 1114

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

Rubi steps

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

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Mathematica [C]  time = 0.10, size = 163, normalized size = 1.83 \[ \frac {\left (2 a^{3/2} \sqrt {b}-i a^2+i a b\right ) \log \left (\sqrt {a} x^2+\sqrt {a}-i \sqrt {b}\right )}{4 \sqrt {a} \sqrt {b} (a+b)^2}+\frac {\left (2 a^{3/2} \sqrt {b}+i a^2-i a b\right ) \log \left (\sqrt {a} x^2+\sqrt {a}+i \sqrt {b}\right )}{4 \sqrt {a} \sqrt {b} (a+b)^2}-\frac {1}{2 x^2 (a+b)}-\frac {2 a \log (x)}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-1/2*1/((a + b)*x^2) - (2*a*Log[x])/(a + b)^2 + (((-I)*a^2 + 2*a^(3/2)*Sqrt[b] + I*a*b)*Log[Sqrt[a] - I*Sqrt[b
] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2) + ((I*a^2 + 2*a^(3/2)*Sqrt[b] - I*a*b)*Log[Sqrt[a] + I*Sqrt[b]
+ Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2)

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fricas [A]  time = 0.86, size = 208, normalized size = 2.34 \[ \left [-\frac {{\left (a - b\right )} x^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, {\left (b x^{2} + b\right )} \sqrt {-\frac {a}{b}} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right ) - 2 \, a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 8 \, a x^{2} \log \relax (x) + 2 \, a + 2 \, b}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}, -\frac {{\left (a - b\right )} x^{2} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a x^{2} + a}\right ) - a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, a x^{2} \log \relax (x) + a + b}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")

[Out]

[-1/4*((a - b)*x^2*sqrt(-a/b)*log((a*x^4 + 2*a*x^2 - 2*(b*x^2 + b)*sqrt(-a/b) + a - b)/(a*x^4 + 2*a*x^2 + a +
b)) - 2*a*x^2*log(a*x^4 + 2*a*x^2 + a + b) + 8*a*x^2*log(x) + 2*a + 2*b)/((a^2 + 2*a*b + b^2)*x^2), -1/2*((a -
b)*x^2*sqrt(a/b)*arctan(b*sqrt(a/b)/(a*x^2 + a)) - a*x^2*log(a*x^4 + 2*a*x^2 + a + b) + 4*a*x^2*log(x) + a +
b)/((a^2 + 2*a*b + b^2)*x^2)]

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giac [A]  time = 0.28, size = 125, normalized size = 1.40 \[ \frac {a \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a^{2} - a b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {2 \, a x^{2} - a - b}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")

[Out]

1/2*a*log(a*x^4 + 2*a*x^2 + a + b)/(a^2 + 2*a*b + b^2) - a*log(x^2)/(a^2 + 2*a*b + b^2) + 1/2*(a^2 - a*b)*arct
an((a*x^2 + a)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b)) + 1/2*(2*a*x^2 - a - b)/((a^2 + 2*a*b + b^2)*x^2)

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maple [A]  time = 0.01, size = 110, normalized size = 1.24 \[ \frac {a^{2} \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \left (a +b \right )^{2} \sqrt {a b}}-\frac {a b \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \left (a +b \right )^{2} \sqrt {a b}}-\frac {2 a \ln \relax (x )}{\left (a +b \right )^{2}}+\frac {a \ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )}{2 \left (a +b \right )^{2}}-\frac {1}{2 \left (a +b \right ) x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a*x^4+2*a*x^2+a+b),x)

[Out]

1/2*a*ln(a*x^4+2*a*x^2+a+b)/(a+b)^2+1/2/(a+b)^2*a^2/(a*b)^(1/2)*arctan(1/2*(2*a*x^2+2*a)/(a*b)^(1/2))-1/2/(a+b
)^2*a/(a*b)^(1/2)*arctan(1/2*(2*a*x^2+2*a)/(a*b)^(1/2))*b-1/2/(a+b)/x^2-2*a*ln(x)/(a+b)^2

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maxima [A]  time = 2.90, size = 104, normalized size = 1.17 \[ \frac {a \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a^{2} - a b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {1}{2 \, {\left (a + b\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")

[Out]

1/2*a*log(a*x^4 + 2*a*x^2 + a + b)/(a^2 + 2*a*b + b^2) - a*log(x^2)/(a^2 + 2*a*b + b^2) + 1/2*(a^2 - a*b)*arct
an((a*x^2 + a)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b)) - 1/2/((a + b)*x^2)

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mupad [B]  time = 7.39, size = 3313, normalized size = 37.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x)

[Out]

(8*a*b*log(((2*a^5)/(a + b)^3 - (a/(2*(a + b)^2) - (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*((12*a^5*x^2)/(a +
b)^2 - (a/(2*(a + b)^2) - (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*((8*a^4*(3*a - b))/(a + b) + 16*a^4*(a/(2*(a
+ b)^2) - (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*(a + b + a*x^2 - 5*b*x^2) + (4*a^4*x^2*(7*a + 5*b))/(a + b)
) + (a^4*(15*a - b))/(a + b)^2) + (a^5*x^2)/(a + b)^3)*((2*a^5)/(a + b)^3 - (a/(2*(a + b)^2) + (-(a*(a - b)^2)
/(b*(a + b)^4))^(1/2)/4)*((12*a^5*x^2)/(a + b)^2 - (a/(2*(a + b)^2) + (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*
((8*a^4*(3*a - b))/(a + b) + 16*a^4*(a/(2*(a + b)^2) + (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*(a + b + a*x^2
- 5*b*x^2) + (4*a^4*x^2*(7*a + 5*b))/(a + b)) + (a^4*(15*a - b))/(a + b)^2) + (a^5*x^2)/(a + b)^3)))/(32*a*b^2
+ 16*a^2*b + 16*b^3) - (2*a*log(x))/(2*a*b + a^2 + b^2) - 1/(2*x^2*(a + b)) + (a^(1/2)*atan(((13*a^2 - 34*a*b
+ b^2)*((8*a*b*((14*a^5*b + 15*a^6 - a^4*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*((40*a^6*b + 24*a^7 -
8*a^4*b^3 + 8*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (8*a*b*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 +
96*a^6*b^2))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(32*a*b^2 + 16*a^2*b + 16*b^3
)))/(32*a*b^2 + 16*a^2*b + 16*b^3) - (2*a^5)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (a^(1/2)*((a^(1/2)*(a - b)*((40
*a^6*b + 24*a^7 - 8*a^4*b^3 + 8*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (8*a*b*(64*a^7*b + 16*a^8 + 16*a^4*
b^4 + 64*a^5*b^3 + 96*a^6*b^2))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(4*b^(1/2)*
(2*a*b + a^2 + b^2)) + (2*a^(3/2)*b^(1/2)*(a - b)*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/
((2*a*b + a^2 + b^2)*(32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)))*(a - b))/(4*b^(1/2)*(2*a
*b + a^2 + b^2)) + (a^2*(a - b)^2*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/(2*(2*a*b + a^2
+ b^2)^2*(32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)))*(24*a*b^(13/2) + 4*b^(15/2) + 4*a^6*
b^(3/2) + 24*a^5*b^(5/2) + 60*a^4*b^(7/2) + 80*a^3*b^(9/2) + 60*a^2*b^(11/2)))/((a + b)^3*(98*a*b + a^2 + b^2)
*(a^(13/2) - 2*a^(11/2)*b + a^(9/2)*b^2)) - (x^2*(((13*a^2 - 34*a*b + b^2)*(a^5/(3*a*b^2 + 3*a^2*b + a^3 + b^3
) - (8*a*b*((12*a^5*b + 12*a^6)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*((76*a^6*b + 28*a^7 + 20*a^4*b^3 + 68
*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*(32*a^7*b - 16*a^8 + 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2
))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(32*a*b^2 + 16*a^2*b + 16*b^3)))/(32*a*b
^2 + 16*a^2*b + 16*b^3) - (a^(1/2)*((a^(1/2)*(a - b)*((76*a^6*b + 28*a^7 + 20*a^4*b^3 + 68*a^5*b^2)/(3*a*b^2 +
3*a^2*b + a^3 + b^3) - (8*a*b*(32*a^7*b - 16*a^8 + 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/((32*a*b^2 + 16*a
^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(4*b^(1/2)*(2*a*b + a^2 + b^2)) - (2*a^(3/2)*b^(1/2)*(a - b)
*(32*a^7*b - 16*a^8 + 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/((2*a*b + a^2 + b^2)*(32*a*b^2 + 16*a^2*b + 16*
b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)))*(a - b))/(4*b^(1/2)*(2*a*b + a^2 + b^2)) + (a^2*(a - b)^2*(32*a^7*b - 1
6*a^8 + 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/(2*(2*a*b + a^2 + b^2)^2*(32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*
b^2 + 3*a^2*b + a^3 + b^3))))/((a + b)^3*(98*a*b + a^2 + b^2)) + (a^(1/2)*(a^2 - 34*a*b + 13*b^2)*((8*a*b*((a^
(1/2)*(a - b)*((76*a^6*b + 28*a^7 + 20*a^4*b^3 + 68*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*(32*a^7*
b - 16*a^8 + 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3
+ b^3))))/(4*b^(1/2)*(2*a*b + a^2 + b^2)) - (2*a^(3/2)*b^(1/2)*(a - b)*(32*a^7*b - 16*a^8 + 80*a^4*b^4 + 224*
a^5*b^3 + 192*a^6*b^2))/((2*a*b + a^2 + b^2)*(32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))
/(32*a*b^2 + 16*a^2*b + 16*b^3) - (a^(1/2)*(a - b)*((12*a^5*b + 12*a^6)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a
*b*((76*a^6*b + 28*a^7 + 20*a^4*b^3 + 68*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*(32*a^7*b - 16*a^8
+ 80*a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/
(32*a*b^2 + 16*a^2*b + 16*b^3)))/(4*b^(1/2)*(2*a*b + a^2 + b^2)) + (a^(3/2)*(a - b)^3*(32*a^7*b - 16*a^8 + 80*
a^4*b^4 + 224*a^5*b^3 + 192*a^6*b^2))/(64*b^(3/2)*(2*a*b + a^2 + b^2)^3*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(b^
(1/2)*(a + b)^3*(98*a*b + a^2 + b^2)))*(24*a*b^(13/2) + 4*b^(15/2) + 4*a^6*b^(3/2) + 24*a^5*b^(5/2) + 60*a^4*b
^(7/2) + 80*a^3*b^(9/2) + 60*a^2*b^(11/2)))/(a^(13/2) - 2*a^(11/2)*b + a^(9/2)*b^2) + (a^(1/2)*((a^(1/2)*(a -
b)*((14*a^5*b + 15*a^6 - a^4*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (8*a*b*((40*a^6*b + 24*a^7 - 8*a^4*b^3 + 8
*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (8*a*b*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))
/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(32*a*b^2 + 16*a^2*b + 16*b^3)))/(4*b^(1/2
)*(2*a*b + a^2 + b^2)) - (8*a*b*((a^(1/2)*(a - b)*((40*a^6*b + 24*a^7 - 8*a^4*b^3 + 8*a^5*b^2)/(3*a*b^2 + 3*a^
2*b + a^3 + b^3) + (8*a*b*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/((32*a*b^2 + 16*a^2*b +
16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(4*b^(1/2)*(2*a*b + a^2 + b^2)) + (2*a^(3/2)*b^(1/2)*(a - b)*(64*a^
7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/((2*a*b + a^2 + b^2)*(32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a
*b^2 + 3*a^2*b + a^3 + b^3))))/(32*a*b^2 + 16*a^2*b + 16*b^3) + (a^(3/2)*(a - b)^3*(64*a^7*b + 16*a^8 + 16*a^4
*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/(64*b^(3/2)*(2*a*b + a^2 + b^2)^3*(3*a*b^2 + 3*a^2*b + a^3 + b^3)))*(a^2 - 34
*a*b + 13*b^2)*(24*a*b^(13/2) + 4*b^(15/2) + 4*a^6*b^(3/2) + 24*a^5*b^(5/2) + 60*a^4*b^(7/2) + 80*a^3*b^(9/2)
+ 60*a^2*b^(11/2)))/(b^(1/2)*(a + b)^3*(98*a*b + a^2 + b^2)*(a^(13/2) - 2*a^(11/2)*b + a^(9/2)*b^2)))*(a - b))
/(2*b^(1/2)*(2*a*b + a^2 + b^2))

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sympy [B]  time = 41.75, size = 386, normalized size = 4.34 \[ - \frac {2 a \log {\relax (x )}}{\left (a + b\right )^{2}} + \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} b \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} + \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} b \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} - \frac {1}{x^{2} \left (2 a + 2 b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a*x**4+2*a*x**2+a+b),x)

[Out]

-2*a*log(x)/(a + b)**2 + (a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2)))*log(x**2 + (4*a**
2*b*(a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a + b)**2) -
sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) - 3*a*b + 4*b**3*(a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(
a**2 + 2*a*b + b**2))))/(a**2 - a*b)) + (a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2)))*lo
g(x**2 + (4*a**2*b*(a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2
*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) - 3*a*b + 4*b**3*(a/(2*(a + b)**2) + sqrt(-a*b)
*(a - b)/(4*b*(a**2 + 2*a*b + b**2))))/(a**2 - a*b)) - 1/(x**2*(2*a + 2*b))

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