\(\int \frac {1}{x^2 (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\) [115]

Optimal result

Integrand size = 26, antiderivative size = 398 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[Out]

455/972/a^4/x/((b*x^3+a)^2)^(1/2)+1/12/a/x/(b*x^3+a)^3/((b*x^3+a)^2)^(1/2)+13/108/a^2/x/(b*x^3+a)^2/((b*x^3+a)
^2)^(1/2)+65/324/a^3/x/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-455/243*(b*x^3+a)/a^5/x/((b*x^3+a)^2)^(1/2)+455/729*b^(1/
3)*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(16/3)/((b*x^3+a)^2)^(1/2)-455/1458*b^(1/3)*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*
b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)/((b*x^3+a)^2)^(1/2)+455/729*b^(1/3)*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)
/a^(1/3)*3^(1/2))/a^(16/3)*3^(1/2)/((b*x^3+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1369, 296, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {13}{108 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac {1}{12 a x \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

[In]

Int[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

455/(972*a^4*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1
3/(108*a^2*x*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 65/(324*a^3*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) - (455*(a + b*x^3))/(243*a^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/
3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*b^(1/3)*(a + b*
x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (13 b^3 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )^4} \, dx}{12 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (65 b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )^3} \, dx}{54 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (455 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )^2} \, dx}{324 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )} \, dx}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (455 b \left (a b+b^2 x^3\right )\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{243 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{1458 a^{16/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (455 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{486 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (455 \left (a b+b^2 x^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{243 a^{16/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{10/3} b x^2-594 a^{7/3} b x^2 \left (a+b x^3\right )-1179 a^{4/3} b x^2 \left (a+b x^3\right )^2-2544 \sqrt [3]{a} b x^2 \left (a+b x^3\right )^3-\frac {2916 \sqrt [3]{a} \left (a+b x^3\right )^4}{x}-1820 \sqrt {3} \sqrt [3]{b} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+1820 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-910 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{16/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

[In]

Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

((a + b*x^3)*(-243*a^(10/3)*b*x^2 - 594*a^(7/3)*b*x^2*(a + b*x^3) - 1179*a^(4/3)*b*x^2*(a + b*x^3)^2 - 2544*a^
(1/3)*b*x^2*(a + b*x^3)^3 - (2916*a^(1/3)*(a + b*x^3)^4)/x - 1820*Sqrt[3]*b^(1/3)*(a + b*x^3)^4*ArcTan[(-a^(1/
3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 1820*b^(1/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 910*b^(1/3)*(a +
b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(16/3)*((a + b*x^3)^2)^(5/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.34

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {455 b^{4} x^{12}}{243 a^{5}}-\frac {2275 b^{3} x^{9}}{324 a^{4}}-\frac {260 b^{2} x^{6}}{27 a^{3}}-\frac {1352 b \,x^{3}}{243 a^{2}}-\frac {1}{a}\right )}{\left (b \,x^{3}+a \right )^{5} x}+\frac {455 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{16} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{16}+3 b \right ) x -a^{11} \textit {\_R}^{2}\right )\right )}{729 \left (b \,x^{3}+a \right )}\)	\(137\)
default	\(-\frac {\left (-1820 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{4} x^{13}-1820 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{13}+910 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{4} x^{13}+5460 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{12}-7280 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a \,b^{3} x^{10}-7280 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{10}+3640 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \,b^{3} x^{10}+20475 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{9}-10920 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} b^{2} x^{7}-10920 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{7}+5460 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} b^{2} x^{7}+28080 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{6}-7280 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{3} b \,x^{4}-7280 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{4}+3640 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{3} b \,x^{4}+16224 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{3}-1820 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{4} x -1820 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4} x +910 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4} x +2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} x \,a^{5} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\)	\(536\)

[In]

int(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(-455/243/a^5*b^4*x^12-2275/324*b^3/a^4*x^9-260/27*b^2/a^3*x^6-1352/243*b/a^2*
x^3-1/a)/x+455/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)*sum(_R*ln((-4*_R^3*a^16+3*b)*x-a^11*_R^2),_R=RootOf(_Z^3*a^16
-b))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {5460 \, b^{4} x^{12} + 20475 \, a b^{3} x^{9} + 28080 \, a^{2} b^{2} x^{6} + 16224 \, a^{3} b x^{3} + 2916 \, a^{4} + 1820 \, \sqrt {3} {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 910 \, {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 1820 \, {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} \]

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/2916*(5460*b^4*x^12 + 20475*a*b^3*x^9 + 28080*a^2*b^2*x^6 + 16224*a^3*b*x^3 + 2916*a^4 + 1820*sqrt(3)*(b^4*
x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*
sqrt(3)) + 910*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/
a)^(2/3) + a*(b/a)^(1/3)) - 1820*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*l
og(b*x + a*(b/a)^(2/3)))/(a^5*b^4*x^13 + 4*a^6*b^3*x^10 + 6*a^7*b^2*x^7 + 4*a^8*b*x^4 + a^9*x)

Sympy [F]

\[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/x**2/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(1/(x**2*((a + b*x**3)**2)**(5/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {1820 \, b^{4} x^{12} + 6825 \, a b^{3} x^{9} + 9360 \, a^{2} b^{2} x^{6} + 5408 \, a^{3} b x^{3} + 972 \, a^{4}}{972 \, {\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} - \frac {455 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {455 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {455 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/972*(1820*b^4*x^12 + 6825*a*b^3*x^9 + 9360*a^2*b^2*x^6 + 5408*a^3*b*x^3 + 972*a^4)/(a^5*b^4*x^13 + 4*a^6*b^
3*x^10 + 6*a^7*b^2*x^7 + 4*a^8*b*x^4 + a^9*x) - 455/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(
1/3))/(a^5*(a/b)^(1/3)) - 455/1458*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(1/3)) + 455/729*log(x +
(a/b)^(1/3))/(a^5*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {455 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {455 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{6} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {455 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{6} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{a^{5} x \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {848 \, b^{4} x^{11} + 2937 \, a b^{3} x^{8} + 3528 \, a^{2} b^{2} x^{5} + 1520 \, a^{3} b x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

455/729*b*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^6*sgn(b*x^3 + a)) + 455/729*sqrt(3)*(-a*b^2)^(2/3)*arctan
(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b*sgn(b*x^3 + a)) - 455/1458*(-a*b^2)^(2/3)*log(x^2 + x*(
-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*b*sgn(b*x^3 + a)) - 1/(a^5*x*sgn(b*x^3 + a)) - 1/972*(848*b^4*x^11 + 2937*a*b
^3*x^8 + 3528*a^2*b^2*x^5 + 1520*a^3*b*x^2)/((b*x^3 + a)^4*a^5*sgn(b*x^3 + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]

[In]

int(1/(x^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)),x)

[Out]

int(1/(x^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)), x)