\(\int \frac {1}{x^2 (a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\) [104]

Optimal result

Integrand size = 26, antiderivative size = 316 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[Out]

7/18/a^2/x/((b*x^3+a)^2)^(1/2)+1/6/a/x/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-14/9*(b*x^3+a)/a^3/x/((b*x^3+a)^2)^(1/2)+
14/27*b^(1/3)*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/((b*x^3+a)^2)^(1/2)-7/27*b^(1/3)*(b*x^3+a)*ln(a^(2/3)-a
^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/((b*x^3+a)^2)^(1/2)+14/27*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^(10/3)*3^(1/2)/((b*x^3+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[In]

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

[Out]

7/(18*a^2*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (14*(a
+ b*x^3))/(9*a^3*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(
Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*b^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(27*a^(10/3)*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^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}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 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}{6 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (14 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^3\right )} \, dx}{9 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (14 b \left (a b+b^2 x^3\right )\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{9 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (14 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (14 \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}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (7 \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}{27 a^{10/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (7 \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}{9 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \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 )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (14 \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 )}{9 a^{10/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7}{18 a^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {14 \left (a+b x^3\right )}{9 a^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \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 )}{9 \sqrt {3} a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \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 )}{27 a^{10/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {-54 a^{7/3}-147 a^{4/3} b x^3-84 \sqrt [3]{a} b^2 x^6+28 \sqrt {3} \sqrt [3]{b} x \left (a+b x^3\right )^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 \sqrt [3]{b} x \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-14 a^2 \sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a b^{4/3} x^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-14 b^{7/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} x \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \]

[In]

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

[Out]

(-54*a^(7/3) - 147*a^(4/3)*b*x^3 - 84*a^(1/3)*b^2*x^6 + 28*Sqrt[3]*b^(1/3)*x*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]] + 28*b^(1/3)*x*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 14*a^2*b^(1/3)*x*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 28*a*b^(4/3)*x^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 14*b^(7
/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*x*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

Maple [C] (warning: unable to verify)

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

Time = 2.42 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.36

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {14 b^{2} x^{6}}{9 a^{3}}-\frac {49 b \,x^{3}}{18 a^{2}}-\frac {1}{a}\right )}{\left (b \,x^{3}+a \right )^{3} x}+\frac {14 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10}+3 b \right ) x -a^{7} \textit {\_R}^{2}\right )\right )}{27 \left (b \,x^{3}+a \right )}\)	\(115\)
default	\(\frac {\left (28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2} x^{7}+28 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{2} x^{7}-14 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{2} x^{7}-84 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2} x^{6}+56 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a b \,x^{4}+56 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a b \,x^{4}-28 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a b \,x^{4}-147 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b \,x^{3}+28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} x +28 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} x -14 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} x -54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} x \,a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}\)	\(316\)

[In]

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

[Out]

((b*x^3+a)^2)^(1/2)/(b*x^3+a)^3*(-14/9*b^2/a^3*x^6-49/18*b/a^2*x^3-1/a)/x+14/27*((b*x^3+a)^2)^(1/2)/(b*x^3+a)*
sum(_R*ln((-4*_R^3*a^10+3*b)*x-a^7*_R^2),_R=RootOf(_Z^3*a^10-b))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {84 \, b^{2} x^{6} + 147 \, a b x^{3} + 28 \, \sqrt {3} {\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, {\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 28 \, {\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 54 \, a^{2}}{54 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} \]

[In]

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

[Out]

-1/54*(84*b^2*x^6 + 147*a*b*x^3 + 28*sqrt(3)*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b
/a)^(1/3) - 1/3*sqrt(3)) + 14*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^
(1/3)) - 28*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) + 54*a^2)/(a^3*b^2*x^7 + 2*a^4*
b*x^4 + a^5*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {28 \, b^{2} x^{6} + 49 \, a b x^{3} + 18 \, a^{2}}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} - \frac {14 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {14 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

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

[Out]

-1/18*(28*b^2*x^6 + 49*a*b*x^3 + 18*a^2)/(a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x) - 14/27*sqrt(3)*arctan(1/3*sqrt(3
)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*(a/b)^(1/3)) - 7/27*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^
(1/3)) + 14/27*log(x + (a/b)^(1/3))/(a^3*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {14 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {14 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {7 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, b^{2} x^{5} + 13 \, a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{a^{3} x \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

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

[Out]

14/27*b*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*sgn(b*x^3 + a)) + 14/27*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3
*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b*sgn(b*x^3 + a)) - 7/27*(-a*b^2)^(2/3)*log(x^2 + x*(-a/b)^(1
/3) + (-a/b)^(2/3))/(a^4*b*sgn(b*x^3 + a)) - 1/18*(10*b^2*x^5 + 13*a*b*x^2)/((b*x^3 + a)^2*a^3*sgn(b*x^3 + a))
- 1/(a^3*x*sgn(b*x^3 + a))

Mupad [F(-1)]

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

[In]

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

[Out]

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