\(\int \frac {1}{(a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\) [102]

Optimal result

Integrand size = 22, antiderivative size = 286 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

[Out]

1/6*x*(b*x^3+a)/a/(b^2*x^6+2*a*b*x^3+a^2)^(3/2)+5/18*x*(b*x^3+a)^2/a^2/(b^2*x^6+2*a*b*x^3+a^2)^(3/2)+5/27*(b*x
^3+a)^3*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(3/2)-5/54*(b*x^3+a)^3*ln(a^(2/3)-a^(1/3
)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(3/2)-5/27*(b*x^3+a)^3*arctan(1/3*(a^(1/3)-2*
b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(3/2)*3^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1357, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

[In]

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

[Out]

(x*(a + b*x^3))/(6*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*x*(a + b*x^3)^2)/(18*a^2*(a^2 + 2*a*b*x^3 + b^2*x
^6)^(3/2)) - (5*(a + b*x^3)^3*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(a
^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*(a + b*x^3)^3*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(1/3)*(a^2 + 2*a*b
*x^3 + b^2*x^6)^(3/2)) - (5*(a + b*x^3)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3)*
(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))

Rule 31

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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 1357

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a b+2 b^2 x^3\right )^3 \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^3} \, dx}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^2} \, dx}{12 a b \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {1}{2 a b+2 b^2 x^3} \, dx}{36 a^2 b^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{108\ 2^{2/3} a^{8/3} b^{8/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}-\sqrt [3]{2} b^{2/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{108\ 2^{2/3} a^{8/3} b^{8/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {-2^{2/3} \sqrt [3]{a} b+2\ 2^{2/3} b^{4/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{432 a^{8/3} b^{10/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac {1}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{72 \sqrt [3]{2} a^{7/3} b^{7/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{72 a^{8/3} b^{10/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ & = \frac {x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac {5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac {5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {24 a^{5/3} \sqrt [3]{b} x+15 a^{2/3} b^{4/3} x^4-10 \sqrt {3} \left (a+b x^3\right )^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-5 a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-5 b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \]

[In]

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

[Out]

(24*a^(5/3)*b^(1/3)*x + 15*a^(2/3)*b^(4/3)*x^4 - 10*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]] + 10*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 5*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 1
0*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 5*b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(54*a^(8/3)*b^(1/3)*(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 = 1.53 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.31

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {5 b \,x^{4}}{18 a^{2}}+\frac {4 x}{9 a}\right )}{\left (b \,x^{3}+a \right )^{3}}+\frac {5 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 \left (b \,x^{3}+a \right ) a^{2} b}\)	\(88\)
default	\(\frac {\left (-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2} x^{6}+10 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{2} x^{6}-5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{2} x^{6}+15 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} x^{4}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a b \,x^{3}+20 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a b \,x^{3}-10 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a b \,x^{3}+24 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b x -10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}+10 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}-5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2}\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b \,a^{2} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}\)	\(299\)

[In]

int(1/(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*(5/18*b/a^2*x^4+4/9*x/a)+5/27*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/a^2/b*sum(1/_R^2*l
n(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\left [\frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}, \frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}\right ] \]

[In]

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

[Out]

[1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 15*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*l
og((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-
(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 5*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (
a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^
5*b^2*x^3 + a^6*b), 1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 30*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(
(a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 5*(b^2*x^
6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^3
+ a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^5*b^2*x^3 + a^6*b)]

Sympy [F]

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

[In]

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

[Out]

Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/18*(5*b*x^4 + 8*a*x)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + 5/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))
/(a/b)^(1/3))/(a^2*b*(a/b)^(2/3)) - 5/54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) + 5/27*log
(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

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

[Out]

-5/27*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*sgn(b*x^3 + a)) + 5/27*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sq
rt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b*sgn(b*x^3 + a)) + 5/54*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^3*b*sgn(b*x^3 + a)) + 1/18*(5*b*x^4 + 8*a*x)/((b*x^3 + a)^2*a^2*sgn(b*x^3 + a))

Mupad [F(-1)]

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

[In]

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

[Out]

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