\(\int \frac {1}{(a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\) [113]

Optimal result

Integrand size = 22, antiderivative size = 364 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]

[Out]

1/12*x*(b*x^3+a)/a/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+11/108*x*(b*x^3+a)^2/a^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+11/81*
x*(b*x^3+a)^3/a^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+55/243*x*(b*x^3+a)^4/a^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+110/729
*(b*x^3+a)^5*ln(a^(1/3)+b^(1/3)*x)/a^(14/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)-55/729*(b*x^3+a)^5*ln(a^(2/3
)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)-110/729*(b*x^3+a)^5*arctan(1/3
*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)*3^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]

[In]

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

[Out]

(x*(a + b*x^3))/(12*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^2)/(108*a^2*(a^2 + 2*a*b*x^3 + b^
2*x^6)^(5/2)) + (11*x*(a + b*x^3)^3)/(81*a^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (55*x*(a + b*x^3)^4)/(243*a^
4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) - (110*(a + b*x^3)^5*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(
243*Sqrt[3]*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (110*(a + b*x^3)^5*Log[a^(1/3) + b^(1/3)*x])
/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) - (55*(a + b*x^3)^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))

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 )^5 \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^5} \, dx}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^4} \, dx}{24 a b \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^3} \, dx}{54 a^2 b^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^3\right )^2} \, dx}{648 a^3 b^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{2 a b+2 b^2 x^3} \, dx}{1944 a^4 b^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\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}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\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}{23328 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\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}{3888 \sqrt [3]{2} a^{13/3} b^{13/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3888 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ & = \frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (243 a^{11/3} x+297 a^{8/3} x \left (a+b x^3\right )+396 a^{5/3} x \left (a+b x^3\right )^2+660 a^{2/3} x \left (a+b x^3\right )^3+\frac {440 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {440 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {220 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}\right )}{2916 a^{14/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

[In]

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

[Out]

((a + b*x^3)*(243*a^(11/3)*x + 297*a^(8/3)*x*(a + b*x^3) + 396*a^(5/3)*x*(a + b*x^3)^2 + 660*a^(2/3)*x*(a + b*
x^3)^3 + (440*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (440*(a + b*
x^3)^4*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (220*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
b^(1/3)))/(2916*a^(14/3)*((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 = 1.53 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.30

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {55 b^{3} x^{10}}{243 a^{4}}+\frac {22 b^{2} x^{7}}{27 a^{3}}+\frac {341 b \,x^{4}}{324 a^{2}}+\frac {133 x}{243 a}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {110 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{729 \left (b \,x^{3}+a \right ) b \,a^{4}}\)	\(110\)
default	\(\frac {\left (-440 \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^{12}+440 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-220 \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^{12}+660 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{10}-1760 \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^{9}+1760 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-880 \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^{9}+2376 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{7}-2640 \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^{6}+2640 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-1320 \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^{6}+3069 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{4}-1760 \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^{3}+1760 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-880 \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^{3}+1596 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b x -440 \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}+440 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-220 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {2}{3}} b \,a^{4} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\)	\(519\)

[In]

int(1/(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*(55/243/a^4*b^3*x^10+22/27*b^2/a^3*x^7+341/324*b/a^2*x^4+133/243*x/a)+110/729*
((b*x^3+a)^2)^(1/2)/(b*x^3+a)/b/a^4*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 660 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} 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 ) - 220 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\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 ) + 440 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}, \frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 1320 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} 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 ) - 220 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\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 ) + 440 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}\right ] \]

[In]

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

[Out]

[1/2916*(660*a^2*b^4*x^10 + 2376*a^3*b^3*x^7 + 3069*a^4*b^2*x^4 + 1596*a^5*b*x + 660*sqrt(1/3)*(a*b^5*x^12 + 4
*a^2*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x^3 + a^5*b)*sqrt(-(a^2*b)^(1/3)/b)*log((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)) -
220*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x
+ (a^2*b)^(1/3)*a) + 440*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(a^2*b)^(2/3)*log(a*b*x
+ (a^2*b)^(2/3)))/(a^6*b^5*x^12 + 4*a^7*b^4*x^9 + 6*a^8*b^3*x^6 + 4*a^9*b^2*x^3 + a^10*b), 1/2916*(660*a^2*b^4
*x^10 + 2376*a^3*b^3*x^7 + 3069*a^4*b^2*x^4 + 1596*a^5*b*x + 1320*sqrt(1/3)*(a*b^5*x^12 + 4*a^2*b^4*x^9 + 6*a^
3*b^3*x^6 + 4*a^4*b^2*x^3 + a^5*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) - 220*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(a^2*b)^(2/3)*
log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 440*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 +
a^4)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^5*x^12 + 4*a^7*b^4*x^9 + 6*a^8*b^3*x^6 + 4*a^9*b^2*x^3
+ a^10*b)]

Sympy [F]

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

[In]

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

[Out]

Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**(-5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} + \frac {110 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {55 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {110 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/972*(220*b^3*x^10 + 792*a*b^2*x^7 + 1023*a^2*b*x^4 + 532*a^3*x)/(a^4*b^4*x^12 + 4*a^5*b^3*x^9 + 6*a^6*b^2*x^
6 + 4*a^7*b*x^3 + a^8) + 110/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b*(a/b)^(2/3
)) - 55/729*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b*(a/b)^(2/3)) + 110/729*log(x + (a/b)^(1/3))/(a^4*b*(
a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {110 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {110 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {55 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

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

[Out]

-110/729*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*sgn(b*x^3 + a)) + 110/729*sqrt(3)*(-a*b^2)^(1/3)*arctan(
1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b*sgn(b*x^3 + a)) + 55/729*(-a*b^2)^(1/3)*log(x^2 + x*(-a/
b)^(1/3) + (-a/b)^(2/3))/(a^5*b*sgn(b*x^3 + a)) + 1/972*(220*b^3*x^10 + 792*a*b^2*x^7 + 1023*a^2*b*x^4 + 532*a
^3*x)/((b*x^3 + a)^4*a^4*sgn(b*x^3 + a))

Mupad [F(-1)]

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

[In]

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

[Out]

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