\(\int \frac {1}{x^3 (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\) [116]

Optimal result

Integrand size = 26, antiderivative size = 398 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {770 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[Out]

154/243/a^4/x^2/((b*x^3+a)^2)^(1/2)+1/12/a/x^2/(b*x^3+a)^3/((b*x^3+a)^2)^(1/2)+7/54/a^2/x^2/(b*x^3+a)^2/((b*x^
3+a)^2)^(1/2)+77/324/a^3/x^2/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-385/243*(b*x^3+a)/a^5/x^2/((b*x^3+a)^2)^(1/2)-770/7
29*b^(2/3)*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(17/3)/((b*x^3+a)^2)^(1/2)+385/729*b^(2/3)*(b*x^3+a)*ln(a^(2/3)-a
^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)/((b*x^3+a)^2)^(1/2)+770/729*b^(2/3)*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(
1/3)*x)/a^(1/3)*3^(1/2))/a^(17/3)*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, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {7}{54 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac {1}{12 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac {770 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

[In]

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

[Out]

154/(243*a^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x^2*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
+ 7/(54*a^2*x^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 77/(324*a^3*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6]) - (385*(a + b*x^3))/(243*a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (770*b^(2/3)*(a + b*x^3)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7
70*b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (385*b^(2/3)
*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(729*a^(17/3)*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 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 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 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^3 \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^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 b^3 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )^4} \, dx}{6 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (77 b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \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^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (154 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )^2} \, dx}{81 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (770 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^3 \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 {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (770 b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a b+b^2 x^3} \, dx}{243 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (770 \sqrt [3]{b} \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^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (770 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac {2 \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^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (385 \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}{729 a^{17/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (385 b^{2/3} \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}{243 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (770 \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^{17/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {154}{243 a^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {770 b^{2/3} \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^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{11/3} b x-621 a^{8/3} b x \left (a+b x^3\right )-1314 a^{5/3} b x \left (a+b x^3\right )^2-3162 a^{2/3} b x \left (a+b x^3\right )^3-\frac {1458 a^{2/3} \left (a+b x^3\right )^4}{x^2}-3080 \sqrt {3} b^{2/3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )-3080 b^{2/3} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+1540 b^{2/3} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{17/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

[In]

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

[Out]

((a + b*x^3)*(-243*a^(11/3)*b*x - 621*a^(8/3)*b*x*(a + b*x^3) - 1314*a^(5/3)*b*x*(a + b*x^3)^2 - 3162*a^(2/3)*
b*x*(a + b*x^3)^3 - (1458*a^(2/3)*(a + b*x^3)^4)/x^2 - 3080*Sqrt[3]*b^(2/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2
*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 3080*b^(2/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 1540*b^(2/3)*(a + b*x^3
)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(17/3)*((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 = 2.86 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.35

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {385 b^{4} x^{12}}{243 a^{5}}-\frac {154 b^{3} x^{9}}{27 a^{4}}-\frac {2387 b^{2} x^{6}}{324 a^{3}}-\frac {931 b \,x^{3}}{243 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{3}+a \right )^{5} x^{2}}+\frac {770 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{17} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{17}-3 b^{2}\right ) x -a^{6} b \textit {\_R} \right )\right )}{729 \left (b \,x^{3}+a \right )}\)	\(138\)
default	\(-\frac {\left (-3080 \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^{14}+3080 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{14}-1540 \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^{14}+4620 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{12}-12320 \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^{11}+12320 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{11}-6160 \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^{11}+16632 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{9}-18480 \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^{8}+18480 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{8}-9240 \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^{8}+21483 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{6}-12320 \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^{5}+12320 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{5}-6160 \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^{5}+11172 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b \,x^{3}-3080 \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^{2}+3080 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4} x^{2}-1540 \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^{2}+1458 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {2}{3}} x^{2} a^{5} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\)	\(542\)

[In]

int(1/x^3/(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*(-385/243/a^5*b^4*x^12-154/27*b^3/a^4*x^9-2387/324*b^2/a^3*x^6-931/243*b/a^2*x
^3-1/2/a)/x^2+770/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)*sum(_R*ln((-4*_R^3*a^17-3*b^2)*x-a^6*b*_R),_R=RootOf(_Z^3*
a^17+b^2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {4620 \, b^{4} x^{12} + 16632 \, a b^{3} x^{9} + 21483 \, a^{2} b^{2} x^{6} + 11172 \, a^{3} b x^{3} + 1458 \, a^{4} - 3080 \, \sqrt {3} {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 1540 \, {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 3080 \, {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{2916 \, {\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} \]

[In]

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

[Out]

-1/2916*(4620*b^4*x^12 + 16632*a*b^3*x^9 + 21483*a^2*b^2*x^6 + 11172*a^3*b*x^3 + 1458*a^4 - 3080*sqrt(3)*(b^4*
x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/
a^2)^(2/3) - sqrt(3)*b)/b) + 1540*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)
^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 3080*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^
2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)))/(a^5*b^4*x^14 + 4*a^6*b^3*x^11
+ 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {1540 \, b^{4} x^{12} + 5544 \, a b^{3} x^{9} + 7161 \, a^{2} b^{2} x^{6} + 3724 \, a^{3} b x^{3} + 486 \, a^{4}}{972 \, {\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} - \frac {770 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {385 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {770 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/972*(1540*b^4*x^12 + 5544*a*b^3*x^9 + 7161*a^2*b^2*x^6 + 3724*a^3*b*x^3 + 486*a^4)/(a^5*b^4*x^14 + 4*a^6*b^
3*x^11 + 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2) - 770/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)
^(1/3))/(a^5*(a/b)^(2/3)) + 385/729*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(2/3)) - 770/729*log(x +
(a/b)^(1/3))/(a^5*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {770 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {770 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {385 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{2 \, a^{5} x^{2} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1054 \, b^{4} x^{10} + 3600 \, a b^{3} x^{7} + 4245 \, a^{2} b^{2} x^{4} + 1780 \, a^{3} b x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

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

[Out]

770/729*b*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^6*sgn(b*x^3 + a)) - 770/729*sqrt(3)*(-a*b^2)^(1/3)*arctan
(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*sgn(b*x^3 + a)) - 385/729*(-a*b^2)^(1/3)*log(x^2 + x*(-a/
b)^(1/3) + (-a/b)^(2/3))/(a^6*sgn(b*x^3 + a)) - 1/2/(a^5*x^2*sgn(b*x^3 + a)) - 1/972*(1054*b^4*x^10 + 3600*a*b
^3*x^7 + 4245*a^2*b^2*x^4 + 1780*a^3*b*x)/((b*x^3 + a)^4*a^5*sgn(b*x^3 + a))

Mupad [F(-1)]

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

[In]

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

[Out]

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