\(\int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx\) [96]

Optimal result

Integrand size = 26, antiderivative size = 243 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {-a-b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[Out]

1/2*(-b*x^3-a)/a/x^2/((b*x^3+a)^2)^(1/2)-1/3*b^(2/3)*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/((b*x^3+a)^2)^(1/
2)+1/6*b^(2/3)*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/((b*x^3+a)^2)^(1/2)+1/3*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^(5/3)*3^(1/2)/((b*x^3+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1369, 331, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[In]

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

[Out]

-1/2*(a + b*x^3)/(a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (b^(2/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^
(1/3)*x])/(3*a^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (b^(2/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(6*a^(5/3)*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 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 (a b+b^2 x^3\right ) \int \frac {1}{x^3 \left (a b+b^2 x^3\right )} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a b+b^2 x^3} \, dx}{a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\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}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\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}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\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}{6 a^{5/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (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}{2 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {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 )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a b+b^2 x^3\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {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 )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {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 )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\left (a+b x^3\right ) \left (3 a^{2/3}-2 \sqrt {3} b^{2/3} x^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b^{2/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 a^{5/3} x^2 \sqrt {\left (a+b x^3\right )^2}} \]

[In]

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

[Out]

-1/6*((a + b*x^3)*(3*a^(2/3) - 2*Sqrt[3]*b^(2/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(2/3)*x
^2*Log[a^(1/3) + b^(1/3)*x] - b^(2/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(5/3)*x^2*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 = 3.57 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.39

method	result	size
risch	\(-\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 \left (b \,x^{3}+a \right ) a \,x^{2}}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5}-3 b^{2}\right ) x -a^{2} b \textit {\_R} \right )\right )}{3 b \,x^{3}+3 a}\)	\(94\)
default	\(-\frac {\left (b \,x^{3}+a \right ) \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) x^{2}+2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) x^{2}-\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) x^{2}+3 \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a \,x^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\)	\(118\)

[In]

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

[Out]

-1/2*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/a/x^2+1/3*((b*x^3+a)^2)^(1/2)/(b*x^3+a)*sum(_R*ln((-4*_R^3*a^5-3*b^2)*x-a^2
*b*_R),_R=RootOf(_Z^3*a^5+b^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3}{6 \, a x^{2}} \]

[In]

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

[Out]

1/6*(2*sqrt(3)*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) - 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)) + 2*x^2*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^
2/a^2)^(1/3)) - 3)/(a*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \]

[In]

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b)^(2/3)) + 1/6*log(x^2 - x*(a/b)^(1/3)
+ (a/b)^(2/3))/(a*(a/b)^(2/3)) - 1/3*log(x + (a/b)^(1/3))/(a*(a/b)^(2/3)) - 1/2/(a*x^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {1}{6} \, {\left (\frac {2 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2}} - \frac {3}{a x^{2}}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]

[In]

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

[Out]

1/6*(2*b*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^2 - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/
b)^(1/3))/(-a/b)^(1/3))/a^2 - (-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^2 - 3/(a*x^2))*sgn(b*x
^3 + a)

Mupad [F(-1)]

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

[In]

int(1/(x^3*((a + b*x^3)^2)^(1/2)),x)

[Out]

int(1/(x^3*((a + b*x^3)^2)^(1/2)), x)