\(\int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx\) [73]

Optimal result

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

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)*arctan(1/3* 
(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/((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)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\left (a+b x^3\right ) \left (3 a^{2/3}-2 \sqrt {3} b^{2/3} x^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b^{2/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 a^{5/3} x^2 \sqrt {\left (a+b x^3\right )^2}} \] Input:

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

Output:

-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])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.65, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1384, 27, 847, 750, 16, 1142, 25, 27, 1082, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 847

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\left (a+b x^3\right ) \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\)

Input:

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

Output:

((a + b*x^3)*(-1/2*1/(a*x^2) - (b*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^( 
1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) 
- Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))) 
)/a))/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
 

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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 847

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] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1384

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[p]))   Int[u*(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] && IntegerQ[p - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

Maple [C] (warning: unable to verify)

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

Time = 1.86 (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	\(\frac {\left (b \,x^{3}+a \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) x^{2}+\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) x^{2}-2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) x^{2}-3 \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a \,x^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\)	\(117\)

Input:

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

Output:

-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)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3}{6 \, a x^{2}} \] Input:

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

Output:

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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \] Input:

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

Output:

-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)

Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {1}{6} \, {\left (\frac {2 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2}} - \frac {3}{a x^{2}}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \] Input:

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

Output:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {2 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{2}+a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{2}-2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{2}-3 b^{\frac {1}{3}} a}{6 b^{\frac {1}{3}} a^{2} x^{2}} \] Input:

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

Output:

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