\(\int \frac {1}{x (a+b x^2)^{4/3}} \, dx\) [778]

Optimal result

Integrand size = 15, antiderivative size = 104 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\frac {3}{2 a \sqrt [3]{a+b x^2}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}} \] Output:

3/2/a/(b*x^2+a)^(1/3)+1/2*3^(1/2)*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))*3 
^(1/2)/a^(1/3))/a^(4/3)-1/2*ln(x)/a^(4/3)+3/4*ln(a^(1/3)-(b*x^2+a)^(1/3))/ 
a^(4/3)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{a}}{\sqrt [3]{a+b x^2}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )}{4 a^{4/3}} \] Input:

Integrate[1/(x*(a + b*x^2)^(4/3)),x]
 

Output:

((6*a^(1/3))/(a + b*x^2)^(1/3) + 2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^2)^(1/3 
))/a^(1/3))/Sqrt[3]] + 2*Log[-a^(1/3) + (a + b*x^2)^(1/3)] - Log[a^(2/3) + 
 a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/(4*a^(4/3))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {243, 61, 67, 16, 1082, 217}

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.

\(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx^2\)

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{2} \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} \left (\frac {-\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\)

\(\Big \downarrow \) 217

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

Input:

Int[1/(x*(a + b*x^2)^(4/3)),x]
 

Output:

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

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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {\sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{2}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\ln \left (\left (b \,x^{2}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}-\frac {\ln \left (a^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\left (b \,x^{2}+a \right )^{\frac {2}{3}}\right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2}+3 a^{\frac {1}{3}}}{2 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{\frac {4}{3}}}\)	\(122\)

Input:

int(1/x/(b*x^2+a)^(4/3),x,method=_RETURNVERBOSE)
 

Output:

1/2*(3^(1/2)*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))*3^(1/2)/a^(1/3))*(b*x^ 
2+a)^(1/3)+ln((b*x^2+a)^(1/3)-a^(1/3))*(b*x^2+a)^(1/3)-1/2*ln(a^(2/3)+a^(1 
/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))*(b*x^2+a)^(1/3)+3*a^(1/3))/(b*x^2+a)^ 
(1/3)/a^(4/3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.14 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\left [\frac {\sqrt {3} {\left (a b x^{2} + a^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{2} + \sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{2}}\right ) - {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, {\left (a^{2} b x^{2} + a^{3}\right )}}, -\frac {{\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {2 \, \sqrt {3} {\left (a b x^{2} + a^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 6 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, {\left (a^{2} b x^{2} + a^{3}\right )}}\right ] \] Input:

integrate(1/x/(b*x^2+a)^(4/3),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/4*(sqrt(3)*(a*b*x^2 + a^2)*sqrt(-1/a^(2/3))*log((2*b*x^2 + sqrt(3)*(2*( 
b*x^2 + a)^(2/3)*a^(2/3) - (b*x^2 + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) 
 - 3*(b*x^2 + a)^(1/3)*a^(2/3) + 3*a)/x^2) - (b*x^2 + a)*a^(2/3)*log((b*x^ 
2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*(b*x^2 + a)*a^(2/3 
)*log((b*x^2 + a)^(1/3) - a^(1/3)) + 6*(b*x^2 + a)^(2/3)*a)/(a^2*b*x^2 + a 
^3), -1/4*((b*x^2 + a)*a^(2/3)*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a 
^(1/3) + a^(2/3)) - 2*(b*x^2 + a)*a^(2/3)*log((b*x^2 + a)^(1/3) - a^(1/3)) 
 - 2*sqrt(3)*(a*b*x^2 + a^2)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^( 
1/3))/a^(1/3))/a^(1/3) - 6*(b*x^2 + a)^(2/3)*a)/(a^2*b*x^2 + a^3)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=- \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac {4}{3}} x^{\frac {8}{3}} \Gamma \left (\frac {7}{3}\right )} \] Input:

integrate(1/x/(b*x**2+a)**(4/3),x)
 

Output:

-gamma(4/3)*hyper((4/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(4 
/3)*x**(8/3)*gamma(7/3))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{2 \, a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{4 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {3}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a} \] Input:

integrate(1/x/(b*x^2+a)^(4/3),x, algorithm="maxima")
 

Output:

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^ 
(4/3) - 1/4*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a 
^(4/3) + 1/2*log((b*x^2 + a)^(1/3) - a^(1/3))/a^(4/3) + 3/2/((b*x^2 + a)^( 
1/3)*a)
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{2 \, a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{4 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{2 \, a^{\frac {4}{3}}} + \frac {3}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a} \] Input:

integrate(1/x/(b*x^2+a)^(4/3),x, algorithm="giac")
 

Output:

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^ 
(4/3) - 1/4*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a 
^(4/3) + 1/2*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(4/3) + 3/2/((b*x^2 + 
 a)^(1/3)*a)
 

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-18\,a^{4/3}\right )}{2\,a^{4/3}}+\frac {3}{2\,a\,{\left (b\,x^2+a\right )}^{1/3}}+\frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,a^{4/3}}-\frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,a^{4/3}} \] Input:

int(1/(x*(a + b*x^2)^(4/3)),x)
 

Output:

log(18*a*(a + b*x^2)^(1/3) - 18*a^(4/3))/(2*a^(4/3)) + 3/(2*a*(a + b*x^2)^ 
(1/3)) + (log(18*a*(a + b*x^2)^(1/3) - (9*a^(4/3)*(3^(1/2)*1i - 1)^2)/2)*( 
3^(1/2)*1i - 1))/(4*a^(4/3)) - (log(18*a*(a + b*x^2)^(1/3) - (9*a^(4/3)*(3 
^(1/2)*1i + 1)^2)/2)*(3^(1/2)*1i + 1))/(4*a^(4/3))
 

Reduce [F]

\[ \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a x +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \] Input:

int(1/x/(b*x^2+a)^(4/3),x)
 

Output:

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