\(\int \frac {(a+b x^2)^{4/3}}{x^3} \, dx\) [743]

Optimal result

Integrand size = 15, antiderivative size = 119 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=\frac {3}{2} b \sqrt [3]{a+b x^2}-\frac {a \sqrt [3]{a+b x^2}}{2 x^2}-\frac {2 \sqrt [3]{a} b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x)+\sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (a-3 b x^2\right ) \sqrt [3]{a+b x^2}}{x^2}-4 \sqrt {3} \sqrt [3]{a} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 \sqrt [3]{a} b \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )-2 \sqrt [3]{a} b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {243, 51, 60, 69, 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.

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

(-((a + b*x^2)^(4/3)/x^2) + (4*b*(3*(a + b*x^2)^(1/3) + a*(-((Sqrt[3]*ArcT 
an[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3)) - Log[x^2]/(2*a^ 
(2/3)) + (3*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(2*a^(2/3)))))/3)/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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60

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

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   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 = 0.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\left (9 b \,x^{2}-3 a \right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}-2 b \,x^{2} a^{\frac {1}{3}} \left (2 \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}+\ln \left (a^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\left (b \,x^{2}+a \right )^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{2}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right )}{6 x^{2}}\)	\(111\)

Input:

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

Output:

1/6*((9*b*x^2-3*a)*(b*x^2+a)^(1/3)-2*b*x^2*a^(1/3)*(2*arctan(2/3*3^(1/2)/a 
^(1/3)*(b*x^2+a)^(1/3)+1/3*3^(1/2))*3^(1/2)+ln(a^(2/3)+a^(1/3)*(b*x^2+a)^( 
1/3)+(b*x^2+a)^(2/3))-2*ln((b*x^2+a)^(1/3)-a^(1/3))))/x^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=-\frac {4 \, \sqrt {3} a^{\frac {1}{3}} b x^{2} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 2 \, a^{\frac {1}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{6 \, x^{2}} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.39 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=- \frac {b^{\frac {4}{3}} x^{\frac {2}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (\frac {2}{3}\right )} \] Input:

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

Output:

-b**(4/3)*x**(2/3)*gamma(-1/3)*hyper((-4/3, -1/3), (2/3,), a*exp_polar(I*p 
i)/(b*x**2))/(2*gamma(2/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=-\frac {2}{3} \, \sqrt {3} a^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {2}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}} a}{2 \, x^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=-\frac {1}{6} \, {\left (4 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2 \, a^{\frac {1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) - 9 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a}{b x^{2}}\right )} b \] Input:

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

Output:

-1/6*(4*sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3)) 
/a^(1/3)) + 2*a^(1/3)*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + 
a^(2/3)) - 4*a^(1/3)*log(abs((b*x^2 + a)^(1/3) - a^(1/3))) - 9*(b*x^2 + a) 
^(1/3) + 3*(b*x^2 + a)^(1/3)*a/(b*x^2))*b
 

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=\frac {3\,b\,{\left (b\,x^2+a\right )}^{1/3}}{2}-\frac {a\,{\left (b\,x^2+a\right )}^{1/3}}{2\,x^2}+\frac {2\,a^{1/3}\,b\,\ln \left (6\,a^{4/3}\,b-6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}\right )}{3}+\frac {a^{1/3}\,b\,\ln \left (6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}-3\,a^{4/3}\,b\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3}-\frac {a^{1/3}\,b\,\ln \left (3\,a^{4/3}\,b\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )+6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3} \] Input:

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

Output:

(3*b*(a + b*x^2)^(1/3))/2 - (a*(a + b*x^2)^(1/3))/(2*x^2) + (2*a^(1/3)*b*l 
og(6*a^(4/3)*b - 6*a*b*(a + b*x^2)^(1/3)))/3 + (a^(1/3)*b*log(6*a*b*(a + b 
*x^2)^(1/3) - 3*a^(4/3)*b*(3^(1/2)*1i - 1))*(3^(1/2)*1i - 1))/3 - (a^(1/3) 
*b*log(3*a^(4/3)*b*(3^(1/2)*1i + 1) + 6*a*b*(a + b*x^2)^(1/3))*(3^(1/2)*1i 
 + 1))/3
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx=\frac {-3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a +9 \left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}+8 \left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {1}{3}}}{b \,x^{3}+a x}d x \right ) a b \,x^{2}}{6 x^{2}} \] Input:

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

Output:

( - 3*(a + b*x**2)**(1/3)*a + 9*(a + b*x**2)**(1/3)*b*x**2 + 8*int((a + b* 
x**2)**(1/3)/(a*x + b*x**3),x)*a*b*x**2)/(6*x**2)