\(\int \frac {1}{(a+b x^2)^{13/6}} \, dx\) [111]

Optimal result

Integrand size = 11, antiderivative size = 604 \[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}+\frac {12 x}{7 a^2 \sqrt [6]{a+b x^2}}+\frac {12 \left (1+\sqrt {3}\right ) x \sqrt [6]{a+b x^2}}{7 a^2 \left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )}+\frac {12 \sqrt [4]{3} \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}}+\frac {2\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/7*x/a/(b*x^2+a)^(7/6)+12/7*x/a^2/(b*x^2+a)^(1/6)+12/7*(1+3^(1/2))*x*(b*x 
^2+a)^(1/6)/a^2/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))+12/7*3^(1/4)*(b*x^2+ 
a)^(1/6)*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^ 
2+a)^(2/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE((1-(a 
^(1/3)-(1-3^(1/2))*(b*x^2+a)^(1/3))^2/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3) 
)^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))/a^(5/3)/b/x/(-(b*x^2+a)^(1/3)*(a^(1/3) 
-(b*x^2+a)^(1/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))^2)^(1/2)+2/7*3^(3/ 
4)*(1-3^(1/2))*(b*x^2+a)^(1/6)*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3) 
*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))^2) 
^(1/2)*InverseJacobiAM(arccos((a^(1/3)-(1-3^(1/2))*(b*x^2+a)^(1/3))/(a^(1/ 
3)-(1+3^(1/2))*(b*x^2+a)^(1/3))),1/4*6^(1/2)+1/4*2^(1/2))/a^(5/3)/b/x/(-(b 
*x^2+a)^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/ 
3))^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 8.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\frac {3 a x+4 x \left (a+b x^2\right ) \sqrt [6]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},-\frac {b x^2}{a}\right )}{7 a^2 \left (a+b x^2\right )^{7/6}} \] Input:

Integrate[(a + b*x^2)^(-13/6),x]
 

Output:

(3*a*x + 4*x*(a + b*x^2)*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 
 3/2, -((b*x^2)/a)])/(7*a^2*(a + b*x^2)^(7/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {215, 214, 233, 833, 760, 2418}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {4 \int \frac {1}{\left (b x^2+a\right )^{7/6}}dx}{7 a}+\frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}\)

\(\Big \downarrow \) 214

\(\displaystyle \frac {4 \int \frac {1}{\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}d\frac {x}{\sqrt {b x^2+a}}}{7 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}+\frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}-\frac {6 \sqrt {-\frac {b x^2}{a+b x^2}} \int \frac {\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{7 a b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}-\frac {6 \sqrt {-\frac {b x^2}{a+b x^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right )}{7 a b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}-\frac {6 \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{7 a b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 x}{7 a \left (a+b x^2\right )^{7/6}}-\frac {6 \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1}\right )}{7 a b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\)

Input:

Int[(a + b*x^2)^(-13/6),x]
 

Output:

(3*x)/(7*a*(a + b*x^2)^(7/6)) - (6*Sqrt[-((b*x^2)/(a + b*x^2))]*((-2*Sqrt[ 
-1 + x^3/(a + b*x^2)^(3/2)])/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3 
)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))*Sqrt 
[(1 + x^2/(a + b*x^2) + (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 
 - (b*x^2)/(a + b*x^2))^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - (b* 
x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))], 
 -7 + 4*Sqrt[3]])/(Sqrt[-1 + x^3/(a + b*x^2)^(3/2)]*Sqrt[-((1 - (1 - (b*x^ 
2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2)] 
) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3 
))*Sqrt[(1 + x^2/(a + b*x^2) + (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[ 
3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - ( 
1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^( 
1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-1 + x^3/(a + b*x^2)^(3/2)]*Sqrt[-( 
(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x 
^2))^(1/3))^2)])))/(7*a*b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6))
 

Defintions of rubi rules used

rule 214

Int[((a_) + (b_.)*(x_)^2)^(-7/6), x_Symbol] :> Simp[1/((a + b*x^2)^(2/3)*(a 
/(a + b*x^2))^(2/3))   Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x 
^2]], x] /; FreeQ[{a, b}, x]
 

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {13}{6}}}d x\]

Input:

int(1/(b*x^2+a)^(13/6),x)
 

Output:

int(1/(b*x^2+a)^(13/6),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {13}{6}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(13/6),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(5/6)/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.04 \[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\frac {x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {13}{6} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {13}{6}}} \] Input:

integrate(1/(b*x**2+a)**(13/6),x)
 

Output:

x*hyper((1/2, 13/6), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(13/6)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {13}{6}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(13/6),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(-13/6), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {13}{6}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(13/6),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(-13/6), x)
 

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.06 \[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\frac {x\,{\left (\frac {b\,x^2}{a}+1\right )}^{13/6}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{6};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^2+a\right )}^{13/6}} \] Input:

int(1/(a + b*x^2)^(13/6),x)
 

Output:

(x*((b*x^2)/a + 1)^(13/6)*hypergeom([1/2, 13/6], 3/2, -(b*x^2)/a))/(a + b* 
x^2)^(13/6)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (a+b x^2\right )^{13/6}} \, dx=\frac {\left (b \,x^{2}+a \right )^{\frac {5}{6}} \left (3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a +3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,x^{2}-b^{2} x^{3}\right )}{3 a^{2} b \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int(1/(b*x^2+a)^(13/6),x)
 

Output:

((a + b*x**2)**(5/6)*(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a + 
3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*x**2 - b**2*x**3))/(3*a* 
*2*b*(a**2 + 2*a*b*x**2 + b**2*x**4))