\(\int \frac {1}{x^4 (a+b x^2)^{7/6}} \, dx\) [1117]

Optimal result

Integrand size = 15, antiderivative size = 627 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac {40 \left (1+\sqrt {3}\right ) b^2 x \sqrt [6]{a+b x^2}}{9 a^3 \left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )}+\frac {40 b \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 3^{3/4} a^{8/3} x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}}+\frac {20 \left (1-\sqrt {3}\right ) b \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} a^{8/3} x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/a/x^3/(b*x^2+a)^(1/6)-10/3*(b*x^2+a)^(5/6)/a^2/x^3+40/9*b*(b*x^2+a)^(5/6 
)/a^3/x+40/9*(1+3^(1/2))*b^2*x*(b*x^2+a)^(1/6)/a^3/(a^(1/3)-(1+3^(1/2))*(b 
*x^2+a)^(1/3))+40/9*b*(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))*3^(1/4)/a 
^(8/3)/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)+20/27*(1-3^(1/2))*b*(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))*3^(3/4)/a^(8/3)/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 = 10.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=-\frac {\sqrt [6]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {7}{6},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 a x^3 \sqrt [6]{a+b x^2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {253, 264, 264, 235, 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.

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {10 \int \frac {1}{x^4 \sqrt [6]{b x^2+a}}dx}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {10 \left (-\frac {4 b \int \frac {1}{x^2 \sqrt [6]{b x^2+a}}dx}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt [6]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 235

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {1}{2} a \int \frac {1}{\left (b x^2+a\right )^{7/6}}dx\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 214

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {a \int \frac {1}{\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}d\frac {x}{\sqrt {b x^2+a}}}{2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \int \frac {\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {10 \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{a}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}}\)

Input:

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

Output:

3/(a*x^3*(a + b*x^2)^(1/6)) + (10*(-1/3*(a + b*x^2)^(5/6)/(a*x^3) - (4*b*( 
-((a + b*x^2)^(5/6)/(a*x)) + (2*b*((3*x)/(2*(a + b*x^2)^(1/6)) + (3*a*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]*Ellip 
ticE[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 - Sq 
rt[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)])))/(4*b*x*(a/(a + b*x^2 
))^(2/3)*(a + b*x^2)^(1/6))))/(3*a)))/(9*a)))/a
 

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 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 235

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

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, 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]

\[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{6}}}d x\]

Input:

int(1/x^4/(b*x^2+a)^(7/6),x)
 

Output:

int(1/x^4/(b*x^2+a)^(7/6),x)
 

Fricas [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(7/6),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(5/6)/(b^2*x^8 + 2*a*b*x^6 + a^2*x^4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {7}{6} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {7}{6}} x^{3}} \] Input:

integrate(1/x**4/(b*x**2+a)**(7/6),x)
 

Output:

-hyper((-3/2, 7/6), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(7/6)*x**3)
 

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(7/6),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(7/6)*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(7/6),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(7/6)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{7/6}} \,d x \] Input:

int(1/(x^4*(a + b*x^2)^(7/6)),x)
 

Output:

int(1/(x^4*(a + b*x^2)^(7/6)), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx=\frac {8 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,x^{3}-a^{2}+4 a b \,x^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {1}{6}} a^{3} x^{3}} \] Input:

int(1/x^4/(b*x^2+a)^(7/6),x)
 

Output:

((a + b*x**2)**(5/6)*(8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*x* 
*3 - a**2 + 4*a*b*x**2))/(3*a**3*x**3*(a + b*x**2))