3.11.38 \(\int \frac {1}{x^6 (a+b x^2)^{7/6}} \, dx\) [1038]

3.11.38.1 Optimal result

Integrand size = 15, antiderivative size = 680 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{7/6}} \, dx=\frac {3}{a x^5 \sqrt [6]{a+b x^2}}+\frac {128 b^3 x}{27 a^4 \sqrt [6]{a+b x^2}}-\frac {16 \left (a+b x^2\right )^{5/6}}{5 a^2 x^5}+\frac {32 b \left (a+b x^2\right )^{5/6}}{9 a^3 x^3}-\frac {128 b^2 \left (a+b x^2\right )^{5/6}}{27 a^4 x}+\frac {128 b^3 x}{27 a^3 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}+\frac {64 \sqrt {2+\sqrt {3}} b^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{9\ 3^{3/4} a^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}}-\frac {128 \sqrt {2} b^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right ),-7+4 \sqrt {3}\right )}{27 \sqrt [4]{3} a^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]

output

3/a/x^5/(b*x^2+a)^(1/6)+128/27*b^3*x/a^4/(b*x^2+a)^(1/6)-16/5*(b*x^2+a)^(5 
/6)/a^2/x^5+32/9*b*(b*x^2+a)^(5/6)/a^3/x^3-128/27*b^2*(b*x^2+a)^(5/6)/a^4/ 
x+128/27*b^3*x/a^3/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^(7/6)/(1-(a/(b*x^2+a))^(1 
/3)-3^(1/2))-128/81*b^2*(1-(a/(b*x^2+a))^(1/3))*EllipticF((1-(a/(b*x^2+a)) 
^(1/3)+3^(1/2))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2)),2*I-I*3^(1/2))*2^(1/2)*((1 
+(a/(b*x^2+a))^(1/3)+(a/(b*x^2+a))^(2/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^ 
2)^(1/2)*3^(3/4)/a^3/x/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^(1/6)/((-1+(a/(b*x^2+ 
a))^(1/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)+64/27*b^2*(1-(a/(b*x^2 
+a))^(1/3))*EllipticE((1-(a/(b*x^2+a))^(1/3)+3^(1/2))/(1-(a/(b*x^2+a))^(1/ 
3)-3^(1/2)),2*I-I*3^(1/2))*((1+(a/(b*x^2+a))^(1/3)+(a/(b*x^2+a))^(2/3))/(1 
-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(1/4)/a 
^3/x/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^(1/6)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/( 
b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)
 

3.11.38.2 Mathematica [C] (verified)

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

Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{7/6}} \, dx=-\frac {\sqrt [6]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {7}{6},-\frac {3}{2},-\frac {b x^2}{a}\right )}{5 a x^5 \sqrt [6]{a+b x^2}} \]

input

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

output

-1/5*((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-5/2, 7/6, -3/2, -((b*x^2)/a 
)])/(a*x^5*(a + b*x^2)^(1/6))
 

3.11.38.3 Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {253, 264, 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.

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {16 \int \frac {1}{x^6 \sqrt [6]{b x^2+a}}dx}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {16 \left (-\frac {2 b \int \frac {1}{x^4 \sqrt [6]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \int \frac {1}{x^2 \sqrt [6]{b x^2+a}}dx}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt [6]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 235

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {1}{2} a \int \frac {1}{\left (b x^2+a\right )^{7/6}}dx\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 214

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {a \int \frac {1}{\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}d\frac {x}{\sqrt {b x^2+a}}}{2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \int \frac {\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (a+b x^2\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{a+b x^2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {16 \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 b \left (\frac {3 x}{2 \sqrt [6]{b x^2+a}}+\frac {3 a \sqrt {-\frac {b x^2}{b x^2+a}} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right ) \sqrt {\frac {\frac {x^2}{b x^2+a}+\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right ) \sqrt {\frac {\frac {x^2}{b x^2+a}+\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )}{4 b \left (\frac {a}{b x^2+a}\right )^{2/3} \sqrt [6]{b x^2+a} x}\right )}{3 a}-\frac {\left (b x^2+a\right )^{5/6}}{a x}\right )}{9 a}-\frac {\left (b x^2+a\right )^{5/6}}{3 a x^3}\right )}{3 a}-\frac {\left (b x^2+a\right )^{5/6}}{5 a x^5}\right )}{a}+\frac {3}{a x^5 \sqrt [6]{b x^2+a}}\)

input

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

output

3/(a*x^5*(a + b*x^2)^(1/6)) + (16*(-1/5*(a + b*x^2)^(5/6)/(a*x^5) - (2*b*( 
-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]*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)])))/(4*b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6))))/(3*a) 
))/(9*a)))/(3*a)))/a
 

3.11.38.3.1 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]
 

3.11.38.4 Maple [F]

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

input

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

output

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

3.11.38.5 Fricas [F]

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

input

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

output

integral((b*x^2 + a)^(5/6)/(b^2*x^10 + 2*a*b*x^8 + a^2*x^6), x)
 

3.11.38.6 Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{7/6}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {7}{6} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {7}{6}} x^{5}} \]

input

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

output

-hyper((-5/2, 7/6), (-3/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(7/6)*x**5)
 

3.11.38.7 Maxima [F]

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

input

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

output

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

3.11.38.8 Giac [F]

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

input

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

output

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

3.11.38.9 Mupad [F(-1)]

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

input

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

output

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