\(\int x^6 (a+b x^2)^{5/6} \, dx\) [1082]

Optimal result

Integrand size = 15, antiderivative size = 655 \[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\frac {405 a^3 x \left (a+b x^2\right )^{5/6}}{11648 b^3}-\frac {45 a^2 x^3 \left (a+b x^2\right )^{5/6}}{1456 b^2}+\frac {3 a x^5 \left (a+b x^2\right )^{5/6}}{104 b}+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}+\frac {1215 \left (1+\sqrt {3}\right ) a^4 x \sqrt [6]{a+b x^2}}{23296 b^3 \left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )}+\frac {1215 \sqrt [4]{3} a^{13/3} \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{23296 b^4 x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}}+\frac {405\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{13/3} \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{46592 b^4 x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:

405/11648*a^3*x*(b*x^2+a)^(5/6)/b^3-45/1456*a^2*x^3*(b*x^2+a)^(5/6)/b^2+3/ 
104*a*x^5*(b*x^2+a)^(5/6)/b+3/26*x^7*(b*x^2+a)^(5/6)+1215/23296*(1+3^(1/2) 
)*a^4*x*(b*x^2+a)^(1/6)/b^3/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))+1215/232 
96*3^(1/4)*a^(13/3)*(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))/b^4/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)+405/46592*3^(3/4)*(1-3^(1/2))*a^(13/3)*(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))/b^4/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 = 9.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.16 \[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\frac {3 x \left (a+b x^2\right )^{5/6} \left (\left (1+\frac {b x^2}{a}\right )^{5/6} \left (27 a^3-15 a^2 b x^2+14 a b^2 x^4+56 b^3 x^6\right )-27 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{1456 b^3 \left (1+\frac {b x^2}{a}\right )^{5/6}} \] Input:

Integrate[x^6*(a + b*x^2)^(5/6),x]
 

Output:

(3*x*(a + b*x^2)^(5/6)*((1 + (b*x^2)/a)^(5/6)*(27*a^3 - 15*a^2*b*x^2 + 14* 
a*b^2*x^4 + 56*b^3*x^6) - 27*a^3*Hypergeometric2F1[-5/6, 1/2, 3/2, -((b*x^ 
2)/a)]))/(1456*b^3*(1 + (b*x^2)/a)^(5/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {248, 262, 262, 262, 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.

\(\displaystyle \int x^6 \left (a+b x^2\right )^{5/6} \, dx\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {5}{26} a \int \frac {x^6}{\sqrt [6]{b x^2+a}}dx+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \int \frac {x^4}{\sqrt [6]{b x^2+a}}dx}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \int \frac {x^2}{\sqrt [6]{b x^2+a}}dx}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \int \frac {1}{\sqrt [6]{b x^2+a}}dx}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 235

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {1}{2} a \int \frac {1}{\left (b x^2+a\right )^{7/6}}dx\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 214

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 x}{2 \sqrt [6]{a+b x^2}}-\frac {a \int \frac {1}{\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}d\frac {x}{\sqrt {b x^2+a}}}{2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \int \frac {\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {5}{26} a \left (\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (a+b x^2\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )+\frac {3}{26} x^7 \left (a+b x^2\right )^{5/6}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3}{26} \left (b x^2+a\right )^{5/6} x^7+\frac {5}{26} a \left (\frac {3 x^5 \left (b x^2+a\right )^{5/6}}{20 b}-\frac {3 a \left (\frac {3 x^3 \left (b x^2+a\right )^{5/6}}{14 b}-\frac {9 a \left (\frac {3 x \left (b x^2+a\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 x}{2 \sqrt [6]{b x^2+a}}+\frac {3 a \sqrt {-\frac {b x^2}{b x^2+a}} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right ) \sqrt {\frac {\frac {x^2}{b x^2+a}+\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}\right ) \sqrt {\frac {\frac {x^2}{b x^2+a}+\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{\left (-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )}{4 b \left (\frac {a}{b x^2+a}\right )^{2/3} \sqrt [6]{b x^2+a} x}\right )}{8 b}\right )}{14 b}\right )}{4 b}\right )\)

Input:

Int[x^6*(a + b*x^2)^(5/6),x]
 

Output:

(3*x^7*(a + b*x^2)^(5/6))/26 + (5*a*((3*x^5*(a + b*x^2)^(5/6))/(20*b) - (3 
*a*((3*x^3*(a + b*x^2)^(5/6))/(14*b) - (9*a*((3*x*(a + b*x^2)^(5/6))/(8*b) 
 - (3*a*((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 - Sqr 
t[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 + Sq 
rt[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) 
)))/(8*b)))/(14*b)))/(4*b)))/26
 

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 248

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) 
  Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ 
p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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]

\[\int x^{6} \left (b \,x^{2}+a \right )^{\frac {5}{6}}d x\]

Input:

int(x^6*(b*x^2+a)^(5/6),x)
 

Output:

int(x^6*(b*x^2+a)^(5/6),x)
 

Fricas [F]

\[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6} \,d x } \] Input:

integrate(x^6*(b*x^2+a)^(5/6),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.96 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.04 \[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\frac {a^{\frac {5}{6}} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} \] Input:

integrate(x**6*(b*x**2+a)**(5/6),x)
 

Output:

a**(5/6)*x**7*hyper((-5/6, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/7
 

Maxima [F]

\[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6} \,d x } \] Input:

integrate(x^6*(b*x^2+a)^(5/6),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(5/6)*x^6, x)
 

Giac [F]

\[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6} \,d x } \] Input:

integrate(x^6*(b*x^2+a)^(5/6),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(5/6)*x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\int x^6\,{\left (b\,x^2+a\right )}^{5/6} \,d x \] Input:

int(x^6*(a + b*x^2)^(5/6),x)
 

Output:

int(x^6*(a + b*x^2)^(5/6), x)
 

Reduce [F]

\[ \int x^6 \left (a+b x^2\right )^{5/6} \, dx=\int \left (b \,x^{2}+a \right )^{\frac {5}{6}} x^{6}d x \] Input:

int(x^6*(b*x^2+a)^(5/6),x)
 

Output:

int((a + b*x**2)**(5/6)*x**6,x)