\(\int \frac {x (-2 q+p x^6) \sqrt {q+p x^6}}{b x^8+a (q+p x^6)^2} \, dx\) [2281]

Optimal result

Integrand size = 41, antiderivative size = 173 \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2 \sqrt {q+p x^6}}{\sqrt {a} q-\sqrt {b} x^4+\sqrt {a} p x^6}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^4}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^6}{\sqrt {2} \sqrt [4]{b}}}{x^2 \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \] Output:

-1/4*arctan(2^(1/2)*a^(1/4)*b^(1/4)*x^2*(p*x^6+q)^(1/2)/(a^(1/2)*q-b^(1/2) 
*x^4+a^(1/2)*p*x^6))*2^(1/2)/a^(3/4)/b^(1/4)-1/4*arctanh((1/2*a^(1/4)*q*2^ 
(1/2)/b^(1/4)+1/2*b^(1/4)*x^4*2^(1/2)/a^(1/4)+1/2*a^(1/4)*p*x^6*2^(1/2)/b^ 
(1/4))/x^2/(p*x^6+q)^(1/2))*2^(1/2)/a^(3/4)/b^(1/4)
 

Mathematica [F]

\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx \] Input:

Integrate[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]
 

Output:

Integrate[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2), x]
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.56, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {7265, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {x \left (p x^6-2 q\right ) \sqrt {p x^6+q}}{a \left (p x^6+q\right )^2+b x^8} \, dx\)

\(\Big \downarrow \) 7265

\(\displaystyle -\int \frac {1}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}\)

\(\Big \downarrow \) 755

\(\displaystyle -\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^4}{p x^6+q}}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {a}}-\frac {\int \frac {\frac {\sqrt {b} x^4}{p x^6+q}+\sqrt {a}}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {\int \frac {1}{\frac {x^4}{p x^6+q}-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b} \sqrt {p x^6+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^4}{p x^6+q}+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b} \sqrt {p x^6+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^4}{p x^6+q}}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {\int \frac {1}{-\frac {x^4}{p x^6+q}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^4}{p x^6+q}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^4}{p x^6+q}}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^4}{p x^6+q}}{\frac {b x^8}{\left (p x^6+q\right )^2}+a}d\frac {x^2}{\sqrt {p x^6+q}}}{2 \sqrt {a}}-\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\)

Input:

Int[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

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]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 7265

Int[(u_)*(v_)^(r_.)*(w_)^(s_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), 
 x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[(-c) 
*(q/(s + 1))   Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + r + 1)*w^ 
(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s 
 + 1) + q*(m*p + r + 1), 0] && NeQ[s, -1] && IntegerQ[q/(s + 1)] && Integer 
Q[m]
 

Maple [A] (verified)

Time = 2.71 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {p \,x^{6}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{6}+q}\, \sqrt {2}\, x^{2}+\sqrt {\frac {b}{a}}\, x^{4}+q}{p \,x^{6}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{6}+q}\, \sqrt {2}\, x^{2}+\sqrt {\frac {b}{a}}\, x^{4}+q}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{6}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {p \,x^{6}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}}+1\right )\right )}{8 \left (\frac {b}{a}\right )^{\frac {1}{4}} a}\)	\(161\)

Input:

int(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x,method=_RETURNVE 
RBOSE)
 

Output:

1/8/(b/a)^(1/4)*2^(1/2)*(ln((p*x^6-(b/a)^(1/4)*(p*x^6+q)^(1/2)*2^(1/2)*x^2 
+(b/a)^(1/2)*x^4+q)/(p*x^6+(b/a)^(1/4)*(p*x^6+q)^(1/2)*2^(1/2)*x^2+(b/a)^( 
1/2)*x^4+q))+2*arctan(2^(1/2)/(b/a)^(1/4)*(p*x^6+q)^(1/2)/x^2+1)-2*arctan( 
-2^(1/2)/(b/a)^(1/4)*(p*x^6+q)^(1/2)/x^2+1))/a
                                                                                    
                                                                                    
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.74 \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} + 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (i \, a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (-i \, a^{3} b p x^{8} - i \, a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} + 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) + \frac {1}{8} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (-i \, a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (i \, a^{3} b p x^{8} + i \, a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} + 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) \] Input:

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm 
="fricas")
 

Output:

1/8*(-1/(a^3*b))^(1/4)*log((a*p^2*x^12 + 2*a*p*q*x^6 - b*x^8 + a*q^2 + 2*( 
a*b*x^6*(-1/(a^3*b))^(1/4) + (a^3*b*p*x^8 + a^3*b*q*x^2)*(-1/(a^3*b))^(3/4 
))*sqrt(p*x^6 + q) - 2*(a^2*b*p*x^10 + a^2*b*q*x^4)*sqrt(-1/(a^3*b)))/(a*p 
^2*x^12 + 2*a*p*q*x^6 + b*x^8 + a*q^2)) - 1/8*(-1/(a^3*b))^(1/4)*log((a*p^ 
2*x^12 + 2*a*p*q*x^6 - b*x^8 + a*q^2 - 2*(a*b*x^6*(-1/(a^3*b))^(1/4) + (a^ 
3*b*p*x^8 + a^3*b*q*x^2)*(-1/(a^3*b))^(3/4))*sqrt(p*x^6 + q) - 2*(a^2*b*p* 
x^10 + a^2*b*q*x^4)*sqrt(-1/(a^3*b)))/(a*p^2*x^12 + 2*a*p*q*x^6 + b*x^8 + 
a*q^2)) - 1/8*I*(-1/(a^3*b))^(1/4)*log((a*p^2*x^12 + 2*a*p*q*x^6 - b*x^8 + 
 a*q^2 - 2*(I*a*b*x^6*(-1/(a^3*b))^(1/4) + (-I*a^3*b*p*x^8 - I*a^3*b*q*x^2 
)*(-1/(a^3*b))^(3/4))*sqrt(p*x^6 + q) + 2*(a^2*b*p*x^10 + a^2*b*q*x^4)*sqr 
t(-1/(a^3*b)))/(a*p^2*x^12 + 2*a*p*q*x^6 + b*x^8 + a*q^2)) + 1/8*I*(-1/(a^ 
3*b))^(1/4)*log((a*p^2*x^12 + 2*a*p*q*x^6 - b*x^8 + a*q^2 - 2*(-I*a*b*x^6* 
(-1/(a^3*b))^(1/4) + (I*a^3*b*p*x^8 + I*a^3*b*q*x^2)*(-1/(a^3*b))^(3/4))*s 
qrt(p*x^6 + q) + 2*(a^2*b*p*x^10 + a^2*b*q*x^4)*sqrt(-1/(a^3*b)))/(a*p^2*x 
^12 + 2*a*p*q*x^6 + b*x^8 + a*q^2))
 

Sympy [F]

\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int \frac {x \left (p x^{6} - 2 q\right ) \sqrt {p x^{6} + q}}{a p^{2} x^{12} + 2 a p q x^{6} + a q^{2} + b x^{8}}\, dx \] Input:

integrate(x*(p*x**6-2*q)*(p*x**6+q)**(1/2)/(b*x**8+a*(p*x**6+q)**2),x)
 

Output:

Integral(x*(p*x**6 - 2*q)*sqrt(p*x**6 + q)/(a*p**2*x**12 + 2*a*p*q*x**6 + 
a*q**2 + b*x**8), x)
 

Maxima [F]

\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a} \,d x } \] Input:

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm 
="maxima")
 

Output:

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)*x/(b*x^8 + (p*x^6 + q)^2*a), x)
 

Giac [F]

\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a} \,d x } \] Input:

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm 
="giac")
 

Output:

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)*x/(b*x^8 + (p*x^6 + q)^2*a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int -\frac {x\,\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{a\,{\left (p\,x^6+q\right )}^2+b\,x^8} \,d x \] Input:

int(-(x*(q + p*x^6)^(1/2)*(2*q - p*x^6))/(a*(q + p*x^6)^2 + b*x^8),x)
 

Output:

int(-(x*(q + p*x^6)^(1/2)*(2*q - p*x^6))/(a*(q + p*x^6)^2 + b*x^8), x)
 

Reduce [F]

\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\left (\int \frac {\sqrt {p \,x^{6}+q}\, x^{7}}{a \,p^{2} x^{12}+2 a p q \,x^{6}+b \,x^{8}+a \,q^{2}}d x \right ) p -2 \left (\int \frac {\sqrt {p \,x^{6}+q}\, x}{a \,p^{2} x^{12}+2 a p q \,x^{6}+b \,x^{8}+a \,q^{2}}d x \right ) q \] Input:

int(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x)
 

Output:

int((sqrt(p*x**6 + q)*x**7)/(a*p**2*x**12 + 2*a*p*q*x**6 + a*q**2 + b*x**8 
),x)*p - 2*int((sqrt(p*x**6 + q)*x)/(a*p**2*x**12 + 2*a*p*q*x**6 + a*q**2 
+ b*x**8),x)*q