Integrand size = 40, antiderivative size = 167 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^3}}{\sqrt {a} q-\sqrt {b} x^2+\sqrt {a} p x^3}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^3}{\sqrt {2} \sqrt [4]{b}}}{x \sqrt {q+p x^3}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \]
-1/2*arctan(2^(1/2)*a^(1/4)*b^(1/4)*x*(p*x^3+q)^(1/2)/(a^(1/2)*q-b^(1/2)*x ^2+a^(1/2)*p*x^3))*2^(1/2)/a^(3/4)/b^(1/4)-1/2*arctanh((1/2*a^(1/4)*q*2^(1 /2)/b^(1/4)+1/2*b^(1/4)*x^2*2^(1/2)/a^(1/4)+1/2*a^(1/4)*p*x^3*2^(1/2)/b^(1 /4))/x/(p*x^3+q)^(1/2))*2^(1/2)/a^(3/4)/b^(1/4)
Time = 1.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^3}}{-\sqrt {b} x^2+\sqrt {a} \left (q+p x^3\right )}\right )+\text {arctanh}\left (\frac {\sqrt {b} x^2+\sqrt {a} \left (q+p x^3\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^3}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \]
-((ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^3])/(-(Sqrt[b]*x^2) + Sq rt[a]*(q + p*x^3))] + ArcTanh[(Sqrt[b]*x^2 + Sqrt[a]*(q + p*x^3))/(Sqrt[2] *a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^3])])/(Sqrt[2]*a^(3/4)*b^(1/4)))
Time = 0.68 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {7263, 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.
| \(\displaystyle \int \frac {\left (p x^3-2 q\right ) \sqrt {p x^3+q}}{a \left (p x^3+q\right )^2+b x^4} \, dx\) |
| \(\Big \downarrow \) 7263 |
| \(\displaystyle -2 \int \frac {1}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -2 \left (\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^3+q}}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {a}}+\frac {\int \frac {\frac {\sqrt {b} x^2}{p x^3+q}+\sqrt {a}}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{\frac {x^2}{p x^3+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^2}{p x^3+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^3+q}}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{-\frac {x^2}{p x^3+q}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^2}{p x^3+q}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^3+q}}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (\frac {\int \frac {\sqrt {a}-\frac {\sqrt {b} x^2}{p x^3+q}}{\frac {b x^4}{\left (p x^3+q\right )^2}+a}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^3+q}}}{\sqrt [4]{b} \left (\frac {x^2}{p x^3+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {x^2}{p x^3+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^3+q}}}{\sqrt [4]{b} \left (\frac {x^2}{p x^3+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {x^2}{p x^3+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} x}{\sqrt {p x^3+q}}}{\frac {x^2}{p x^3+q}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt [4]{a}}{\frac {x^2}{p x^3+q}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b} \sqrt {p x^3+q}}+\frac {\sqrt {a}}{\sqrt {b}}}d\frac {x}{\sqrt {p x^3+q}}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )\) |
-2*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a^(1/4)*Sqrt[q + p*x^3])]/(Sqrt[2]* a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a^(1/4)*Sqrt[q + p*x^3 ])]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] + (Sqrt[b]* x^2)/(q + p*x^3) - (Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[q + p*x^3]]/(Sqrt[2]*a ^(1/4)*b^(1/4)) + Log[Sqrt[a] + (Sqrt[b]*x^2)/(q + p*x^3) + (Sqrt[2]*a^(1/ 4)*b^(1/4)*x)/Sqrt[q + p*x^3]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]))
3.23.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]]))
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]
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]
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]
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]
Int[(u_)*(v_)^(r_.)*((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 Subst[ Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{ a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && Inte gerQ[m]
Time = 1.86 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{3}+q}\, \sqrt {2}\, x +p \,x^{3}+\sqrt {\frac {b}{a}}\, x^{2}+q}{\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{3}+q}\, \sqrt {2}\, x +p \,x^{3}+\sqrt {\frac {b}{a}}\, x^{2}+q}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{3}+q}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{3}+q}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}\right )\right )}{4 \left (\frac {b}{a}\right )^{\frac {1}{4}} a}\) | \(175\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{3}+q}\, \sqrt {2}\, x +p \,x^{3}+\sqrt {\frac {b}{a}}\, x^{2}+q}{\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{3}+q}\, \sqrt {2}\, x +p \,x^{3}+\sqrt {\frac {b}{a}}\, x^{2}+q}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{3}+q}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {p \,x^{3}+q}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x}\right )\right )}{4 \left (\frac {b}{a}\right )^{\frac {1}{4}} a}\) | \(175\) |
| elliptic | \(\text {Expression too large to display}\) | \(1164\) |
1/4/(b/a)^(1/4)*2^(1/2)*(ln((-(b/a)^(1/4)*(p*x^3+q)^(1/2)*2^(1/2)*x+p*x^3+ (b/a)^(1/2)*x^2+q)/((b/a)^(1/4)*(p*x^3+q)^(1/2)*2^(1/2)*x+p*x^3+(b/a)^(1/2 )*x^2+q))+2*arctan((2^(1/2)*(p*x^3+q)^(1/2)+(b/a)^(1/4)*x)/(b/a)^(1/4)/x)+ 2*arctan((2^(1/2)*(p*x^3+q)^(1/2)-(b/a)^(1/4)*x)/(b/a)^(1/4)/x))/a
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.83 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} + 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} - 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} - 2 \, {\left (i \, a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (-i \, a^{3} b p x^{4} - i \, a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} + 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} - 2 \, {\left (-i \, a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (i \, a^{3} b p x^{4} + i \, a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} + 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) \]
1/4*(-1/(a^3*b))^(1/4)*log((a*p^2*x^6 + 2*a*p*q*x^3 - b*x^4 + a*q^2 + 2*(a *b*x^3*(-1/(a^3*b))^(1/4) + (a^3*b*p*x^4 + a^3*b*q*x)*(-1/(a^3*b))^(3/4))* sqrt(p*x^3 + q) - 2*(a^2*b*p*x^5 + a^2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2*x ^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)) - 1/4*(-1/(a^3*b))^(1/4)*log((a*p^2*x^6 + 2*a*p*q*x^3 - b*x^4 + a*q^2 - 2*(a*b*x^3*(-1/(a^3*b))^(1/4) + (a^3*b*p* x^4 + a^3*b*q*x)*(-1/(a^3*b))^(3/4))*sqrt(p*x^3 + q) - 2*(a^2*b*p*x^5 + a^ 2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)) - 1/4*I*(-1/(a^3*b))^(1/4)*log((a*p^2*x^6 + 2*a*p*q*x^3 - b*x^4 + a*q^2 - 2* (I*a*b*x^3*(-1/(a^3*b))^(1/4) + (-I*a^3*b*p*x^4 - I*a^3*b*q*x)*(-1/(a^3*b) )^(3/4))*sqrt(p*x^3 + q) + 2*(a^2*b*p*x^5 + a^2*b*q*x^2)*sqrt(-1/(a^3*b))) /(a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)) + 1/4*I*(-1/(a^3*b))^(1/4)*log ((a*p^2*x^6 + 2*a*p*q*x^3 - b*x^4 + a*q^2 - 2*(-I*a*b*x^3*(-1/(a^3*b))^(1/ 4) + (I*a^3*b*p*x^4 + I*a^3*b*q*x)*(-1/(a^3*b))^(3/4))*sqrt(p*x^3 + q) + 2 *(a^2*b*p*x^5 + a^2*b*q*x^2)*sqrt(-1/(a^3*b)))/(a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2))
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}}\, dx \]
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{b x^{4} + {\left (p x^{3} + q\right )}^{2} a} \,d x } \]
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{b x^{4} + {\left (p x^{3} + q\right )}^{2} a} \,d x } \]
Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^4+a \left (q+p x^3\right )^2} \, dx=\int -\frac {\sqrt {p\,x^3+q}\,\left (2\,q-p\,x^3\right )}{a\,{\left (p\,x^3+q\right )}^2+b\,x^4} \,d x \]