Integrand size = 24, antiderivative size = 144 \[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{b x+a x^4}}{-\sqrt {a} x^2+\sqrt {b x+a x^4}}\right )}{3 \sqrt [4]{a} b}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a} x^2}{\sqrt {2}}+\frac {\sqrt {b x+a x^4}}{\sqrt {2} \sqrt [4]{a}}}{x \sqrt [4]{b x+a x^4}}\right )}{3 \sqrt [4]{a} b} \]
1/3*2^(1/2)*arctan(2^(1/2)*a^(1/4)*x*(a*x^4+b*x)^(1/4)/(-a^(1/2)*x^2+(a*x^ 4+b*x)^(1/2)))/a^(1/4)/b+1/3*2^(1/2)*arctanh((1/2*a^(1/4)*x^2*2^(1/2)+1/2* (a*x^4+b*x)^(1/2)*2^(1/2)/a^(1/4))/x/(a*x^4+b*x)^(1/4))/a^(1/4)/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 x \sqrt [4]{1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {a x^3}{b+2 a x^3}\right )}{3 b \sqrt [4]{x \left (b+a x^3\right )} \sqrt [4]{1+\frac {2 a x^3}{b}}} \]
(4*x*(1 + (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (a*x^3)/(b + 2 *a*x^3)])/(3*b*(x*(b + a*x^3))^(1/4)*(1 + (2*a*x^3)/b)^(1/4))
Time = 0.54 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.65, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2467, 966, 965, 902, 755, 27, 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.
| \(\displaystyle \int \frac {1}{\left (2 a x^3+b\right ) \sqrt [4]{a x^4+b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a x^3+b} \left (2 a x^3+b\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\sqrt {x}}{\sqrt [4]{a x^3+b} \left (2 a x^3+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{\sqrt [4]{b+a x} (b+2 a x)}dx^{3/4}}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{a x b+b}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {1}{2} \int \frac {1-\sqrt {a} \sqrt {x}}{b (a x+1)}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}+\frac {1}{2} \int \frac {\sqrt {a} \sqrt {x}+1}{b (a x+1)}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\int \frac {1-\sqrt {a} \sqrt {x}}{a x+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 b}+\frac {\int \frac {\sqrt {a} \sqrt {x}+1}{a x+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\int \frac {1-\sqrt {a} \sqrt {x}}{a x+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 b}+\frac {\frac {\int \frac {1}{-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}+\frac {\int \frac {1}{\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\frac {\int \frac {1}{-\sqrt {x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {1}{-\sqrt {x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}+1\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}+\frac {\int \frac {1-\sqrt {a} \sqrt {x}}{a x+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\int \frac {1-\sqrt {a} \sqrt {x}}{a x+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 b}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}}{\sqrt [4]{a} \left (-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}\right )}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}+1\right )}{\sqrt [4]{a} \left (\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}\right )}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt [4]{a}}}{2 b}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}}{\sqrt [4]{a} \left (-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}\right )}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt [4]{a}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}+1\right )}{\sqrt [4]{a} \left (\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}\right )}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt [4]{a}}}{2 b}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}}{-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x}}+1}{\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{a} \sqrt [4]{b+a x}}+\sqrt {x}+\frac {1}{\sqrt {a}}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}}{2 b}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 b}+\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+\sqrt {a} \sqrt {x}+1\right )}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}+\sqrt {a} \sqrt {x}+1\right )}{2 \sqrt {2} \sqrt [4]{a}}}{2 b}\right )}{3 \sqrt [4]{a x^4+b x}}\) |
(4*x^(1/4)*(b + a*x^3)^(1/4)*((-(ArcTan[1 - (Sqrt[2]*a^(1/4)*x^(3/4))/(b + a*x)^(1/4)]/(Sqrt[2]*a^(1/4))) + ArcTan[1 + (Sqrt[2]*a^(1/4)*x^(3/4))/(b + a*x)^(1/4)]/(Sqrt[2]*a^(1/4)))/(2*b) + (-1/2*Log[1 + Sqrt[a]*Sqrt[x] - ( Sqrt[2]*a^(1/4)*x^(3/4))/(b + a*x)^(1/4)]/(Sqrt[2]*a^(1/4)) + Log[1 + Sqrt [a]*Sqrt[x] + (Sqrt[2]*a^(1/4)*x^(3/4))/(b + a*x)^(1/4)]/(2*Sqrt[2]*a^(1/4 )))/(2*b)))/(3*(b*x + a*x^4)^(1/4))
3.21.25.3.1 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_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
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[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\frac {\ln \left (\frac {-a^{\frac {1}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}+b \right )}}{a^{\frac {1}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}+b \right )}}\right )}{2}+\arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} \sqrt {2}+a^{\frac {1}{4}} x}{x \,a^{\frac {1}{4}}}\right )+\arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} \sqrt {2}-a^{\frac {1}{4}} x}{x \,a^{\frac {1}{4}}}\right )\right )}{3 a^{\frac {1}{4}} b}\) | \(155\) |
-1/3/a^(1/4)*2^(1/2)*(1/2*ln((-a^(1/4)*(x*(a*x^3+b))^(1/4)*2^(1/2)*x+a^(1/ 2)*x^2+(x*(a*x^3+b))^(1/2))/(a^(1/4)*(x*(a*x^3+b))^(1/4)*2^(1/2)*x+a^(1/2) *x^2+(x*(a*x^3+b))^(1/2)))+arctan(((x*(a*x^3+b))^(1/4)*2^(1/2)+a^(1/4)*x)/ x/a^(1/4))+arctan(((x*(a*x^3+b))^(1/4)*2^(1/2)-a^(1/4)*x)/x/a^(1/4)))/b
Result contains complex when optimal does not.
Time = 134.33 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.96 \[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {1}{6} \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a b^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \sqrt {a x^{4} + b x} a b x \sqrt {-\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a x^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} - 1}{2 \, a x^{3} + b}\right ) - \frac {1}{6} \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a b^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 2 \, \sqrt {a x^{4} + b x} a b x \sqrt {-\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a x^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 1}{2 \, a x^{3} + b}\right ) + \frac {1}{6} i \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a b^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \sqrt {a x^{4} + b x} a b x \sqrt {-\frac {1}{a b^{4}}} - 2 i \, {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a x^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 1}{2 \, a x^{3} + b}\right ) - \frac {1}{6} i \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a b^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \sqrt {a x^{4} + b x} a b x \sqrt {-\frac {1}{a b^{4}}} + 2 i \, {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a x^{2} \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 1}{2 \, a x^{3} + b}\right ) \]
1/6*(-1/(a*b^4))^(1/4)*log(-(2*(a*x^4 + b*x)^(3/4)*a*b^2*(-1/(a*b^4))^(3/4 ) + 2*sqrt(a*x^4 + b*x)*a*b*x*sqrt(-1/(a*b^4)) + 2*(a*x^4 + b*x)^(1/4)*a*x ^2*(-1/(a*b^4))^(1/4) - 1)/(2*a*x^3 + b)) - 1/6*(-1/(a*b^4))^(1/4)*log((2* (a*x^4 + b*x)^(3/4)*a*b^2*(-1/(a*b^4))^(3/4) - 2*sqrt(a*x^4 + b*x)*a*b*x*s qrt(-1/(a*b^4)) + 2*(a*x^4 + b*x)^(1/4)*a*x^2*(-1/(a*b^4))^(1/4) + 1)/(2*a *x^3 + b)) + 1/6*I*(-1/(a*b^4))^(1/4)*log((2*I*(a*x^4 + b*x)^(3/4)*a*b^2*( -1/(a*b^4))^(3/4) + 2*sqrt(a*x^4 + b*x)*a*b*x*sqrt(-1/(a*b^4)) - 2*I*(a*x^ 4 + b*x)^(1/4)*a*x^2*(-1/(a*b^4))^(1/4) + 1)/(2*a*x^3 + b)) - 1/6*I*(-1/(a *b^4))^(1/4)*log((-2*I*(a*x^4 + b*x)^(3/4)*a*b^2*(-1/(a*b^4))^(3/4) + 2*sq rt(a*x^4 + b*x)*a*b*x*sqrt(-1/(a*b^4)) + 2*I*(a*x^4 + b*x)^(1/4)*a*x^2*(-1 /(a*b^4))^(1/4) + 1)/(2*a*x^3 + b))
\[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x \left (a x^{3} + b\right )} \left (2 a x^{3} + b\right )}\, dx \]
\[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (2 \, a x^{3} + b\right )}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a^{\frac {1}{4}} b} - \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a^{\frac {1}{4}} b} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a^{\frac {1}{4}} b} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a^{\frac {1}{4}} b} \]
-1/3*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(a + b/x^3)^(1/4))/a^ (1/4))/(a^(1/4)*b) - 1/3*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2* (a + b/x^3)^(1/4))/a^(1/4))/(a^(1/4)*b) + 1/6*sqrt(2)*log(sqrt(2)*(a + b/x ^3)^(1/4)*a^(1/4) + sqrt(a + b/x^3) + sqrt(a))/(a^(1/4)*b) - 1/6*sqrt(2)*l og(-sqrt(2)*(a + b/x^3)^(1/4)*a^(1/4) + sqrt(a + b/x^3) + sqrt(a))/(a^(1/4 )*b)
Timed out. \[ \int \frac {1}{\left (b+2 a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {1}{{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (2\,a\,x^3+b\right )} \,d x \]