Integrand size = 26, antiderivative size = 236 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {\text {RootSum}\left [b^2 c+a^2 d-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)-a^2 d \log (x)+b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 c d} \]
Time = 0.00 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (16 \sqrt [4]{a} d \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )+\text {RootSum}\left [b^2 c+a^2 d-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)-a^2 d \log (x)+4 b^2 c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 c d \left (x^3 (-b+a x)\right )^{3/4}} \]
-1/8*(x^(9/4)*(-b + a*x)^(3/4)*(16*a^(1/4)*d*(ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)] - ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]) + RootSum[b^ 2*c + a^2*d - 2*a*d*#1^4 + d*#1^8 & , (-(b^2*c*Log[x]) - a^2*d*Log[x] + 4* b^2*c*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + 4*a^2*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + a*d*Log[x]*#1^4 - 4*a*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1]*# 1^4)/(-(a*#1^3) + #1^7) & ]))/(c*d*(x^3*(-b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(847\) vs. \(2(236)=472\).
Time = 2.49 (sec) , antiderivative size = 847, normalized size of antiderivative = 3.59, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2467, 609, 73, 854, 827, 216, 219, 2035, 7276, 2009}
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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4-b x^3}}{c x^2+d} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \int \frac {x^{3/4} \sqrt [4]{a x-b}}{c x^2+d}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 609 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {a \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4}}dx}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}\right )}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-\frac {\int \frac {a d+b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (c x^2+d\right )}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-\frac {4 \int \frac {\sqrt {x} (a d+b c x)}{(a x-b)^{3/4} \left (c x^2+d\right )}d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-\frac {4 \int \left (\frac {b c x^{3/2}}{(a x-b)^{3/4} \left (c x^2+d\right )}+\frac {a d \sqrt {x}}{(a x-b)^{3/4} \left (c x^2+d\right )}\right )d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-\frac {4 \left (-\frac {a (-c)^{3/8} d^{3/8} \arctan \left (\frac {\sqrt [4]{a \sqrt {-c} \sqrt {d}-b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {-c} \sqrt {d}-b c\right )^{3/4}}-\frac {b (-c)^{7/8} \arctan \left (\frac {\sqrt [4]{a \sqrt {-c} \sqrt {d}-b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {-c} \sqrt {d}-b c\right )^{3/4} \sqrt [8]{d}}-\frac {a (-c)^{3/8} d^{3/8} \arctan \left (\frac {\sqrt [4]{\sqrt {-c} \sqrt {d} a+b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (\sqrt {-c} \sqrt {d} a+b c\right )^{3/4}}+\frac {b (-c)^{7/8} \arctan \left (\frac {\sqrt [4]{\sqrt {-c} \sqrt {d} a+b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (\sqrt {-c} \sqrt {d} a+b c\right )^{3/4} \sqrt [8]{d}}+\frac {a (-c)^{3/8} d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {-c} \sqrt {d}-b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {-c} \sqrt {d}-b c\right )^{3/4}}+\frac {b (-c)^{7/8} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {-c} \sqrt {d}-b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {-c} \sqrt {d}-b c\right )^{3/4} \sqrt [8]{d}}+\frac {a (-c)^{3/8} d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-c} \sqrt {d} a+b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (\sqrt {-c} \sqrt {d} a+b c\right )^{3/4}}-\frac {b (-c)^{7/8} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-c} \sqrt {d} a+b c} \sqrt [4]{x}}{\sqrt [8]{-c} \sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (\sqrt {-c} \sqrt {d} a+b c\right )^{3/4} \sqrt [8]{d}}\right )}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
((-(b*x^3) + a*x^4)^(1/4)*((4*a*(-1/2*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^ (1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]/(2*a^(3/4)))) /c - (4*(-1/4*(b*(-c)^(7/8)*ArcTan[((-(b*c) + a*Sqrt[-c]*Sqrt[d])^(1/4)*x^ (1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/((-(b*c) + a*Sqrt[-c]*Sqrt[ d])^(3/4)*d^(1/8)) - (a*(-c)^(3/8)*d^(3/8)*ArcTan[((-(b*c) + a*Sqrt[-c]*Sq rt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/(4*(-(b*c) + a*Sqrt[-c]*Sqrt[d])^(3/4)) + (b*(-c)^(7/8)*ArcTan[((b*c + a*Sqrt[-c]*Sqrt [d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/(4*(b*c + a*Sq rt[-c]*Sqrt[d])^(3/4)*d^(1/8)) - (a*(-c)^(3/8)*d^(3/8)*ArcTan[((b*c + a*Sq rt[-c]*Sqrt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/(4* (b*c + a*Sqrt[-c]*Sqrt[d])^(3/4)) + (b*(-c)^(7/8)*ArcTanh[((-(b*c) + a*Sqr t[-c]*Sqrt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/(4*( -(b*c) + a*Sqrt[-c]*Sqrt[d])^(3/4)*d^(1/8)) + (a*(-c)^(3/8)*d^(3/8)*ArcTan h[((-(b*c) + a*Sqrt[-c]*Sqrt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a*x)^(1/4))])/(4*(-(b*c) + a*Sqrt[-c]*Sqrt[d])^(3/4)) - (b*(-c)^(7/8)*ArcT anh[((b*c + a*Sqrt[-c]*Sqrt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^(1/8)*(-b + a *x)^(1/4))])/(4*(b*c + a*Sqrt[-c]*Sqrt[d])^(3/4)*d^(1/8)) + (a*(-c)^(3/8)* d^(3/8)*ArcTanh[((b*c + a*Sqrt[-c]*Sqrt[d])^(1/4)*x^(1/4))/((-c)^(1/8)*d^( 1/8)*(-b + a*x)^(1/4))])/(4*(b*c + a*Sqrt[-c]*Sqrt[d])^(3/4))))/c))/(x^(3/ 4)*(-b + a*x)^(1/4))
3.27.52.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[d*(e/b) Int[(e*x)^(m - 1)*(c + d*x)^(n - 1), x], x] - Simp [e/b Int[(e*x)^(m - 1)*(c + d*x)^(n - 1)*((a*d - b*c*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[0, n, 1] && LtQ[0, m, 1] && !Integ erQ[m] && !IntegerQ[n]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(\frac {2 a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right ) d +4 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) d +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-2 a d \,\textit {\_Z}^{4}+a^{2} d +b^{2} c \right )}{\sum }\frac {\left (-\textit {\_R}^{4} a d +a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right )}{2 c d}\) | \(167\) |
1/2*(2*a^(1/4)*ln((a^(1/4)*x+(x^3*(a*x-b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x-b) )^(1/4)))*d+4*a^(1/4)*arctan(1/a^(1/4)/x*(x^3*(a*x-b))^(1/4))*d+sum((-_R^4 *a*d+a^2*d+b^2*c)*ln((-_R*x+(x^3*(a*x-b))^(1/4))/x)/_R^3/(-_R^4+a),_R=Root Of(_Z^8*d-2*_Z^4*a*d+a^2*d+b^2*c)))/c/d
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 829, normalized size of antiderivative = 3.51 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {1}{2} \, \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} \log \left (\frac {c x \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} \log \left (-\frac {c x \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} \log \left (\frac {c x \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} \log \left (-\frac {c x \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/2*sqrt(-sqrt((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4))*log((c*x*sqrt(-sqrt((c^ 4*sqrt(-b^2/(c^7*d)) + a)/c^4)) + (a*x^4 - b*x^3)^(1/4))/x) + 1/2*sqrt(-sq rt((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4))*log(-(c*x*sqrt(-sqrt((c^4*sqrt(-b^2/ (c^7*d)) + a)/c^4)) - (a*x^4 - b*x^3)^(1/4))/x) - 1/2*sqrt(-sqrt(-(c^4*sqr t(-b^2/(c^7*d)) - a)/c^4))*log((c*x*sqrt(-sqrt(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)) + (a*x^4 - b*x^3)^(1/4))/x) + 1/2*sqrt(-sqrt(-(c^4*sqrt(-b^2/(c^7 *d)) - a)/c^4))*log(-(c*x*sqrt(-sqrt(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)) - (a*x^4 - b*x^3)^(1/4))/x) - 1/2*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4)* log((c*x*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4)) /x) + 1/2*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4)*log(-(c*x*((c^4*sqrt(-b ^2/(c^7*d)) + a)/c^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) - 1/2*(-(c^4*sqrt( -b^2/(c^7*d)) - a)/c^4)^(1/4)*log((c*x*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4) ^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) + 1/2*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^ 4)^(1/4)*log(-(c*x*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(1/4) - (a*x^4 - b* x^3)^(1/4))/x) + (a/c^4)^(1/4)*log((c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1 /4))/x) - (a/c^4)^(1/4)*log(-(c*x*(a/c^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x ) + I*(a/c^4)^(1/4)*log((I*c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - I*(a/c^4)^(1/4)*log((-I*c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x)
Not integrable
Time = 0.73 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{c x^{2} + d}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{c x^{2} + d} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 116.42 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {2 \, \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, c} \]
sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x)^(1 /4))/(-a)^(1/4))/c + sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^ (1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/c + 1/2*sqrt(2)*(-a)^(1/4)*log(sqrt (2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - 1/2*sqrt(2) *(-a)^(1/4)*log(-sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - 1/2*(2*((a*d + sqrt(-c*d)*b)/d)^(1/4)*arctan((a - b/x)^(1/4)*d/( a*d^4 + sqrt(-c*d)*b*d^3)^(1/4)) + 2*((a*d - sqrt(-c*d)*b)/d)^(1/4)*arctan ((a - b/x)^(1/4)*d/(a*d^4 - sqrt(-c*d)*b*d^3)^(1/4)) + ((a*d + sqrt(-c*d)* b)/d)^(1/4)*log(abs((a - b/x)^(1/4)*d + (a*d^4 + sqrt(-c*d)*b*d^3)^(1/4))) + ((a*d - sqrt(-c*d)*b)/d)^(1/4)*log(abs((a - b/x)^(1/4)*d + (a*d^4 - sqr t(-c*d)*b*d^3)^(1/4))) - ((a*d + sqrt(-c*d)*b)/d)^(1/4)*log(abs(-(a - b/x) ^(1/4)*d + (a*d^4 + sqrt(-c*d)*b*d^3)^(1/4))) - ((a*d - sqrt(-c*d)*b)/d)^( 1/4)*log(abs(-(a - b/x)^(1/4)*d + (a*d^4 - sqrt(-c*d)*b*d^3)^(1/4))))/c
Not integrable
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{c\,x^2+d} \,d x \]