Integrand size = 31, antiderivative size = 191 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 d^2} \]
Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {32 d x^2 (-b+a x)-x^{9/4} (-b+a x)^{3/4} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 d^2 \left (x^3 (-b+a x)\right )^{3/4}} \]
(32*d*x^2*(-b + a*x) - x^(9/4)*(-b + a*x)^(3/4)*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*L og[x]*#1^4 - 4*a*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1]*#1^4)/(-(a*#1^3) + # 1^7) & ])/(8*d^2*(x^3*(-b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(658\) vs. \(2(191)=382\).
Time = 2.18 (sec) , antiderivative size = 658, normalized size of antiderivative = 3.45, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2467, 25, 611, 48, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4-b x^3}}{x^2 \left (c x^2-d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \int -\frac {\sqrt [4]{a x-b}}{x^{5/4} \left (d-c x^2\right )}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \int \frac {\sqrt [4]{a x-b}}{x^{5/4} \left (d-c x^2\right )}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 611 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {\int \frac {a d-b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (d-c x^2\right )}dx}{d}-\frac {b \int \frac {1}{x^{5/4} (a x-b)^{3/4}}dx}{d}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {\int \frac {a d-b c x}{\sqrt [4]{x} (a x-b)^{3/4} \left (d-c x^2\right )}dx}{d}-\frac {4 \sqrt [4]{a x-b}}{d \sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 \int \frac {\sqrt {x} (a d-b c x)}{(a x-b)^{3/4} \left (d-c x^2\right )}d\sqrt [4]{x}}{d}-\frac {4 \sqrt [4]{a x-b}}{d \sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 \int \left (\frac {b c x^{3/2}}{(a x-b)^{3/4} \left (c x^2-d\right )}+\frac {a d \sqrt {x}}{(a x-b)^{3/4} \left (d-c x^2\right )}\right )d\sqrt [4]{x}}{d}-\frac {4 \sqrt [4]{a x-b}}{d \sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {4 \left (-\frac {a d^{3/8} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}-\frac {a d^{3/8} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}+\frac {b \sqrt {c} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \sqrt [8]{d} \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}-\frac {b \sqrt {c} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \sqrt [8]{d} \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}+\frac {a d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}+\frac {a d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}-\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \sqrt [8]{d} \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}+\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{4 \sqrt [8]{d} \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}\right )}{d}-\frac {4 \sqrt [4]{a x-b}}{d \sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
-(((-(b*x^3) + a*x^4)^(1/4)*((-4*(-b + a*x)^(1/4))/(d*x^(1/4)) + (4*((b*Sq rt[c]*ArcTan[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x )^(1/4))])/(4*(-(b*Sqrt[c]) + a*Sqrt[d])^(3/4)*d^(1/8)) - (a*d^(3/8)*ArcTa n[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/ (4*(-(b*Sqrt[c]) + a*Sqrt[d])^(3/4)) - (b*Sqrt[c]*ArcTan[((b*Sqrt[c] + a*S qrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(4*(b*Sqrt[c] + a*Sqrt [d])^(3/4)*d^(1/8)) - (a*d^(3/8)*ArcTan[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^( 1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(4*(b*Sqrt[c] + a*Sqrt[d])^(3/4)) - (b* Sqrt[c]*ArcTanh[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(4*(-(b*Sqrt[c]) + a*Sqrt[d])^(3/4)*d^(1/8)) + (a*d^(3/8)*Ar cTanh[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4) )])/(4*(-(b*Sqrt[c]) + a*Sqrt[d])^(3/4)) + (b*Sqrt[c]*ArcTanh[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(4*(b*Sqrt[c] + a *Sqrt[d])^(3/4)*d^(1/8)) + (a*d^(3/8)*ArcTanh[((b*Sqrt[c] + a*Sqrt[d])^(1/ 4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(4*(b*Sqrt[c] + a*Sqrt[d])^(3/4)) ))/d))/(x^(3/4)*(-b + a*x)^(1/4)))
3.24.84.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[c/a Int[(e*x)^m*(c + d*x)^(n - 1), x], x] + Simp[1/(a*e) 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] && GtQ[n, 0] && LtQ[m, -1] && !IntegerQ[m] && !IntegerQ[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 = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {8 \left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}} d +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} d -2 \textit {\_Z}^{4} a d +a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x}{2 d^{2} x}\) | \(110\) |
1/2*(8*(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=RootOf(_Z^8*d-2*_Z^4*a*d+a^2*d-b^2*c))*x) /d^2/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, d x} \]
-1/2*(d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d^9) + a)/d^4))*log((d*x*sqrt(-sqrt(( d^4*sqrt(b^2*c/d^9) + a)/d^4)) + (a*x^4 - b*x^3)^(1/4))/x) - d*x*sqrt(-sqr t((d^4*sqrt(b^2*c/d^9) + a)/d^4))*log(-(d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d^9 ) + a)/d^4)) - (a*x^4 - b*x^3)^(1/4))/x) + d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c /d^9) - a)/d^4))*log((d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)) + (a *x^4 - b*x^3)^(1/4))/x) - d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4))* log(-(d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)) - (a*x^4 - b*x^3)^(1 /4))/x) + d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4)*log((d*x*((d^4*sqrt(b^ 2*c/d^9) + a)/d^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - d*x*((d^4*sqrt(b^2* c/d^9) + a)/d^4)^(1/4)*log(-(d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4) - ( a*x^4 - b*x^3)^(1/4))/x) + d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log( (d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log(-(d*x*(-(d^4*sqrt(b^2*c/d^9 ) - a)/d^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) - 8*(a*x^4 - b*x^3)^(1/4))/( d*x)
Not integrable
Time = 2.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x^{2} \left (c x^{2} - d\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x^{2} - d\right )} x^{2}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 117.64 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {2 \, \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, d} + \frac {4 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}}{d} \]
-1/2*(2*((a*d + sqrt(c*d)*b)/d)^(1/4)*arctan((a - b/x)^(1/4)*d/(a*d^4 + sq rt(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)*lo g(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))) - ((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))))/d + 4*(a - b/x)^(1/4)/d
Not integrable
Time = 6.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,d x \]