Integrand size = 37, antiderivative size = 241 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{d} \]
Time = 0.01 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {x^2 (b+a x)^{3/4} \left (2 d \left (2 \sqrt [4]{b+a x}-\sqrt [4]{a} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\sqrt [4]{a} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )-\frac {1}{4} \sqrt [4]{x} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{d \left (x^3 (b+a x)\right )^{3/4}} \]
(x^2*(b + a*x)^(3/4)*(2*d*(2*(b + a*x)^(1/4) - a^(1/4)*x^(1/4)*ArcTan[(a^( 1/4)*x^(1/4))/(b + a*x)^(1/4)] + a^(1/4)*x^(1/4)*ArcTanh[(a^(1/4)*x^(1/4)) /(b + a*x)^(1/4)]) - (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*L og[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/4))/(d*(x^3 *(b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(688\) vs. \(2(241)=482\).
Time = 2.26 (sec) , antiderivative size = 688, normalized size of antiderivative = 2.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2467, 25, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4+b x^3} \left (c x^2+d\right )}{x^2 \left (c x^2-d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {\sqrt [4]{b+a x} \left (c x^2+d\right )}{x^{5/4} \left (d-c x^2\right )}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {\sqrt [4]{b+a x} \left (c x^2+d\right )}{x^{5/4} \left (d-c x^2\right )}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \frac {\sqrt [4]{b+a x} \left (c x^2+d\right )}{\sqrt {x} \left (d-c x^2\right )}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \left (\frac {\sqrt [4]{b+a x}}{\sqrt {x}}-\frac {2 c x^{3/2} \sqrt [4]{b+a x}}{c x^2-d}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \left (-\frac {a d^{3/8} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}-\frac {a d^{3/8} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}+\frac {b \sqrt {c} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \sqrt [8]{d} \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}-\frac {b \sqrt {c} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \sqrt [8]{d} \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}+\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )+\frac {a d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}+\frac {a d^{3/8} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}-\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \sqrt [8]{d} \left (a \sqrt {d}-b \sqrt {c}\right )^{3/4}}+\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{2 \sqrt [8]{d} \left (a \sqrt {d}+b \sqrt {c}\right )^{3/4}}-\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )-\frac {\sqrt [4]{a x+b}}{\sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
(-4*(b*x^3 + a*x^4)^(1/4)*(-((b + a*x)^(1/4)/x^(1/4)) + (a^(1/4)*ArcTan[(a ^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/2 + (b*Sqrt[c]*ArcTan[((-(b*Sqrt[c]) + a *Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])/(2*(-(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))])/(2*(-(b*Sqrt[c]) + a*Sqrt[d])^(3/ 4)) - (b*Sqrt[c]*ArcTan[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*( b + a*x)^(1/4))])/(2*(b*Sqrt[c] + a*Sqrt[d])^(3/4)*d^(1/8)) - (a*d^(3/8)*A rcTan[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])/ (2*(b*Sqrt[c] + a*Sqrt[d])^(3/4)) - (a^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/2 - (b*Sqrt[c]*ArcTanh[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^ (1/4))/(d^(1/8)*(b + a*x)^(1/4))])/(2*(-(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))])/(2*(-(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))] )/(2*(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))])/(2*(b*Sqrt[c] + a*Sqrt[d])^(3/4))))/(x^(3/4)*(b + a*x)^(1/4))
3.27.73.3.1 Defintions of rubi rules used
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 = 0.00 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-2 a d \,\textit {\_Z}^{4}+a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x +2 \left (\arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) x \,a^{\frac {1}{4}}+\frac {\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) x \,a^{\frac {1}{4}}}{2}+2 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}\right ) d}{d x}\) | \(177\) |
(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+2*(arctan(1/a^(1/4)/x*(x^3*( a*x+b))^(1/4))*x*a^(1/4)+1/2*ln((a^(1/4)*x+(x^3*(a*x+b))^(1/4))/(-a^(1/4)* x+(x^3*(a*x+b))^(1/4)))*x*a^(1/4)+2*(x^3*(a*x+b))^(1/4))*d)/d/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.58 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{\frac {1}{4}} x \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{\frac {1}{4}} x \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \]
-(x*sqrt(-sqrt(a + sqrt(b^2*c/d)))*log(2*(x*sqrt(-sqrt(a + sqrt(b^2*c/d))) + (a*x^4 + b*x^3)^(1/4))/x) - x*sqrt(-sqrt(a + sqrt(b^2*c/d)))*log(-2*(x* sqrt(-sqrt(a + sqrt(b^2*c/d))) - (a*x^4 + b*x^3)^(1/4))/x) + x*sqrt(-sqrt( a - sqrt(b^2*c/d)))*log(2*(x*sqrt(-sqrt(a - sqrt(b^2*c/d))) + (a*x^4 + b*x ^3)^(1/4))/x) - x*sqrt(-sqrt(a - sqrt(b^2*c/d)))*log(-2*(x*sqrt(-sqrt(a - sqrt(b^2*c/d))) - (a*x^4 + b*x^3)^(1/4))/x) + (a + sqrt(b^2*c/d))^(1/4)*x* log(2*((a + sqrt(b^2*c/d))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - (a + sqrt (b^2*c/d))^(1/4)*x*log(-2*((a + sqrt(b^2*c/d))^(1/4)*x - (a*x^4 + b*x^3)^( 1/4))/x) + (a - sqrt(b^2*c/d))^(1/4)*x*log(2*((a - sqrt(b^2*c/d))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - (a - sqrt(b^2*c/d))^(1/4)*x*log(-2*((a - sqr t(b^2*c/d))^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) - a^(1/4)*x*log((a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) + a^(1/4)*x*log(-(a^(1/4)*x - (a*x^4 + b*x^3) ^(1/4))/x) - I*a^(1/4)*x*log((I*a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) + I* a^(1/4)*x*log((-I*a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 4*(a*x^4 + b*x^3 )^(1/4))/x
Not integrable
Time = 4.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (c x^{2} + d\right )}{x^{2} \left (c x^{2} - d\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (c x^{2} + d\right )}}{{\left (c x^{2} - d\right )} x^{2}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 116.02 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.04 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\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 ) + \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 ) + \frac {1}{2} \, \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 ) - \frac {1}{2} \, \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 \, \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 ) + 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} \]
sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1 /4))/(-a)^(1/4)) + sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1 /4) - 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 1/2*sqrt(2)*(-a)^(1/4)*log(sqrt(2)* (-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 1/2*sqrt(2)*(-a)^ (1/4)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 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(ab s((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 + sq rt(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))) + 4*(a + b/x)^(1/4)
Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\int \frac {\left (c\,x^2+d\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,d x \]