Integrand size = 37, antiderivative size = 215 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+a \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+b \log (x)+a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ] \]
Time = 0.49 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {x^2 (b+a x)^{3/4} \left (4 \sqrt [4]{b+a x}-2 \sqrt [4]{a} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+2 \sqrt [4]{a} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\frac {1}{4} a \sqrt [4]{x} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+b \log (x)+4 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-4 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{\left (x^3 (b+a x)\right )^{3/4}} \]
(x^2*(b + a*x)^(3/4)*(4*(b + a*x)^(1/4) - 2*a^(1/4)*x^(1/4)*ArcTan[(a^(1/4 )*x^(1/4))/(b + a*x)^(1/4)] + 2*a^(1/4)*x^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/ (b + a*x)^(1/4)] - (a*x^(1/4)*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(a *Log[x]) + b*Log[x] + 4*a*Log[(b + a*x)^(1/4) - x^(1/4)*#1] - 4*b*Log[(b + a*x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(b + a*x)^(1/4) - x^(1/4)* #1]*#1^4)/(-(a*#1^3) + #1^7) & ])/4))/(x^3*(b + a*x))^(3/4)
Leaf count is larger than twice the leaf count of optimal. \(624\) vs. \(2(215)=430\).
Time = 2.26 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.90, 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.
| \(\displaystyle \int \frac {\left (a x^2+b\right ) \sqrt [4]{a x^4+b x^3}}{x^2 \left (a x^2-b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {\sqrt [4]{b+a x} \left (a x^2+b\right )}{x^{5/4} \left (b-a x^2\right )}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {\sqrt [4]{b+a x} \left (a x^2+b\right )}{x^{5/4} \left (b-a x^2\right )}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \frac {\sqrt [4]{b+a x} \left (a x^2+b\right )}{\sqrt {x} \left (b-a x^2\right )}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \left (\frac {\sqrt [4]{b+a x}}{\sqrt {x}}-\frac {2 a x^{3/2} \sqrt [4]{b+a x}}{a x^2-b}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \left (-\frac {a^{5/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}-\frac {a^{5/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}+\frac {a^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}+\frac {a^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}+\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )+\frac {\sqrt [8]{a} \sqrt {b} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}-\frac {\sqrt [8]{a} \sqrt {b} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}-\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )-\frac {\sqrt [8]{a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}+\frac {\sqrt [8]{a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}-\frac {\sqrt [4]{a x+b}}{\sqrt [4]{x}}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
(-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 - (a^(5/8)*ArcTan[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^(3/4)) + (a^(1/8)*Sqrt[b]*ArcTan[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x^(1/4))/(b + a *x)^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^(3/4)) - (a^(5/8)*ArcTan[(a^(1/8)*(Sqrt [a] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(3/ 4)) - (a^(1/8)*Sqrt[b]*ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/ (b + a*x)^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(3/4)) - (a^(1/4)*ArcTanh[(a^(1/4 )*x^(1/4))/(b + a*x)^(1/4)])/2 + (a^(5/8)*ArcTanh[(a^(1/8)*(Sqrt[a] - Sqrt [b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^(3/4)) - (a^( 1/8)*Sqrt[b]*ArcTanh[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x^(1/4))/(b + a*x) ^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^(3/4)) + (a^(5/8)*ArcTanh[(a^(1/8)*(Sqrt[a ] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(3/4) ) + (a^(1/8)*Sqrt[b]*ArcTanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/( b + a*x)^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(3/4))))/(x^(3/4)*(b + a*x)^(1/4))
3.26.52.3.1 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 = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72
| method | result | size |
| pseudoelliptic | \(\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a +b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x +a^{\frac {1}{4}} x \ln \left (\frac {x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} x \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\) | \(154\) |
(a*sum((_R^4-a+b)*ln((-_R*x+(x^3*(a*x+b))^(1/4))/x)/_R^3/(_R^4-a),_R=RootO f(_Z^8-2*_Z^4*a+a^2-a*b))*x+a^(1/4)*x*ln((x*a^(1/4)+(x^3*(a*x+b))^(1/4))/( -x*a^(1/4)+(x^3*(a*x+b))^(1/4)))+2*a^(1/4)*x*arctan(1/a^(1/4)/x*(x^3*(a*x+ b))^(1/4))+4*(x^3*(a*x+b))^(1/4))/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.31 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.52 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\frac {x \sqrt {-\sqrt {a + \sqrt {a b}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {a b}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a + \sqrt {a b}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {a b}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + x \sqrt {-\sqrt {a - \sqrt {a b}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {a b}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a - \sqrt {a b}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {a b}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {a b}\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(a*b)))*log(2*(x*sqrt(-sqrt(a + sqrt(a*b))) + (a*x^ 4 + b*x^3)^(1/4))/x) - x*sqrt(-sqrt(a + sqrt(a*b)))*log(-2*(x*sqrt(-sqrt(a + sqrt(a*b))) - (a*x^4 + b*x^3)^(1/4))/x) + x*sqrt(-sqrt(a - sqrt(a*b)))* log(2*(x*sqrt(-sqrt(a - sqrt(a*b))) + (a*x^4 + b*x^3)^(1/4))/x) - x*sqrt(- sqrt(a - sqrt(a*b)))*log(-2*(x*sqrt(-sqrt(a - sqrt(a*b))) - (a*x^4 + b*x^3 )^(1/4))/x) + (a + sqrt(a*b))^(1/4)*x*log(2*((a + sqrt(a*b))^(1/4)*x + (a* x^4 + b*x^3)^(1/4))/x) - (a + sqrt(a*b))^(1/4)*x*log(-2*((a + sqrt(a*b))^( 1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) + (a - sqrt(a*b))^(1/4)*x*log(2*((a - s qrt(a*b))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - (a - sqrt(a*b))^(1/4)*x*lo g(-2*((a - sqrt(a*b))^(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 = 4.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.13 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (a x^{2} + b\right )}{x^{2} \left (a x^{2} - b\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]
Not integrable
Time = 0.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]
Not integrable
Time = 8.60 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.18 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\int \frac {\left (a\,x^2+b\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (b-a\,x^2\right )} \,d x \]