Integrand size = 34, antiderivative size = 327 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=2 a \sqrt [4]{b x^3+a x^4}+\left (4 a^{9/4}-3 \sqrt [4]{a} b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\left (-4 a^{9/4}+3 \sqrt [4]{a} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )-\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-4 a^4 \log (x)+4 a^2 b \log (x)-b^2 \log (x)+4 a^4 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-4 a^2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+2 a^3 \log (x) \text {$\#$1}^4-3 a b \log (x) \text {$\#$1}^4-2 a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ] \]
Time = 1.01 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.03 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (2 a x^{3/4} \sqrt [4]{b+a x}+\sqrt [4]{a} \left (4 a^2-3 b\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\sqrt [4]{a} \left (-4 a^2+3 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {4 a^4 \log (x)-4 a^2 b \log (x)+b^2 \log (x)-16 a^4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+16 a^2 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-4 b^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-2 a^3 \log (x) \text {$\#$1}^4+3 a b \log (x) \text {$\#$1}^4+8 a^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4-12 a b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{\left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(b + a*x)^(3/4)*(2*a*x^(3/4)*(b + a*x)^(1/4) + a^(1/4)*(4*a^2 - 3 *b)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + a^(1/4)*(-4*a^2 + 3*b)*Arc Tanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + RootSum[2*a^2 - b - 3*a*#1^4 + # 1^8 & , (4*a^4*Log[x] - 4*a^2*b*Log[x] + b^2*Log[x] - 16*a^4*Log[(b + a*x) ^(1/4) - x^(1/4)*#1] + 16*a^2*b*Log[(b + a*x)^(1/4) - x^(1/4)*#1] - 4*b^2* Log[(b + a*x)^(1/4) - x^(1/4)*#1] - 2*a^3*Log[x]*#1^4 + 3*a*b*Log[x]*#1^4 + 8*a^3*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^4 - 12*a*b*Log[(b + a*x)^(1/4 ) - x^(1/4)*#1]*#1^4)/(3*a*#1^3 - 2*#1^7) & ]/4))/(x^3*(b + a*x))^(3/4)
Leaf count is larger than twice the leaf count of optimal. \(1400\) vs. \(2(327)=654\).
Time = 4.70 (sec) , antiderivative size = 1400, normalized size of antiderivative = 4.28, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2467, 25, 2035, 7279, 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.
| \(\displaystyle \int \frac {(2 a x+b) \sqrt [4]{a x^4+b x^3}}{a x-b+x^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {x^{3/4} \sqrt [4]{b+a x} (b+2 a x)}{-x^2-a x+b}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{3/4} \sqrt [4]{b+a x} (b+2 a x)}{-x^2-a x+b}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \frac {x^{3/2} \sqrt [4]{b+a x} (b+2 a x)}{-x^2-a x+b}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \left (\frac {\sqrt {x} \sqrt [4]{b+a x} \left (2 a b-\left (2 a^2-b\right ) x\right )}{-x^2-a x+b}-2 a \sqrt {x} \sqrt [4]{b+a x}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \left (\frac {4 b x^{3/4} \sqrt [4]{b+a x} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {2 x}{a-\sqrt {a^2+4 b}},-\frac {a x}{b}\right ) a}{3 \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{\frac {a x}{b}+1}}+\frac {4 b x^{3/4} \sqrt [4]{b+a x} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {2 x}{a+\sqrt {a^2+4 b}},-\frac {a x}{b}\right ) a}{3 \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{\frac {a x}{b}+1}}-\frac {\left (2 a^2-b\right ) b \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) a}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (2 a^2-b\right ) b \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) a}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (2 a^2-b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) a}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {\left (2 a^2-b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) a}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {1}{2} x^{3/4} \sqrt [4]{b+a x} a-\frac {1}{2} \left (2 a^2-b\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \sqrt [4]{a}+\frac {1}{4} b \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \sqrt [4]{a}+\frac {1}{2} \left (2 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \sqrt [4]{a}-\frac {1}{4} b \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \sqrt [4]{a}-\frac {\left (a^2-b\right ) \left (2 a^2-b\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (a^2-b\right ) \left (2 a^2-b\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (a^2-b\right ) \left (2 a^2-b\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {\left (a^2-b\right ) \left (2 a^2-b\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\) |
(-4*(b*x^3 + a*x^4)^(1/4)*(-1/2*(a*x^(3/4)*(b + a*x)^(1/4)) + (4*a*b*x^(3/ 4)*(b + a*x)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (-2*x)/(a - Sqrt[a^2 + 4*b] ), -((a*x)/b)])/(3*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*(1 + (a*x)/b)^(1/4)) + (4*a*b*x^(3/4)*(b + a*x)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (-2*x)/(a + Sqr t[a^2 + 4*b]), -((a*x)/b)])/(3*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*(1 + (a*x)/ b)^(1/4)) - (a^(1/4)*(2*a^2 - b)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] )/2 + (a^(1/4)*b*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/4 - (a*(2*a^2 - b)*b*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a ^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(Sqrt[a^2 + 4*b]*(a - Sqrt[a^2 + 4*b]) ^(1/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) - ((a^2 - b)*(2*a^2 - b)*(a - Sqrt[a^2 + 4*b])^(3/4)*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^( 1/4))/((a - Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(2*Sqrt[a^2 + 4*b]*( a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) + (a*(2*a^2 - b)*b*ArcTan[((a^2 - 2* b + a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 4*b])^(1/4)*(b + a* x)^(1/4))])/(Sqrt[a^2 + 4*b]*(a + Sqrt[a^2 + 4*b])^(1/4)*(a^2 - 2*b + a*Sq rt[a^2 + 4*b])^(3/4)) + ((a^2 - b)*(2*a^2 - b)*(a + Sqrt[a^2 + 4*b])^(3/4) *ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 4 *b])^(1/4)*(b + a*x)^(1/4))])/(2*Sqrt[a^2 + 4*b]*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(3/4)) + (a^(1/4)*(2*a^2 - b)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^( 1/4)])/2 - (a^(1/4)*b*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/4 + (...
3.30.10.3.1 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_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.60 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.60
| method | result | size |
| pseudoelliptic | \(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{4} a^{3}+3 \textit {\_R}^{4} a b +4 a^{4}-4 a^{2} b +b^{2}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{7}+3 \textit {\_R}^{3} a}\right )+\frac {\left (3 a^{\frac {1}{4}} b -4 a^{\frac {9}{4}}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{2}+\left (3 a^{\frac {1}{4}} b -4 a^{\frac {9}{4}}\right ) \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+2 a \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}\) | \(195\) |
-sum((-2*_R^4*a^3+3*_R^4*a*b+4*a^4-4*a^2*b+b^2)*ln((-_R*x+(x^3*(a*x+b))^(1 /4))/x)/(-2*_R^7+3*_R^3*a),_R=RootOf(_Z^8-3*_Z^4*a+2*a^2-b))+1/2*(3*a^(1/4 )*b-4*a^(9/4))*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a*x+b) )^(1/4)))+(3*a^(1/4)*b-4*a^(9/4))*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))+ 2*a*(x^3*(a*x+b))^(1/4)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.07 (sec) , antiderivative size = 6380, normalized size of antiderivative = 19.51 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\text {Too large to display} \]
Not integrable
Time = 4.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.08 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (2 a x + b\right )}{a x - b + x^{2}}\, dx \]
Not integrable
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.10 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (2 \, a x + b\right )}}{a x + x^{2} - b} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.05 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.74 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=-\frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a x \]
-1/2*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*arctan(1/2*sqrt(2)*(sqrt( 2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4)) - 1/2*sqrt(2)*(4*(-a)^(1/4) *a^2 - 3*(-a)^(1/4)*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/ x)^(1/4))/(-a)^(1/4)) - 1/4*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*lo g(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) + 1/4*sqr t(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x) ^(1/4) + sqrt(-a) + sqrt(a + b/x)) + 2*(a + b/x)^(1/4)*a*x
Not integrable
Time = 7.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.10 \[ \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (b+2\,a\,x\right )}{x^2+a\,x-b} \,d x \]