Integrand size = 36, antiderivative size = 276 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right )}{30720 a^5}+\frac {\left (19712 a^4 b^2-9843 b^5\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}}+\frac {\left (-19712 a^4 b^2+9843 b^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}} \]
1/30720*(a*x^4+b*x^3)^(1/4)*(6144*a^4*x^4+8064*a^3*b*x^3-15360*a^5*x+10400 *a^2*b^2*x^2-65280*a^4*b+16420*a*b^3*x+32705*b^4)/a^5+1/4096*(19712*a^4*b^ 2-9843*b^5)*arctan(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)-2*2^(1/4)*(2*a^ 4*b^2-b^5)*arctan(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)+1/4096*( -19712*a^4*b^2+9843*b^5)*arctanh(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)+2 *2^(1/4)*(2*a^4*b^2-b^5)*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^ (23/4)
Time = 1.76 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (2 a^{3/4} x^{3/4} \sqrt [4]{b+a x} \left (32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+768 a^4 \left (-85 b+8 x^4\right )\right )+15 b^2 \left (19712 a^4-9843 b^3\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+122880 \sqrt [4]{2} b^2 \left (-2 a^4+b^3\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+15 b^2 \left (-19712 a^4+9843 b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-122880 \sqrt [4]{2} b^2 \left (-2 a^4+b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{61440 a^{23/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(b + a*x)^(3/4)*(2*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)*(32705*b^4 - 1 5360*a^5*x + 16420*a*b^3*x + 10400*a^2*b^2*x^2 + 8064*a^3*b*x^3 + 768*a^4* (-85*b + 8*x^4)) + 15*b^2*(19712*a^4 - 9843*b^3)*ArcTan[(a^(1/4)*x^(1/4))/ (b + a*x)^(1/4)] + 122880*2^(1/4)*b^2*(-2*a^4 + b^3)*ArcTan[(2^(1/4)*a^(1/ 4)*x^(1/4))/(b + a*x)^(1/4)] + 15*b^2*(-19712*a^4 + 9843*b^3)*ArcTanh[(a^( 1/4)*x^(1/4))/(b + a*x)^(1/4)] - 122880*2^(1/4)*b^2*(-2*a^4 + b^3)*ArcTanh [(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]))/(61440*a^(23/4)*(x^3*(b + a* x))^(3/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.38 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2467, 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.
| \(\displaystyle \int \frac {\left (-a x-b+x^4\right ) \sqrt [4]{a x^4+b x^3}}{a x-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-x^4+a x+b\right )}{b-a x}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{a x^4+b x^3} \int \frac {x^{3/2} \sqrt [4]{b+a x} \left (-x^4+a x+b\right )}{b-a x}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{a x^4+b x^3} \int \left (\frac {\sqrt [4]{b+a x} x^{9/2}}{a}+\frac {b \sqrt [4]{b+a x} x^{7/2}}{a^2}+\frac {b^2 \sqrt [4]{b+a x} x^{5/2}}{a^3}-\left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b+a x} x^{3/2}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b+a x} \sqrt {x}}{a^5}+\frac {\left (2 a^4 b^2-b^5\right ) \sqrt [4]{b+a x} \sqrt {x}}{a^5 (b-a x)}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt [4]{a x^4+b x^3} \left (-\frac {371 b^5 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16384 a^{23/4}}+\frac {371 b^5 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16384 a^{23/4}}-\frac {371 b^4 x^{3/4} \sqrt [4]{a x+b}}{24576 a^5}+\frac {53 b^3 x^{7/4} \sqrt [4]{a x+b}}{6144 a^4}-\frac {1}{8} x^{7/4} \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{a x+b}+\frac {65 b^2 x^{11/4} \sqrt [4]{a x+b}}{768 a^3}+\frac {21 b x^{15/4} \sqrt [4]{a x+b}}{320 a^2}+\frac {b^2 \left (2 a^4-b^3\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{23/4}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{7/4}}-\frac {b^2 \left (2 a^4-b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{23/4}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{7/4}}+\frac {b x^{3/4} \left (2 a^4-b^3\right ) \sqrt [4]{a x+b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{\frac {a x}{b}+1}}-\frac {b x^{3/4} \left (a^4-b^3\right ) \sqrt [4]{a x+b}}{32 a^5}-\frac {b x^{3/4} \left (2 a^4-b^3\right ) \sqrt [4]{a x+b}}{4 a^5}+\frac {x^{19/4} \sqrt [4]{a x+b}}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
(4*(b*x^3 + a*x^4)^(1/4)*((-371*b^4*x^(3/4)*(b + a*x)^(1/4))/(24576*a^5) - (b*(a^4 - b^3)*x^(3/4)*(b + a*x)^(1/4))/(32*a^5) - (b*(2*a^4 - b^3)*x^(3/ 4)*(b + a*x)^(1/4))/(4*a^5) + (53*b^3*x^(7/4)*(b + a*x)^(1/4))/(6144*a^4) - ((1 - b^3/a^4)*x^(7/4)*(b + a*x)^(1/4))/8 + (65*b^2*x^(11/4)*(b + a*x)^( 1/4))/(768*a^3) + (21*b*x^(15/4)*(b + a*x)^(1/4))/(320*a^2) + (x^(19/4)*(b + a*x)^(1/4))/(20*a) + (b*(2*a^4 - b^3)*x^(3/4)*(b + a*x)^(1/4)*AppellF1[ 3/4, 1, -1/4, 7/4, (a*x)/b, -((a*x)/b)])/(3*a^5*(1 + (a*x)/b)^(1/4)) - (37 1*b^5*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16384*a^(23/4)) + (b^2*( 2*a^4 - b^3)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(23/4)) - (3* b^2*(1 - b^3/a^4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(64*a^(7/4)) + (371*b^5*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16384*a^(23/4)) - (b^2*(2*a^4 - b^3)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(23/4) ) + (3*b^2*(1 - b^3/a^4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(64*a ^(7/4))))/(x^(3/4)*(b + a*x)^(1/4))
3.29.11.3.1 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 = 0.76 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(-\frac {77 \left (-\frac {32 \left (a^{4}-\frac {b^{3}}{2}\right ) b^{2} 2^{\frac {1}{4}} \ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{77}+\frac {\left (a^{4} b^{2}-\frac {9843}{19712} b^{5}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{2}-\frac {64 \left (a^{4}-\frac {b^{3}}{2}\right ) b^{2} 2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )}{77}+\left (a^{4} b^{2}-\frac {9843}{19712} b^{5}\right ) \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\frac {65 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} \left (\frac {24 \left (\frac {8 x^{4}}{5}-17 b \right ) a^{\frac {19}{4}}}{65}+a^{\frac {11}{4}} b^{2} x^{2}+\frac {252 a^{\frac {15}{4}} b \,x^{3}}{325}+\frac {6541 a^{\frac {3}{4}} b^{4}}{2080}+\frac {821 a^{\frac {7}{4}} b^{3} x}{520}-\frac {96 a^{\frac {23}{4}} x}{65}\right )}{924}\right )}{16 a^{\frac {23}{4}}}\) | \(274\) |
-77/16/a^(23/4)*(-32/77*(a^4-1/2*b^3)*b^2*2^(1/4)*ln((-2^(1/4)*a^(1/4)*x-( x^3*(a*x+b))^(1/4))/(2^(1/4)*a^(1/4)*x-(x^3*(a*x+b))^(1/4)))+1/2*(a^4*b^2- 9843/19712*b^5)*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a*x+b ))^(1/4)))-64/77*(a^4-1/2*b^3)*b^2*2^(1/4)*arctan(1/2*(x^3*(a*x+b))^(1/4)/ x*2^(3/4)/a^(1/4))+(a^4*b^2-9843/19712*b^5)*arctan(1/a^(1/4)/x*(x^3*(a*x+b ))^(1/4))-65/924*(x^3*(a*x+b))^(1/4)*(24/65*(8/5*x^4-17*b)*a^(19/4)+a^(11/ 4)*b^2*x^2+252/325*a^(15/4)*b*x^3+6541/2080*a^(3/4)*b^4+821/520*a^(7/4)*b^ 3*x-96/65*a^(23/4)*x))
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 1152, normalized size of antiderivative = 4.17 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\text {Too large to display} \]
1/122880*(122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*log(-(2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12 *b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) + (2*a^4*b^2 - b^5)*( a*x^4 + b*x^3)^(1/4))/x) - 122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*log((2^(1/4)*a^6*x*((16*a^ 16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) - (2* a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 122880*I*2^(1/4)*a^5*((16*a^16* b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*log((I*2 ^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^2 0)/a^23)^(1/4) - (2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) + 122880*I*2^ (1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/ a^23)^(1/4)*log((-I*2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^ 14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) - (2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1 /4))/x) - 15*a^5*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b ^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 938663521 1853201*b^20)/a^23)^(1/4)*log(-(a^6*x*((150981161449947136*a^16*b^8 - 3015 64036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736 *a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) + (19712*a^4*b^2 - 9843*b^5 )*(a*x^4 + b*x^3)^(1/4))/x) + 15*a^5*((150981161449947136*a^16*b^8 - 30156 4036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 751922597972367...
\[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- a x - b + x^{4}\right )}{a x - b}\, dx \]
\[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - a x - b\right )}}{a x - b} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (236) = 472\).
Time = 0.38 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.55 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\text {Too large to display} \]
1/8192*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a) ^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^5) + 1/8192*sqrt(2)* (19712*a^4*b^2 - 9843*b^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^5) + 1/16384*sqrt(2)*(19712*a^4*b^ 2 - 9843*b^5)*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) - 1/16384*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*log(- sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4) *a^5) + 1/2*sqrt(2)*(2*2^(1/4)*(-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^5 )*log(2^(3/4)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x ))/a^6 - 1/2*sqrt(2)*(2*2^(1/4)*(-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^ 5)*log(-2^(3/4)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b /x))/a^6 - 1/30720*(65280*(a + b/x)^(17/4)*a^4*b^2 - 245760*(a + b/x)^(13/ 4)*a^5*b^2 + 345600*(a + b/x)^(9/4)*a^6*b^2 - 215040*(a + b/x)^(5/4)*a^7*b ^2 + 49920*(a + b/x)^(1/4)*a^8*b^2 - 32705*(a + b/x)^(17/4)*b^5 + 114400*( a + b/x)^(13/4)*a*b^5 - 157370*(a + b/x)^(9/4)*a^2*b^5 + 94296*(a + b/x)^( 5/4)*a^3*b^5 - 24765*(a + b/x)^(1/4)*a^4*b^5)*x^5/(a^5*b^5) + (2*2^(3/4)*( -a)^(1/4)*a^4*b^2 - 2^(3/4)*(-a)^(1/4)*b^5)*arctan(1/2*2^(1/4)*(2^(3/4)*(- a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^6 + (2*2^(3/4)*(-a)^(1/4)*a^4* b^2 - 2^(3/4)*(-a)^(1/4)*b^5)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2* (a + b/x)^(1/4))/(-a)^(1/4))/a^6
Timed out. \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (-x^4+a\,x+b\right )}{b-a\,x} \,d x \]