Integrand size = 43, antiderivative size = 192 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=-\frac {4 \left (b-a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{21 b^2 x^6}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b x+a x^4}}{-\sqrt {a} x^2+\sqrt {-b x+a x^4}}\right )}{3 \sqrt [4]{a} b^2}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \text {arctanh}\left (\frac {\sqrt {a} x^2+\sqrt {-b x+a x^4}}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b x+a x^4}}\right )}{3 \sqrt [4]{a} b^2} \]
-4/21*(-a*x^3+b)*(a*x^4-b*x)^(3/4)/b^2/x^6+1/3*2^(1/2)*(2*a^2-3*b)*arctan( 2^(1/2)*a^(1/4)*x*(a*x^4-b*x)^(1/4)/(-a^(1/2)*x^2+(a*x^4-b*x)^(1/2)))/a^(1 /4)/b^2+1/3*2^(1/2)*(2*a^2-3*b)*arctanh(1/2*(a^(1/2)*x^2+(a*x^4-b*x)^(1/2) )*2^(1/2)/a^(1/4)/x/(a*x^4-b*x)^(1/4))/a^(1/4)/b^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.68 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.27 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {\frac {210 b x^3 \left (-b+a x^3\right )}{a}+15 \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right ) \left (3 b+4 a x^3\right )+\frac {\left (2 a^2-3 b\right ) \left (5 \left (3 b^3+13 a b^2 x^3-144 a^2 b x^6+128 a^3 x^9\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {a x^3}{b-a x^3}\right )+8 a x^3 \left (b^2+10 a b x^3-24 a^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{4},2,\frac {9}{4},\frac {a x^3}{b-a x^3}\right )-16 a x^3 \left (b-2 a x^3\right )^2 \, _3F_2\left (\frac {5}{4},2,2;1,\frac {9}{4};\frac {a x^3}{b-a x^3}\right )\right )}{a^2 \left (-b+a x^3\right )}}{315 b^2 x^5 \sqrt [4]{-b x+a x^4}} \]
((210*b*x^3*(-b + a*x^3))/a + 15*(2 - b/a^2)*(b - a*x^3)*(3*b + 4*a*x^3) + ((2*a^2 - 3*b)*(5*(3*b^3 + 13*a*b^2*x^3 - 144*a^2*b*x^6 + 128*a^3*x^9)*Hy pergeometric2F1[1/4, 1, 5/4, (a*x^3)/(b - a*x^3)] + 8*a*x^3*(b^2 + 10*a*b* x^3 - 24*a^2*x^6)*Hypergeometric2F1[5/4, 2, 9/4, (a*x^3)/(b - a*x^3)] - 16 *a*x^3*(b - 2*a*x^3)^2*HypergeometricPFQ[{5/4, 2, 2}, {1, 9/4}, (a*x^3)/(b - a*x^3)]))/(a^2*(-b + a*x^3)))/(315*b^2*x^5*(-(b*x) + a*x^4)^(1/4))
Time = 1.66 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, 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.
| \(\displaystyle \int \frac {-3 a x^3+b+3 x^6}{x^6 \left (2 a x^3-b\right ) \sqrt [4]{a x^4-b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3-b} \int -\frac {3 x^6-3 a x^3+b}{x^{25/4} \left (b-2 a x^3\right ) \sqrt [4]{a x^3-b}}dx}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {3 x^6-3 a x^3+b}{x^{25/4} \left (b-2 a x^3\right ) \sqrt [4]{a x^3-b}}dx}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {3 x^6-3 a x^3+b}{x^{11/2} \left (b-2 a x^3\right ) \sqrt [4]{a x^3-b}}d\sqrt [4]{x}}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \left (-\frac {a}{b x^{5/2} \sqrt [4]{a x^3-b}}+\frac {\left (3 b-2 a^2\right ) \sqrt {x}}{b \left (b-2 a x^3\right ) \sqrt [4]{a x^3-b}}+\frac {1}{x^{11/2} \sqrt [4]{a x^3-b}}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \left (\frac {\left (2 a^2-3 b\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{6 \sqrt {2} \sqrt [4]{a} b^2}-\frac {\left (2 a^2-3 b\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{6 \sqrt {2} \sqrt [4]{a} b^2}+\frac {\left (2 a^2-3 b\right ) \log \left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3-b}}-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{12 \sqrt {2} \sqrt [4]{a} b^2}-\frac {\left (2 a^2-3 b\right ) \log \left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3-b}}+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{12 \sqrt {2} \sqrt [4]{a} b^2}-\frac {a \left (a x^3-b\right )^{3/4}}{21 b^2 x^{9/4}}+\frac {\left (a x^3-b\right )^{3/4}}{21 b x^{21/4}}\right )}{\sqrt [4]{a x^4-b x}}\) |
(-4*x^(1/4)*(-b + a*x^3)^(1/4)*((-b + a*x^3)^(3/4)/(21*b*x^(21/4)) - (a*(- b + a*x^3)^(3/4))/(21*b^2*x^(9/4)) + ((2*a^2 - 3*b)*ArcTan[1 - (Sqrt[2]*a^ (1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(6*Sqrt[2]*a^(1/4)*b^2) - ((2*a^2 - 3* b)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(6*Sqrt[2]*a^ (1/4)*b^2) + ((2*a^2 - 3*b)*Log[1 + (Sqrt[a]*x^(3/2))/Sqrt[-b + a*x^3] - ( Sqrt[2]*a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(12*Sqrt[2]*a^(1/4)*b^2) - ( (2*a^2 - 3*b)*Log[1 + (Sqrt[a]*x^(3/2))/Sqrt[-b + a*x^3] + (Sqrt[2]*a^(1/4 )*x^(3/4))/(-b + a*x^3)^(1/4)])/(12*Sqrt[2]*a^(1/4)*b^2)))/(-(b*x) + a*x^4 )^(1/4)
3.24.92.3.1 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 = 1.68 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \ln \left (\frac {-a^{\frac {1}{4}} {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}-b \right )}}{a^{\frac {1}{4}} {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}-b \right )}}\right )}{2}+\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \arctan \left (\frac {\sqrt {2}\, {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}-a^{\frac {1}{4}} x}{a^{\frac {1}{4}} x}\right )+\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \arctan \left (\frac {\sqrt {2}\, {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}+a^{\frac {1}{4}} x}{a^{\frac {1}{4}} x}\right )-\frac {2 {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {7}{4}} a^{\frac {1}{4}}}{7}\right )}{3 a^{\frac {1}{4}} x^{7} b^{2}}\) | \(226\) |
-2/3/a^(1/4)*(1/2*2^(1/2)*x^7*(a^2-3/2*b)*ln((-a^(1/4)*(x*(a*x^3-b))^(1/4) *2^(1/2)*x+a^(1/2)*x^2+(x*(a*x^3-b))^(1/2))/(a^(1/4)*(x*(a*x^3-b))^(1/4)*2 ^(1/2)*x+a^(1/2)*x^2+(x*(a*x^3-b))^(1/2)))+2^(1/2)*x^7*(a^2-3/2*b)*arctan( (2^(1/2)*(x*(a*x^3-b))^(1/4)-a^(1/4)*x)/a^(1/4)/x)+2^(1/2)*x^7*(a^2-3/2*b) *arctan((2^(1/2)*(x*(a*x^3-b))^(1/4)+a^(1/4)*x)/a^(1/4)/x)-2/7*(x*(a*x^3-b ))^(7/4)*a^(1/4))/x^7/b^2
Timed out. \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {- 3 a x^{3} + b + 3 x^{6}}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )} \left (2 a x^{3} - b\right )}\, dx \]
\[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {3 \, x^{6} - 3 \, a x^{3} + b}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (2 \, a x^{3} - b\right )} x^{6}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} + \frac {\sqrt {2} {\left (2 \, a^{\frac {11}{4}} - 3 \, a^{\frac {3}{4}} b\right )} \log \left (\sqrt {2} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a - \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a b^{2}} - \frac {\sqrt {2} {\left (2 \, a^{\frac {11}{4}} - 3 \, a^{\frac {3}{4}} b\right )} \log \left (-\sqrt {2} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a - \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a b^{2}} - \frac {{\left (2 \, \sqrt {2} a^{\frac {11}{4}} - 3 \, \sqrt {2} a^{\frac {3}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a b^{2}} - \frac {{\left (2 \, \sqrt {2} a^{\frac {11}{4}} - 3 \, \sqrt {2} a^{\frac {3}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a b^{2}} \]
4/21*(a - b/x^3)^(7/4)/b^2 + 1/6*sqrt(2)*(2*a^(11/4) - 3*a^(3/4)*b)*log(sq rt(2)*(a - b/x^3)^(1/4)*a^(1/4) + sqrt(a - b/x^3) + sqrt(a))/(a*b^2) - 1/6 *sqrt(2)*(2*a^(11/4) - 3*a^(3/4)*b)*log(-sqrt(2)*(a - b/x^3)^(1/4)*a^(1/4) + sqrt(a - b/x^3) + sqrt(a))/(a*b^2) - 1/3*(2*sqrt(2)*a^(11/4) - 3*sqrt(2 )*a^(3/4)*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(a - b/x^3)^(1/4))/a^ (1/4))/(a*b^2) - 1/3*(2*sqrt(2)*a^(11/4) - 3*sqrt(2)*a^(3/4)*b)*arctan(-1/ 2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(a - b/x^3)^(1/4))/a^(1/4))/(a*b^2)
Timed out. \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=-\int \frac {3\,x^6-3\,a\,x^3+b}{x^6\,{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (b-2\,a\,x^3\right )} \,d x \]