Integrand size = 35, antiderivative size = 149 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (b+a x^3\right ) \left (b x+a x^4\right )^{3/4}}{21 b^2 x^6}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2} \]
4/21*(a*x^3+b)*(a*x^4+b*x)^(3/4)/b^2/x^6-1/3*2^(3/4)*(a^(7/4)+a^(3/4)*b)*a rctan(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/b^2-1/3*2^(3/4)*(a^(7/4 )+a^(3/4)*b)*arctanh(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/b^2
Time = 15.85 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {\frac {4 \left (x \left (b+a x^3\right )\right )^{7/4}}{x}-7\ 2^{3/4} a^{3/4} (a+b) x^6 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{x \left (b+a x^3\right )}}\right )-7\ 2^{3/4} a^{3/4} (a+b) x^6 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{x \left (b+a x^3\right )}}\right )}{21 b^2 x^6} \]
((4*(x*(b + a*x^3))^(7/4))/x - 7*2^(3/4)*a^(3/4)*(a + b)*x^6*ArcTan[(2^(1/ 4)*a^(1/4)*x)/(x*(b + a*x^3))^(1/4)] - 7*2^(3/4)*a^(3/4)*(a + b)*x^6*ArcTa nh[(2^(1/4)*a^(1/4)*x)/(x*(b + a*x^3))^(1/4)])/(21*b^2*x^6)
Time = 1.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
\(\displaystyle \int \frac {a x^6+b}{x^6 \left (a x^3-b\right ) \sqrt [4]{a x^4+b x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int -\frac {a x^6+b}{x^{25/4} \left (b-a x^3\right ) \sqrt [4]{a x^3+b}}dx}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {a x^6+b}{x^{25/4} \left (b-a x^3\right ) \sqrt [4]{a x^3+b}}dx}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {a x^6+b}{x^{11/2} \left (b-a x^3\right ) \sqrt [4]{a x^3+b}}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \left (\frac {(a+b) \sqrt {x} a}{b \left (b-a x^3\right ) \sqrt [4]{a x^3+b}}+\frac {a}{b x^{5/2} \sqrt [4]{a x^3+b}}+\frac {1}{x^{11/2} \sqrt [4]{a x^3+b}}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {a^{3/4} (a+b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 \sqrt [4]{2} b^2}+\frac {a^{3/4} (a+b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 \sqrt [4]{2} b^2}-\frac {a \left (a x^3+b\right )^{3/4}}{21 b^2 x^{9/4}}-\frac {\left (a x^3+b\right )^{3/4}}{21 b x^{21/4}}\right )}{\sqrt [4]{a x^4+b x}}\) |
(-4*x^(1/4)*(b + a*x^3)^(1/4)*(-1/21*(b + a*x^3)^(3/4)/(b*x^(21/4)) - (a*( b + a*x^3)^(3/4))/(21*b^2*x^(9/4)) + (a^(3/4)*(a + b)*ArcTan[(2^(1/4)*a^(1 /4)*x^(3/4))/(b + a*x^3)^(1/4)])/(6*2^(1/4)*b^2) + (a^(3/4)*(a + b)*ArcTan h[(2^(1/4)*a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(6*2^(1/4)*b^2)))/(b*x + a *x^4)^(1/4)
3.21.66.3.1 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 = 0.70 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-7 \,2^{\frac {3}{4}} x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right )+14 \,2^{\frac {3}{4}} x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+8 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {7}{4}}}{42 b^{2} x^{7}}\) | \(125\) |
1/42*(-7*2^(3/4)*x^7*(a^(7/4)+a^(3/4)*b)*ln((x*2^(1/4)*a^(1/4)+(x*(a*x^3+b ))^(1/4))/(-x*2^(1/4)*a^(1/4)+(x*(a*x^3+b))^(1/4)))+14*2^(3/4)*x^7*(a^(7/4 )+a^(3/4)*b)*arctan(1/2*(x*(a*x^3+b))^(1/4)/x*2^(3/4)/a^(1/4))+8*(x*(a*x^3 +b))^(7/4))/b^2/x^7
Timed out. \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{6} + b}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \]
\[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{6}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (117) = 234\).
Time = 0.31 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.83 \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} + \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} \]
-1/6*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(1/2*2^(1 /4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/b^2 - 1/6*sqrt( 2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(-1/2*2^(1/4)*(2^(3 /4)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/b^2 + 1/12*sqrt(2)*(2^(3 /4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*log(2^(3/4)*(-a)^(1/4)*(a + b/x^3 )^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/b^2 - 1/12*sqrt(2)*(2^(3/4)* (-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*log(-2^(3/4)*(-a)^(1/4)*(a + b/x^3)^( 1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/b^2 + 4/21*(a + b/x^3)^(7/4)/b^ 2
Timed out. \[ \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {a\,x^6+b}{x^6\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]