Integrand size = 34, antiderivative size = 157 \[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (b-a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{21 b^2 x^6}+\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 b^2}+\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 b^2} \] Output:
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)* arctan(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 = 15.83 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92 \[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (b-a x^3\right ) \left (-b x+a x^4\right )^{3/4}+7\ 2^{3/4} a^{3/4} (a+b) x^6 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )+7\ 2^{3/4} a^{3/4} (a+b) x^6 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{21 b^2 x^6} \] Input:
Integrate[(b + a*x^6)/(x^6*(b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]
Output:
(4*(b - a*x^3)*(-(b*x) + a*x^4)^(3/4) + 7*2^(3/4)*a^(3/4)*(a + b)*x^6*ArcT an[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)] + 7*2^(3/4)*a^(3 /4)*(a + b)*x^6*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x ^3)])/(21*b^2*x^6)
Time = 1.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {a x^6+b}{x^6 \left (a x^3+b\right ) \sqrt [4]{a x^4-b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {a x^6+b}{x^{25/4} \sqrt [4]{a x^3-b} \left (a x^3+b\right )}dx}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {a x^6+b}{x^{11/2} \sqrt [4]{a x^3-b} \left (a x^3+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \left (-\frac {a}{b x^{5/2} \sqrt [4]{a x^3-b}}+\frac {(a+b) \sqrt {x} a}{b \sqrt [4]{a x^3-b} \left (a x^3+b\right )}+\frac {1}{x^{11/2} \sqrt [4]{a x^3-b}}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \left (\frac {a^{3/4} (a+b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{6 \sqrt [4]{2} b^2}+\frac {a^{3/4} (a+b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{6 \sqrt [4]{2} b^2}-\frac {a \left (a x^3-b\right )^{3/4}}{21 b^2 x^{9/4}}+\frac {\left (a x^3-b\right )^{3/4}}{21 b x^{21/4}}\right )}{\sqrt [4]{a x^4-b x}}\) |
Input:
Int[(b + a*x^6)/(x^6*(b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]
Output:
(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)) + (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)
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 = 137, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {7 x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) 2^{\frac {3}{4}} \ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right )-14 x^{7} \left (a^{\frac {7}{4}}+a^{\frac {3}{4}} b \right ) 2^{\frac {3}{4}} \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )-8 {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {7}{4}}}{42 b^{2} x^{7}}\) | \(137\) |
Input:
int((a*x^6+b)/x^6/(a*x^3+b)/(a*x^4-b*x)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/42*(7*x^7*(a^(7/4)+a^(3/4)*b)*2^(3/4)*ln((-2^(1/4)*a^(1/4)*x-(x*(a*x^3-b ))^(1/4))/(2^(1/4)*a^(1/4)*x-(x*(a*x^3-b))^(1/4)))-14*x^7*(a^(7/4)+a^(3/4) *b)*2^(3/4)*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. \[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \] Input:
integrate((a*x^6+b)/x^6/(a*x^3+b)/(a*x^4-b*x)^(1/4),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {a x^{6} + b}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )} \left (a x^{3} + b\right )}\, dx \] Input:
integrate((a*x**6+b)/x**6/(a*x**3+b)/(a*x**4-b*x)**(1/4),x)
Output:
Integral((a*x**6 + b)/(x**6*(x*(a*x**3 - b))**(1/4)*(a*x**3 + b)), x)
\[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )} x^{6}} \,d x } \] Input:
integrate((a*x^6+b)/x^6/(a*x^3+b)/(a*x^4-b*x)^(1/4),x, algorithm="maxima")
Output:
integrate((a*x^6 + b)/((a*x^4 - b*x)^(1/4)*(a*x^3 + b)*x^6), x)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (126) = 252\).
Time = 0.24 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.78 \[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} + \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{12 \, b^{2}} + \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{12 \, b^{2}} - \frac {4 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} \] Input:
integrate((a*x^6+b)/x^6/(a*x^3+b)/(a*x^4-b*x)^(1/4),x, algorithm="giac")
Output:
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. \[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {a\,x^6+b}{x^6\,{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,d x \] Input:
int((b + a*x^6)/(x^6*(a*x^4 - b*x)^(1/4)*(b + a*x^3)),x)
Output:
int((b + a*x^6)/(x^6*(a*x^4 - b*x)^(1/4)*(b + a*x^3)), x)
\[ \int \frac {b+a x^6}{x^6 \left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\left (\int \frac {1}{x^{\frac {37}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} a +x^{\frac {25}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} b}d x \right ) b +\left (\int \frac {1}{x^{\frac {13}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} a +x^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} b}d x \right ) a \] Input:
int((a*x^6+b)/x^6/(a*x^3+b)/(a*x^4-b*x)^(1/4),x)
Output:
int(1/(x**(1/4)*(a*x**3 - b)**(1/4)*a*x**9 + x**(1/4)*(a*x**3 - b)**(1/4)* b*x**6),x)*b + int(1/(x**(1/4)*(a*x**3 - b)**(1/4)*a*x**3 + x**(1/4)*(a*x* *3 - b)**(1/4)*b),x)*a