Integrand size = 25, antiderivative size = 117 \[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (3 b+4 a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{63 b x^6}+\frac {2}{3} a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )+\frac {2}{3} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right ) \] Output:
4/63*(4*a*x^3+3*b)*(a*x^4-b*x)^(3/4)/b/x^6+2/3*a^(3/4)*arctan(a^(1/4)*(a*x ^4-b*x)^(3/4)/(a*x^3-b))+2/3*a^(3/4)*arctanh(a^(1/4)*(a*x^4-b*x)^(3/4)/(a* x^3-b))
Time = 10.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.26 \[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {2 \left (-6 b^2-2 a b x^3+8 a^2 x^6+21 a^{3/4} b x^{21/4} \sqrt [4]{-b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )+21 a^{3/4} b x^{21/4} \sqrt [4]{-b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )\right )}{63 b x^5 \sqrt [4]{-b x+a x^4}} \] Input:
Integrate[(b + a*x^6)/(x^6*(-(b*x) + a*x^4)^(1/4)),x]
Output:
(2*(-6*b^2 - 2*a*b*x^3 + 8*a^2*x^6 + 21*a^(3/4)*b*x^(21/4)*(-b + a*x^3)^(1 /4)*ArcTan[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)] + 21*a^(3/4)*b*x^(21/4)*( -b + a*x^3)^(1/4)*ArcTanh[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)]))/(63*b*x^ 5*(-(b*x) + a*x^4)^(1/4))
Time = 0.40 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2449, 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.
\(\displaystyle \int \frac {a x^6+b}{x^6 \sqrt [4]{a x^4-b x}} \, dx\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (\frac {a}{\sqrt [4]{a x^4-b x}}+\frac {b}{x^6 \sqrt [4]{a x^4-b x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 \left (a x^4-b x\right )^{3/4}}{21 x^6}+\frac {16 a \left (a x^4-b x\right )^{3/4}}{63 b x^3}\) |
Input:
Int[(b + a*x^6)/(x^6*(-(b*x) + a*x^4)^(1/4)),x]
Output:
(4*(-(b*x) + a*x^4)^(3/4))/(21*x^6) + (16*a*(-(b*x) + a*x^4)^(3/4))/(63*b* x^3) + (2*a^(3/4)*x^(1/4)*(-b + a*x^3)^(1/4)*ArcTan[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(3*(-(b*x) + a*x^4)^(1/4)) + (2*a^(3/4)*x^(1/4)*(-b + a*x ^3)^(1/4)*ArcTanh[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(3*(-(b*x) + a*x^ 4)^(1/4))
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Time = 0.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(\frac {-42 \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{x \,a^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} b \,x^{6}+21 \ln \left (\frac {a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} b \,x^{6}+16 {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {3}{4}} a \,x^{3}+12 b {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {3}{4}}}{63 b \,x^{6}}\) | \(127\) |
Input:
int((a*x^6+b)/x^6/(a*x^4-b*x)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/63*(-42*arctan((x*(a*x^3-b))^(1/4)/x/a^(1/4))*a^(3/4)*b*x^6+21*ln((a^(1/ 4)*x+(x*(a*x^3-b))^(1/4))/(-a^(1/4)*x+(x*(a*x^3-b))^(1/4)))*a^(3/4)*b*x^6+ 16*(x*(a*x^3-b))^(3/4)*a*x^3+12*b*(x*(a*x^3-b))^(3/4))/b/x^6
Timed out. \[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \] Input:
integrate((a*x^6+b)/x^6/(a*x^4-b*x)^(1/4),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {a x^{6} + b}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )}}\, dx \] Input:
integrate((a*x**6+b)/x**6/(a*x**4-b*x)**(1/4),x)
Output:
Integral((a*x**6 + b)/(x**6*(x*(a*x**3 - b))**(1/4)), x)
\[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} x^{6}} \,d x } \] Input:
integrate((a*x^6+b)/x^6/(a*x^4-b*x)^(1/4),x, algorithm="maxima")
Output:
integrate((a*x^6 + b)/((a*x^4 - b*x)^(1/4)*x^6), x)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (97) = 194\).
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.85 \[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) + \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) - \frac {4 \, {\left (3 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}} b^{6} - 7 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a b^{6}\right )}}{63 \, b^{7}} \] Input:
integrate((a*x^6+b)/x^6/(a*x^4-b*x)^(1/4),x, algorithm="giac")
Output:
1/3*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x ^3)^(1/4))/(-a)^(1/4)) + 1/3*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt( 2)*(-a)^(1/4) - 2*(a - b/x^3)^(1/4))/(-a)^(1/4)) - 1/6*sqrt(2)*(-a)^(3/4)* log(sqrt(2)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(-a) + sqrt(a - b/x^3)) + 1 /6*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(-a) + sqrt(a - b/x^3)) - 4/63*(3*(a - b/x^3)^(7/4)*b^6 - 7*(a - b/x^3)^(3/4)* a*b^6)/b^7
Time = 7.99 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4\,{\left (a\,x^4-b\,x\right )}^{3/4}\,\left (4\,a\,x^3+3\,b\right )}{63\,b\,x^6}+\frac {4\,a\,x\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4-b\,x\right )}^{1/4}} \] Input:
int((b + a*x^6)/(x^6*(a*x^4 - b*x)^(1/4)),x)
Output:
(4*(a*x^4 - b*x)^(3/4)*(3*b + 4*a*x^3))/(63*b*x^6) + (4*a*x*(1 - (a*x^3)/b )^(1/4)*hypergeom([1/4, 1/4], 5/4, (a*x^3)/b))/(3*(a*x^4 - b*x)^(1/4))
\[ \int \frac {b+a x^6}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\left (\int \frac {1}{x^{\frac {25}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}}}d x \right ) b +\left (\int \frac {1}{x^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}}}d x \right ) a \] Input:
int((a*x^6+b)/x^6/(a*x^4-b*x)^(1/4),x)
Output:
int(1/(x**(1/4)*(a*x**3 - b)**(1/4)*x**6),x)*b + int(1/(x**(1/4)*(a*x**3 - b)**(1/4)),x)*a