Integrand size = 30, antiderivative size = 231 \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\sqrt {-\left (\left (\sqrt {c}+\sqrt {d}\right ) \sqrt {d}\right )} (b c+a d) \arctan \left (\frac {\sqrt {-\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {\sqrt {\left (\sqrt {c}-\sqrt {d}\right ) \sqrt {d}} (b c+a d) \arctan \left (\frac {\sqrt {\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}-\frac {a \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 c} \]
[Out]
Time = 0.51 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.16, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2081, 6847, 1707, 201, 223, 212, 1189, 399, 385, 211, 214} \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x^4-x} \sqrt {\sqrt {c}-\sqrt {d}} (a d+b c) \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {c}-\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt {x^4-x} \sqrt {\sqrt {c}+\sqrt {d}} (a d+b c) \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {c}+\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}-\frac {a \sqrt {x^4-x} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 c \sqrt {x^3-1} \sqrt {x}}+\frac {a \sqrt {x^4-x} x}{3 c} \]
[In]
[Out]
Rule 201
Rule 211
Rule 212
Rule 214
Rule 223
Rule 385
Rule 399
Rule 1189
Rule 1707
Rule 2081
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3} \left (b+a x^6\right )}{-d+c x^6} \, dx}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (b+a x^4\right )}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \left (\frac {a \sqrt {-1+x^2}}{c}+\frac {(b c+a d) \sqrt {-1+x^2}}{c \left (-d+c x^4\right )}\right ) \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (2 (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}-c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}+c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}-c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}+c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {a \sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (-c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}+\frac {\sqrt {\sqrt {c}-\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \arctan \left (\frac {\sqrt {\sqrt {c}-\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {-1+x^3}}-\frac {a \sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\sqrt {\sqrt {c}+\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \text {arctanh}\left (\frac {\sqrt {\sqrt {c}+\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {-1+x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x} \sqrt {-1+x^3} \left (a \left (x^{3/2} \sqrt {-1+x^3}-\log \left (x^{3/2}+\sqrt {-1+x^3}\right )\right )+(b c+a d) \text {RootSum}\left [16 c-16 d+32 c \text {$\#$1}-32 d \text {$\#$1}+24 c \text {$\#$1}^2-16 d \text {$\#$1}^2+8 c \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (-2+2 x^3+2 x^{3/2} \sqrt {-1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{8 c-8 d+12 c \text {$\#$1}-8 d \text {$\#$1}+6 c \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]\right )}{3 c \sqrt {x \left (-1+x^3\right )}} \]
[In]
[Out]
Time = 1.79 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {a \,x^{2} \left (x^{3}-1\right )}{3 c \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{3}+\frac {2 \left (a d +b c \right ) \left (\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{3 \sqrt {c d}}}{2 c}\) | \(166\) |
default | \(\frac {a \left (\frac {x \sqrt {x^{4}-x}}{3}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )}{c}-\frac {\left (a d +b c \right ) \left (\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{3 c \sqrt {c d}}\) | \(177\) |
pseudoelliptic | \(\frac {\left (2 \left (a d +b c \right ) \sqrt {\left (\sqrt {c d}+d \right ) d}\, \left (-\sqrt {c d}+d \right ) \arctan \left (\frac {d \sqrt {x^{4}-x}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )+\sqrt {\left (\sqrt {c d}-d \right ) d}\, \left (2 \left (\sqrt {c d}+d \right ) \left (a d +b c \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x^{4}-x}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )+a \sqrt {\left (\sqrt {c d}+d \right ) d}\, \left (2 x \sqrt {x^{4}-x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right ) \sqrt {c d}\right )\right ) x}{6 \sqrt {\left (\sqrt {c d}+d \right ) d}\, \sqrt {c d}\, \sqrt {\left (\sqrt {c d}-d \right ) d}\, c \left (x^{2}+\sqrt {x^{4}-x}\right ) \left (x^{2}-\sqrt {x^{4}-x}\right )}\) | \(254\) |
elliptic | \(\text {Expression too large to display}\) | \(681\) |
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{6} + b\right )}{c x^{6} - d}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int { \frac {{\left (a x^{6} + b\right )} \sqrt {x^{4} - x}}{c x^{6} - d} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (179) = 358\).
Time = 1.83 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.99 \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x^{4} - x} a x}{3 \, c} - \frac {a \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{6 \, c} + \frac {a \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, c} + \frac {{\left ({\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} d^{2}\right )} a c^{2} {\left | d \right |} + {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d\right )} b c^{2} {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} d^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d^{3}\right )} a {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{3} d + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} d^{2}\right )} b {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} + {\left (c^{2} - c d\right )} c d}}{c d}}}\right )}{3 \, {\left (4 \, c^{4} d^{3} + c^{3} d^{4} - 5 \, c^{2} d^{5}\right )} {\left | c \right |}} + \frac {{\left ({\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} d^{2}\right )} a c^{2} {\left | d \right |} + {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d\right )} b c^{2} {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} d^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d^{3}\right )} a {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{3} d + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} d^{2}\right )} b {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} + {\left (c^{2} - c d\right )} c d}}{c d}}}\right )}{3 \, {\left (4 \, c^{4} d^{3} + c^{3} d^{4} - 5 \, c^{2} d^{5}\right )} {\left | c \right |}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int -\frac {\sqrt {x^4-x}\,\left (a\,x^6+b\right )}{d-c\,x^6} \,d x \]
[In]
[Out]