Optimal. Leaf size=113 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{b-x}\right )}{d^{3/4}} \]
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Rubi [C] time = 65.52, antiderivative size = 3261, normalized size of antiderivative = 28.86, number of steps used = 21, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6719, 6728, 107, 106, 490, 1217, 220, 1707}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 106
Rule 107
Rule 220
Rule 490
Rule 1217
Rule 1707
Rule 6719
Rule 6728
Rubi steps
\begin {align*} \int \frac {-2 a+b+x}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {-2 a+b+x}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}+\frac {1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt {-b+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \int \frac {1}{\sqrt [4]{-a+x} \left (-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right ) \sqrt {\frac {b}{-a+b}-\frac {x}{-a+b}}} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \int \frac {1}{\sqrt [4]{-a+x} \left (-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}+2 x\right ) \sqrt {\frac {b}{-a+b}-\frac {x}{-a+b}}} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (4 \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}-2 x^4\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}-2 x^4\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\frac {\sqrt {2} \left (-\frac {a}{-a+b}+\frac {b}{-a+b}\right )}{\sqrt {a-b}}-\frac {\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}{-a+b}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (-2 \left (-\frac {a}{-a+b}+\frac {b}{-a+b}\right )-\frac {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}{-a+b}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (\sqrt {2} a-\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt {a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}-\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (\sqrt {2} \left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {-b+x}{-a+b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {a-b}}}{\left (\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}+\sqrt {2} x^2\right ) \sqrt {-\frac {a}{-a+b}+\frac {b}{-a+b}-\frac {x^4}{-a+b}}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}-\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {-\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {-\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \sqrt [4]{-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {a-b} \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} d^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {-a+b} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt {-\frac {b-x}{a-b}} \sqrt [4]{-a+x} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {\sqrt {d}+\sqrt {-4 a+4 b+d}} \sqrt [4]{-a+x}}{\sqrt [4]{2} \sqrt {-a+b} \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {-\frac {b-x}{a-b}}}\right )}{2^{3/4} \sqrt {a-b} d^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (\sqrt {d}-\sqrt {-4 a+4 b+d}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \sqrt {d} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b-\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (2 a-\sqrt {2} \left (\sqrt {2} b+\sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1+\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d-\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1+\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b+\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {a-b}-\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (1+\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (1-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}}\right ) \left (2 a-2 b-\sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right ) \sqrt [4]{-a+x} \sqrt {-\frac {b-x}{(a-b) \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right )^2}} \left (1+\frac {\sqrt {-a+x}}{\sqrt {a-b}}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {a-b}+\sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}\right )^2}{4 \sqrt {2} \sqrt {a-b} \sqrt {-2 a+2 b+d+\sqrt {d} \sqrt {-4 a+4 b+d}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-a+x}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a-b} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [C] time = 25.69, size = 675, normalized size = 5.97 \begin {gather*} -\frac {i \sqrt {2} (x-a)^{3/4} \sqrt {\frac {b-x}{a-x}} \left (\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (-\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d-\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\left (\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \Pi \left (\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d-\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\left (-\sqrt {d (-4 a+4 b+d)}+4 a-4 b-d\right ) \sqrt {\frac {-\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \left (\Pi \left (-\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2}}{\sqrt {b-a} \sqrt {\frac {-2 a+2 b+d+\sqrt {d (-4 a+4 b+d)}}{(a-b)^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b-a}}}{\sqrt [4]{x-a}}\right )\right |-1\right )\right )\right )}{(a-b) \sqrt {-\sqrt {b-a}} \sqrt {d (-4 a+4 b+d)} \sqrt {\frac {-\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \sqrt {\frac {\sqrt {d (-4 a+4 b+d)}-2 a+2 b+d}{(a-b)^2}} \sqrt [4]{(x-a) (b-x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 113, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )}{d^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, a - b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {-2 a +b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{2}+a d -\left (2 b +d \right ) x +x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, a - b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {b-2\,a+x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-x\,\left (2\,b+d\right )+b^2+x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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