Optimal. Leaf size=188 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{x^4-x^3} \sqrt [4]{a-b}}{x^2 \sqrt {a-b}-\sqrt {b} \sqrt {x^4-x^3}}\right )}{b^{3/4} \sqrt [4]{a-b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2 \sqrt [4]{a-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {x^4-x^3}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{x^4-x^3}}\right )}{b^{3/4} \sqrt [4]{a-b}} \]
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Rubi [A] time = 0.11, antiderivative size = 139, normalized size of antiderivative = 0.74, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2056, 93, 212, 208, 205} \begin {gather*} -\frac {2 \sqrt [4]{x-1} x^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{b^{3/4} \sqrt [4]{x^4-x^3} \sqrt [4]{b-a}}-\frac {2 \sqrt [4]{x-1} x^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{b^{3/4} \sqrt [4]{x^4-x^3} \sqrt [4]{b-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 205
Rule 208
Rule 212
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx &=\frac {\left (\sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {1}{\sqrt [4]{-1+x} x^{3/4} (-b+a x)} \, dx}{\sqrt [4]{-x^3+x^4}}\\ &=\frac {\left (4 \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(a-b) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-x^3+x^4}}\\ &=-\frac {\left (2 \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {-a+b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {b} \sqrt [4]{-x^3+x^4}}-\frac {\left (2 \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {-a+b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {b} \sqrt [4]{-x^3+x^4}}\\ &=-\frac {2 \sqrt [4]{-1+x} x^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{b^{3/4} \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4}}-\frac {2 \sqrt [4]{-1+x} x^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{b^{3/4} \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.26 \begin {gather*} \frac {4 (x-1) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b (x-1)}{(a-b) x}\right )}{3 \sqrt [4]{(x-1) x^3} (a-b)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.30, size = 188, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} x \sqrt [4]{-x^3+x^4}}{\sqrt {a-b} x^2-\sqrt {b} \sqrt {-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a-b} x^2}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 262, normalized size = 1.39 \begin {gather*} -4 \, \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {b x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \sqrt {-\frac {{\left (a b - b^{2}\right )} x^{2} \sqrt {-\frac {1}{a b^{3} - b^{4}}} - \sqrt {x^{4} - x^{3}}}{x^{2}}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} b \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}}}{x}\right ) + \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 317, normalized size = 1.69 \begin {gather*} -\frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} - \frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} + \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} - \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (-\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x -b \right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\, dx\]
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 {1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (a x - b\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 {1}{{\left (x^4-x^3\right )}^{1/4}\,\left (b-a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{3} \left (x - 1\right )} \left (a x - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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