Optimal. Leaf size=237 \[ -\frac {\sqrt {2} \sqrt [4]{a-b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{x^4-x^3} \sqrt [4]{a-b}}{x^2 \sqrt {a-b}-\sqrt {b} \sqrt {x^4-x^3}}\right )}{a \sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [4]{a-b} \tanh ^{-1}\left (\frac {\frac {x^2 \sqrt [4]{a-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {x^4-x^3}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{x^4-x^3}}\right )}{a \sqrt [4]{b}}-\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{a}+\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{a} \]
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Rubi [A] time = 0.25, antiderivative size = 233, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2042, 105, 63, 240, 212, 206, 203, 93, 298, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{x^4-x^3} \sqrt [4]{b-a} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{a \sqrt [4]{b} \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [4]{x^4-x^3} \sqrt [4]{b-a} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{a \sqrt [4]{b} \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{x-1} x^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 298
Rule 2042
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} (-b+a x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{a \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((a-b) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (-b+a x)} \, dx}{a \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{a \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 (a-b) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-b-(a-b) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 (a-b) \sqrt [4]{-x^3+x^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 )}{a \sqrt {-a+b} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 (a-b) \sqrt [4]{-x^3+x^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 )}{a \sqrt {-a+b} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {2 \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{a \sqrt [4]{b} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{a \sqrt [4]{b} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{a \sqrt [4]{b} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{a \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-a+b} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-a+b} \sqrt [4]{x}}{\sqrt [4]{b} \sqrt [4]{-1+x}}\right )}{a \sqrt [4]{b} \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 68, normalized size = 0.29 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (\sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {b-b x}{a x-b x}\right )\right )}{a x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.08, size = 237, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \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 )}{a \sqrt [4]{b}}+\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \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 )}{a \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 297, normalized size = 1.25 \begin {gather*} \frac {4 \, a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \arctan \left (-\frac {a^{3} b x \sqrt {\frac {a^{2} x^{2} \sqrt {-\frac {a - b}{a^{4} b}} + \sqrt {x^{4} - x^{3}}}{x^{2}}} \left (-\frac {a - b}{a^{4} b}\right )^{\frac {3}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} a^{3} b \left (-\frac {a - b}{a^{4} b}\right )^{\frac {3}{4}}}{{\left (a - b\right )} x}\right ) - a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (-\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.00, size = 325, normalized size = 1.37 \begin {gather*} -\frac {2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}{a} - \frac {\log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right )}{a} + \frac {\log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right )}{a} + \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{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 )}{a b} + \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{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 )}{a b} + \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{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 )}{2 \, a b} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{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 )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x \left (a x -b \right )}\, dx\]
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 {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (a x - b\right )} x}\,{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 = -0.00 \begin {gather*} \int -\frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\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 {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (a x - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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