Optimal. Leaf size=291 \[ -\frac {(1-i) \sqrt [4]{a d-b c} \tan ^{-1}\left (\frac {(1+i) \sqrt [4]{d} x \sqrt [4]{a x^4-b x^3} \sqrt [4]{a d-b c}}{x^2 \sqrt {a d-b c}-i \sqrt {d} \sqrt {a x^4-b x^3}}\right )}{c \sqrt [4]{d}}-\frac {(1-i) \sqrt [4]{a d-b c} \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \sqrt [4]{a d-b c}}{\sqrt [4]{d}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{d} \sqrt {a x^4-b x^3}}{\sqrt [4]{a d-b c}}}{x \sqrt [4]{a x^4-b x^3}}\right )}{c \sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c} \]
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Rubi [A] time = 0.40, antiderivative size = 309, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2042, 105, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a d-b c} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a d-b c} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}} \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 298
Rule 331
Rule 2042
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} (-d+c x)} \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (a \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left ((b c-a d) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} (-d+c x)} \, dx}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 (b c-a d) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-d-(b c-a d) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 (b c-a d) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d}-\sqrt {-b c+a d} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {-b c+a d} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 (b c-a d) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d}+\sqrt {-b c+a d} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {-b c+a d} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {2 \sqrt [4]{-b c+a d} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-b c+a d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{-b c+a d} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b c+a d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt {a} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt {a} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{-b c+a d} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-b c+a d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{-b c+a d} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b c+a d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 105, normalized size = 0.36 \begin {gather*} -\frac {4 \sqrt [4]{x^3 (a x-b)} \left ((b c-a d) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {b c x-a d x}{b d-a d x}\right )+a d \left (1-\frac {a x}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {a x}{b}\right )\right )}{3 c d (b-a x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.38, size = 293, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {\sqrt {2} \sqrt [4]{b c-a d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} x \sqrt [4]{-b x^3+a x^4}}{\sqrt {b c-a d} x^2-\sqrt {d} \sqrt {-b x^3+a x^4}}\right )}{c \sqrt [4]{d}}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {\sqrt {2} \sqrt [4]{b c-a d} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b c-a d} x^2}{\sqrt {2} \sqrt [4]{d}}+\frac {\sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt {2} \sqrt [4]{b c-a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 441, normalized size = 1.52 \begin {gather*} 4 \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \arctan \left (-\frac {c^{3} d x \sqrt {\frac {c^{2} x^{2} \sqrt {-\frac {b c - a d}{c^{4} d}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} d \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {3}{4}}}{{\left (b c - a d\right )} x}\right ) - 4 \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {c^{3} x \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}}}{a x}\right ) + \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} - {\left (a x^{4} - b 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 = 0.51, size = 505, normalized size = 1.74 \begin {gather*} \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} + \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} \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 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{x \left (c x -d \right )}\, dx\]
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 (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x - d\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 (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (d-c\,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 (a x - b\right )}}{x \left (c x - d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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