Optimal. Leaf size=291 \[ -\frac {2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \text {ArcTan}\left (\frac {(1+i) \sqrt [4]{d} \sqrt [4]{-b c+a d} x \sqrt [4]{-b x^3+a x^4}}{\sqrt {-b c+a d} x^2-i \sqrt {d} \sqrt {-b x^3+a x^4}}\right )}{c \sqrt [4]{d}}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{-b c+a d} x^2}{\sqrt [4]{d}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt [4]{-b c+a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.19, antiderivative size = 59, normalized size of antiderivative = 0.20, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2067, 129, 525,
524} \begin {gather*} -\frac {4 \sqrt [4]{a x^4-b x^3} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};\frac {a x}{b},\frac {c x}{d}\right )}{3 d \sqrt [4]{1-\frac {a x}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 524
Rule 525
Rule 2067
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 1.03, size = 303, normalized size = 1.04 \begin {gather*} -\frac {x^{9/4} (-b+a x)^{3/4} \left (2 \sqrt [4]{a} \sqrt [4]{d} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}{\sqrt {b c-a d} \sqrt {x}-\sqrt {d} \sqrt {-b+a x}}\right )-2 \sqrt [4]{a} \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {x}+\sqrt {d} \sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}\right )\right )}{c \sqrt [4]{d} \left (x^3 (-b+a x)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, 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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs.
\(2 (228) = 456\).
time = 0.45, 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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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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