Integrand size = 26, antiderivative size = 236 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {\text {RootSum}\left [b^2 c+a^2 d-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)-a^2 d \log (x)+b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 c d} \]
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Leaf count is larger than twice the leaf count of optimal. \(561\) vs. \(2(236)=472\).
Time = 1.03 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.38, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2081, 920, 65, 338, 304, 209, 212, 6857, 95, 211, 214} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}+\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}} \]
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Rule 65
Rule 95
Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 338
Rule 920
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{-b+a x}}{d+c x^2} \, dx}{x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {\sqrt [4]{-b x^3+a x^4} \int \frac {-a d-b c x}{\sqrt [4]{x} (-b+a x)^{3/4} \left (d+c x^2\right )} \, dx}{c 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 {\sqrt [4]{-b x^3+a x^4} \int \left (\frac {-\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, 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 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 (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {-c} x\right )} \, dx}{2 c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (-b \sqrt {-c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {-c} x\right )} \, dx}{2 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 {\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 \left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (b \sqrt {-c}+a \sqrt {d}\right ) 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 \left (-b \sqrt {-c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (-b \sqrt {-c}+a \sqrt {d}\right ) x^4} \, 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} \arctan \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]{a} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\sqrt {-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {-b \sqrt {-c}+a \sqrt {d}} 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 (\sqrt {-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {-b \sqrt {-c}+a \sqrt {d}} 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 (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} x^{3/4} \sqrt [4]{-b+a x}} \\ & = -\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (16 \sqrt [4]{a} d \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )+\text {RootSum}\left [b^2 c+a^2 d-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)-a^2 d \log (x)+4 b^2 c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 c d \left (x^3 (-b+a x)\right )^{3/4}} \]
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Time = 0.50 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {2 a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right ) d +4 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) d +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-2 a d \,\textit {\_Z}^{4}+a^{2} d +b^{2} c \right )}{\sum }\frac {\left (-\textit {\_R}^{4} a d +a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right )}{2 c d}\) | \(167\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 829, normalized size of antiderivative = 3.51 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=-\frac {1}{2} \, \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} \log \left (\frac {c x \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} \log \left (-\frac {c x \sqrt {-\sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} \log \left (\frac {c x \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} \log \left (-\frac {c x \sqrt {-\sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - 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 {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 ) + i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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Not integrable
Time = 0.72 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{c x^{2} + d}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{c x^{2} + d} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 121.01 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\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 {2 \, \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {-c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {-c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, c} \]
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Not integrable
Time = 6.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx=\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{c\,x^2+d} \,d x \]
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