Integrand size = 31, antiderivative size = 191 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 d^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(191)=382\).
Time = 0.67 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.24, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2081, 922, 37, 6857, 95, 304, 211, 214} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {4 \sqrt [4]{a x^4-b x^3}}{d x} \]
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Rule 37
Rule 95
Rule 211
Rule 214
Rule 304
Rule 922
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{x^{5/4} \left (-d+c x^2\right )} \, 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}{d x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (b \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (-b+a x)^{3/4}} \, dx}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}-\frac {\sqrt [4]{-b x^3+a x^4} \int \left (-\frac {b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {c} x\right )}-\frac {-b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )}\right ) \, dx}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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 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 ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )} \, dx}{2 d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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 )}{d 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 )}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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 )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} 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 )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} d 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 )}{d 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 )}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d 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 )}{d^{9/8} 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 )}{d^{9/8} 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 )}{d^{9/8} 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 )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {32 d x^2 (-b+a x)-x^{9/4} (-b+a x)^{3/4} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 d^2 \left (x^3 (-b+a x)\right )^{3/4}} \]
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Time = 0.00 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {8 \left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}} d +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} d -2 \textit {\_Z}^{4} a d +a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x}{2 d^{2} x}\) | \(110\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, d x} \]
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Not integrable
Time = 2.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x^{2} \left (c x^{2} - d\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x^{2} - d\right )} x^{2}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 121.09 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {2 \, \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, d} + \frac {4 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}}{d} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,d x \]
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