Integrand size = 30, antiderivative size = 187 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 d^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(406\) vs. \(2(187)=374\).
Time = 0.91 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.17, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, 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 = 0.40 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {32 d x^2 (b+a x)-x^{9/4} (b+a x)^{3/4} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 d^2 \left (x^3 (b+a x)\right )^{3/4}} \]
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Time = 0.44 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-2 a d \,\textit {\_Z}^{4}+a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x +8 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} d}{2 d^{2} x}\) | \(106\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.47 \[ \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 = 1.81 (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 = 0.22 (sec) , antiderivative size = 30, 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 = 118.85 (sec) , antiderivative size = 329, 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 = 6.39 (sec) , antiderivative size = 31, 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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