Integrand size = 37, antiderivative size = 241 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{d} \]
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Leaf count is larger than twice the leaf count of optimal. \(519\) vs. \(2(241)=482\).
Time = 1.02 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.15, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2081, 6857, 49, 65, 338, 304, 209, 212, 922, 37, 95, 211, 214} \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {4 \sqrt [4]{a x^4+b x^3}}{x} \]
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Rule 37
Rule 49
Rule 65
Rule 95
Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 338
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} \left (d+c x^2\right )}{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 \left (\frac {\sqrt [4]{b+a x}}{x^{5/4}}+\frac {2 d \sqrt [4]{b+a x}}{x^{5/4} \left (-d+c x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x}}{x^{5/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 d \sqrt [4]{b x^3+a x^4}\right ) \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 {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {-a d-b c x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-d+c x^2\right )} \, 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}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \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}{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 )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{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 )}{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}{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}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{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 )}{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 )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 \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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 \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 )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{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 )}{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 )}{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 {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}} 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 {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}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{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 )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \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 )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \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 )}{\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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \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 )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \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 )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {x^2 (b+a x)^{3/4} \left (2 d \left (2 \sqrt [4]{b+a x}-\sqrt [4]{a} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\sqrt [4]{a} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )-\frac {1}{4} \sqrt [4]{x} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{d \left (x^3 (b+a x)\right )^{3/4}} \]
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Time = 0.00 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-2 a d \,\textit {\_Z}^{4}+a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x +2 \left (\arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) x \,a^{\frac {1}{4}}+\frac {\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) x \,a^{\frac {1}{4}}}{2}+2 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}\right ) d}{d x}\) | \(177\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.58 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {\frac {b^{2} c}{d}}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {\frac {b^{2} c}{d}}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{\frac {1}{4}} x \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{\frac {1}{4}} x \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \]
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Not integrable
Time = 4.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (c x^{2} + d\right )}{x^{2} \left (c x^{2} - d\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (c x^{2} + d\right )}}{{\left (c x^{2} - d\right )} x^{2}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 116.02 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.04 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\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 ) + \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 ) + \frac {1}{2} \, \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 ) - \frac {1}{2} \, \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 \, \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 ) + 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16 \[ \int \frac {\left (d+c x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\int \frac {\left (c\,x^2+d\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,d x \]
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