Integrand size = 15, antiderivative size = 80 \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 338, 304, 209, 212} \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {3 a \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac {x^3 \sqrt [4]{a+b x^4}}{4 b} \]
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 327
Rule 338
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \int \frac {x^2}{\left (a+b x^4\right )^{3/4}} \, dx}{4 b} \\ & = \frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 b} \\ & = \frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/2}}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/2}} \\ & = \frac {x^3 \sqrt [4]{a+b x^4}}{4 b}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {2 b^{3/4} x^3 \sqrt [4]{a+b x^4}+3 a \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-3 a \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}} \]
[In]
[Out]
Time = 4.50 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {3 \left (-\frac {4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} x^{3} b^{\frac {3}{4}}}{3}+\ln \left (\frac {-b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right ) a +2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) a \right )}{16 b^{\frac {7}{4}}}\) | \(85\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.51 \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{3} - 3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right ) + 3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right ) + 3 i \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (i \, b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right ) - 3 i \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (-i \, b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right )}{16 \, b} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.46 \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.38 \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=-\frac {3 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{16 \, b} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a}{4 \, {\left (b^{2} - \frac {{\left (b x^{4} + a\right )} b}{x^{4}}\right )} x} \]
[In]
[Out]
\[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx=\int \frac {x^6}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \]
[In]
[Out]