Integrand size = 15, antiderivative size = 92 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=\frac {\sqrt [4]{b+a x^3} \left (-4 b+7 a x^3\right )}{24 b^2 x^6}-\frac {7 a^2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}}-\frac {7 a^2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 44, 65, 218, 212, 209} \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=-\frac {7 a^2 \arctan \left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{16 b^{11/4}}-\frac {7 a^2 \text {arctanh}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{16 b^{11/4}}+\frac {7 a \sqrt [4]{a x^3+b}}{24 b^2 x^3}-\frac {\sqrt [4]{a x^3+b}}{6 b x^6} \]
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Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 (b+a x)^{3/4}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [4]{b+a x^3}}{6 b x^6}-\frac {(7 a) \text {Subst}\left (\int \frac {1}{x^2 (b+a x)^{3/4}} \, dx,x,x^3\right )}{24 b} \\ & = -\frac {\sqrt [4]{b+a x^3}}{6 b x^6}+\frac {7 a \sqrt [4]{b+a x^3}}{24 b^2 x^3}+\frac {\left (7 a^2\right ) \text {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^3\right )}{32 b^2} \\ & = -\frac {\sqrt [4]{b+a x^3}}{6 b x^6}+\frac {7 a \sqrt [4]{b+a x^3}}{24 b^2 x^3}+\frac {(7 a) \text {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )}{8 b^2} \\ & = -\frac {\sqrt [4]{b+a x^3}}{6 b x^6}+\frac {7 a \sqrt [4]{b+a x^3}}{24 b^2 x^3}-\frac {\left (7 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )}{16 b^{5/2}}-\frac {\left (7 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )}{16 b^{5/2}} \\ & = -\frac {\sqrt [4]{b+a x^3}}{6 b x^6}+\frac {7 a \sqrt [4]{b+a x^3}}{24 b^2 x^3}-\frac {7 a^2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}}-\frac {7 a^2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=\frac {\sqrt [4]{b+a x^3} \left (-4 b+7 a x^3\right )}{24 b^2 x^6}-\frac {7 a^2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}}-\frac {7 a^2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{16 b^{11/4}} \]
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Time = 1.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(\frac {-21 \ln \left (\frac {\left (a \,x^{3}+b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{\left (a \,x^{3}+b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}\right ) a^{2} x^{6}-42 \arctan \left (\frac {\left (a \,x^{3}+b \right )^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) a^{2} x^{6}-16 \left (a \,x^{3}+b \right )^{\frac {1}{4}} b^{\frac {7}{4}}+28 a \,x^{3} \left (a \,x^{3}+b \right )^{\frac {1}{4}} b^{\frac {3}{4}}}{96 b^{\frac {11}{4}} x^{6}}\) | \(104\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.37 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=-\frac {21 \, b^{2} x^{6} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (7 \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 7 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{2}\right ) + 21 i \, b^{2} x^{6} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (7 i \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 7 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 21 i \, b^{2} x^{6} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-7 i \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 7 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 21 \, b^{2} x^{6} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-7 \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 7 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 4 \, {\left (7 \, a x^{3} - 4 \, b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}}{96 \, b^{2} x^{6}} \]
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Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{4}} x^{\frac {33}{4}} \Gamma \left (\frac {15}{4}\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=\frac {7 \, {\left (a x^{3} + b\right )}^{\frac {5}{4}} a^{2} - 11 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{2} b}{24 \, {\left ({\left (a x^{3} + b\right )}^{2} b^{2} - 2 \, {\left (a x^{3} + b\right )} b^{3} + b^{4}\right )}} - \frac {7 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \log \left (\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{3} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}}\right )}}{32 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (72) = 144\).
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=\frac {\frac {42 \, \sqrt {2} a^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {42 \, \sqrt {2} a^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \log \left (\sqrt {2} {\left (a x^{3} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{3} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \left (-b\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{3} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{3} + b} + \sqrt {-b}\right )}{b^{3}} + \frac {8 \, {\left (7 \, {\left (a x^{3} + b\right )}^{\frac {5}{4}} a^{3} - 11 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a^{3} b\right )}}{a^{2} b^{2} x^{6}}}{192 \, a} \]
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Time = 6.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^7 \left (b+a x^3\right )^{3/4}} \, dx=\frac {7\,{\left (a\,x^3+b\right )}^{5/4}}{24\,b^2\,x^6}-\frac {11\,{\left (a\,x^3+b\right )}^{1/4}}{24\,b\,x^6}-\frac {7\,a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^3+b\right )}^{1/4}}{b^{1/4}}\right )}{16\,b^{11/4}}+\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^3+b\right )}^{1/4}\,1{}\mathrm {i}}{b^{1/4}}\right )\,7{}\mathrm {i}}{16\,b^{11/4}} \]
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