Integrand size = 40, antiderivative size = 56 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {2 \sqrt {q+p x^6} \left (3 a q^2+5 b q x^4+6 a p q x^6+5 b p x^{10}+3 a p^2 x^{12}\right )}{15 x^{10}} \]
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Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.70, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {12, 1847, 1598, 457, 75, 1492, 460} \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {2 a \left (p x^6+q\right )^{5/2}}{5 x^{10}}+\frac {2 b \left (p x^6+q\right )^{3/2}}{3 x^6} \]
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Rule 12
Rule 75
Rule 457
Rule 460
Rule 1492
Rule 1598
Rule 1847
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx \\ & = 2 \int \left (\frac {\sqrt {q+p x^6} \left (-2 b q x^3+b p x^9\right )}{x^{10}}+\frac {\sqrt {q+p x^6} \left (-2 a q^2-a p q x^6+a p^2 x^{12}\right )}{x^{11}}\right ) \, dx \\ & = 2 \int \frac {\sqrt {q+p x^6} \left (-2 b q x^3+b p x^9\right )}{x^{10}} \, dx+2 \int \frac {\sqrt {q+p x^6} \left (-2 a q^2-a p q x^6+a p^2 x^{12}\right )}{x^{11}} \, dx \\ & = 2 \int \frac {\left (q+p x^6\right )^{3/2} \left (-2 a q+a p x^6\right )}{x^{11}} \, dx+2 \int \frac {\sqrt {q+p x^6} \left (-2 b q+b p x^6\right )}{x^7} \, dx \\ & = \frac {2 a \left (q+p x^6\right )^{5/2}}{5 x^{10}}+\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {q+p x} (-2 b q+b p x)}{x^2} \, dx,x,x^6\right ) \\ & = \frac {2 b \left (q+p x^6\right )^{3/2}}{3 x^6}+\frac {2 a \left (q+p x^6\right )^{5/2}}{5 x^{10}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
Time = 10.48 (sec) , antiderivative size = 248, normalized size of antiderivative = 4.43 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {12 a q^2 \left (q+p x^6\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{2},-\frac {2}{3},-\frac {p x^6}{q}\right )+5 x^4 \left (-4 b p \sqrt {q} x^6 \sqrt {q+p x^6} \sqrt {1+\frac {p x^6}{q}} \text {arctanh}\left (\frac {\sqrt {q+p x^6}}{\sqrt {q}}\right )+4 b \left (q+p x^6\right ) \left (\left (q+p x^6\right ) \sqrt {1+\frac {p x^6}{q}}+p x^6 \text {arctanh}\left (\sqrt {1+\frac {p x^6}{q}}\right )\right )+3 a p q x^2 \left (q+p x^6\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {p x^6}{q}\right )+6 a p^2 x^8 \left (q+p x^6\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {p x^6}{q}\right )\right )}{30 x^{10} \sqrt {q+p x^6} \sqrt {1+\frac {p x^6}{q}}} \]
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Time = 2.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(\frac {2 \left (p \,x^{6}+q \right )^{\frac {3}{2}} \left (a p \,x^{6}+\frac {5}{3} b \,x^{4}+a q \right )}{5 x^{10}}\) | \(31\) |
gosper | \(\frac {2 \left (p \,x^{6}+q \right )^{\frac {3}{2}} \left (3 a p \,x^{6}+5 b \,x^{4}+3 a q \right )}{15 x^{10}}\) | \(33\) |
trager | \(\frac {2 \sqrt {p \,x^{6}+q}\, \left (3 a \,p^{2} x^{12}+5 b p \,x^{10}+6 a p q \,x^{6}+5 b q \,x^{4}+3 a \,q^{2}\right )}{15 x^{10}}\) | \(53\) |
risch | \(\frac {2 \sqrt {p \,x^{6}+q}\, \left (3 a \,p^{2} x^{12}+5 b p \,x^{10}+6 a p q \,x^{6}+5 b q \,x^{4}+3 a \,q^{2}\right )}{15 x^{10}}\) | \(53\) |
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Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {2 \, {\left (3 \, a p^{2} x^{12} + 5 \, b p x^{10} + 6 \, a p q x^{6} + 5 \, b q x^{4} + 3 \, a q^{2}\right )} \sqrt {p x^{6} + q}}{15 \, x^{10}} \]
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Result contains complex when optimal does not.
Time = 5.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.98 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {a p^{2} \sqrt {q} x^{2} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {p x^{6} e^{i \pi }}{q}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {a p q^{\frac {3}{2}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {p x^{6} e^{i \pi }}{q}} \right )}}{3 x^{4} \Gamma \left (\frac {1}{3}\right )} - \frac {2 a q^{\frac {5}{2}} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {p x^{6} e^{i \pi }}{q}} \right )}}{3 x^{10} \Gamma \left (- \frac {2}{3}\right )} + \frac {2 b p^{\frac {3}{2}} x^{3}}{3 \sqrt {1 + \frac {q}{p x^{6}}}} + \frac {2 b \sqrt {p} q \sqrt {1 + \frac {q}{p x^{6}}}}{3 x^{3}} + \frac {2 b \sqrt {p} q}{3 x^{3} \sqrt {1 + \frac {q}{p x^{6}}}} \]
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Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {2 \, {\left (3 \, a p^{2} x^{12} + 5 \, b p x^{10} + 6 \, a p q x^{6} + 5 \, b q x^{4} + 3 \, a q^{2}\right )} \sqrt {p x^{6} + q}}{15 \, x^{10}} \]
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Time = 0.55 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\frac {2}{15} \, \sqrt {p x^{6} + q} {\left (3 \, a p^{2} x^{2} + 5 \, b p\right )} + \frac {2}{15} \, {\left (6 \, a p q + \frac {5 \, b q + \frac {3 \, a q^{2}}{x^{4}}}{x^{2}}\right )} \sqrt {\frac {p}{x^{2}} + \frac {q}{x^{8}}} \]
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Time = 6.71 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36 \[ \int \frac {2 \left (-2 q+p x^6\right ) \sqrt {q+p x^6} \left (a q+b x^4+a p x^6\right )}{x^{11}} \, dx=\sqrt {p\,x^6+q}\,\left (\frac {2\,a\,p^2\,x^2}{5}+\frac {2\,b\,p}{3}\right )+\frac {2\,a\,q^2\,\sqrt {p\,x^6+q}}{5\,x^{10}}+\frac {2\,b\,q\,\sqrt {p\,x^6+q}}{3\,x^6}+\frac {4\,a\,p\,q\,\sqrt {p\,x^6+q}}{5\,x^4} \]
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