Integrand size = 21, antiderivative size = 323 \[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=-\frac {4 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}+\frac {4 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}}-\frac {2 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.23 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2045, 2050, 2057, 335, 311, 226, 1210} \[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=-\frac {2 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}}+\frac {4 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}}-\frac {4 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}-\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2045
Rule 2050
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}+\frac {1}{9} (2 c) \int \frac {1}{x^{5/2} \sqrt {b x^2+c x^4}} \, dx \\ & = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}-\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx}{15 b} \\ & = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac {\left (2 c^3\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 b^2} \\ & = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac {\left (2 c^3 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 b^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac {\left (4 c^3 x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac {\left (4 c^{5/2} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{3/2} \sqrt {b x^2+c x^4}}+\frac {\left (4 c^{5/2} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{3/2} \sqrt {b x^2+c x^4}} \\ & = -\frac {4 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{9 x^{11/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{45 b x^{7/2}}+\frac {4 c^2 \sqrt {b x^2+c x^4}}{15 b^2 x^{3/2}}+\frac {4 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}}-\frac {2 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {b x^2+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=-\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {1}{2},-\frac {5}{4},-\frac {c x^2}{b}\right )}{9 x^{11/2} \sqrt {1+\frac {c x^2}{b}}} \]
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Time = 0.41 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (6 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-3 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-6 c^{3} x^{6}-4 b \,c^{2} x^{4}+7 b^{2} c \,x^{2}+5 b^{3}\right )}{45 x^{\frac {11}{2}} \left (c \,x^{2}+b \right ) b^{2}}\) | \(239\) |
risch | \(-\frac {2 \left (-6 c^{2} x^{4}+2 b c \,x^{2}+5 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{45 x^{\frac {11}{2}} b^{2}}-\frac {2 c^{2} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{15 b^{2} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(241\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=\frac {2 \, {\left (6 \, c^{\frac {5}{2}} x^{6} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (6 \, c^{2} x^{4} - 2 \, b c x^{2} - 5 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{45 \, b^{2} x^{6}} \]
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\[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=\int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{\frac {13}{2}}}\, dx \]
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\[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {b x^2+c x^4}}{x^{13/2}} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2}}{x^{13/2}} \,d x \]
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