Integrand size = 21, antiderivative size = 323 \[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\frac {28 b^3 x^{3/2} \left (b+c x^2\right )}{195 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}-\frac {28 b^{13/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {14 b^{13/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{195 c^{11/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.24 (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 = {2046, 2049, 2057, 335, 311, 226, 1210} \[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\frac {14 b^{13/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{195 c^{11/4} \sqrt {b x^2+c x^4}}-\frac {28 b^{13/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {28 b^3 x^{3/2} \left (b+c x^2\right )}{195 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {1}{13} (2 b) \int \frac {x^{11/2}}{\sqrt {b x^2+c x^4}} \, dx \\ & = \frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}-\frac {\left (14 b^2\right ) \int \frac {x^{7/2}}{\sqrt {b x^2+c x^4}} \, dx}{117 c} \\ & = -\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {\left (14 b^3\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{195 c^2} \\ & = -\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {\left (14 b^3 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{195 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {\left (28 b^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 )}{195 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}+\frac {\left (28 b^{7/2} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{195 c^{5/2} \sqrt {b x^2+c x^4}}-\frac {\left (28 b^{7/2} 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 )}{195 c^{5/2} \sqrt {b x^2+c x^4}} \\ & = \frac {28 b^3 x^{3/2} \left (b+c x^2\right )}{195 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {28 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^2}+\frac {4 b x^{5/2} \sqrt {b x^2+c x^4}}{117 c}+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}-\frac {28 b^{13/4} 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 )}{195 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {14 b^{13/4} 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 )}{195 c^{11/4} \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 = 10.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.32 \[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {1+\frac {c x^2}{b}} \left (-7 b^2+2 b c x^2+9 c^2 x^4\right )+7 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{117 c^2 \sqrt {1+\frac {c x^2}{b}}} \]
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Time = 0.53 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (45 c^{4} x^{8}+55 c^{3} x^{6} b +42 b^{4} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-21 b^{4} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-4 b^{2} c^{2} x^{4}-14 x^{2} c \,b^{3}\right )}{585 x^{\frac {3}{2}} \left (c \,x^{2}+b \right ) c^{3}}\) | \(237\) |
risch | \(-\frac {2 \sqrt {x}\, \left (-45 c^{2} x^{4}-10 b c \,x^{2}+14 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{585 c^{2}}+\frac {14 b^{3} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{195 c^{3} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(241\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.23 \[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=-\frac {2 \, {\left (42 \, b^{3} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (45 \, c^{3} x^{4} + 10 \, b c^{2} x^{2} - 14 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{585 \, c^{3}} \]
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\[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\int x^{\frac {7}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}\, dx \]
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\[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2}} x^{\frac {7}{2}} \,d x } \]
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\[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2}} x^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int x^{7/2} \sqrt {b x^2+c x^4} \, dx=\int x^{7/2}\,\sqrt {c\,x^4+b\,x^2} \,d x \]
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