Integrand size = 28, antiderivative size = 204 \[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=-\frac {4 b (5 b B-11 A c) \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}-\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}+\frac {2 b^{7/4} (5 b B-11 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2064, 2046, 2049, 2057, 335, 226} \[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\frac {2 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-11 A c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{231 c^{9/4} \sqrt {b x^2+c x^4}}-\frac {4 b \sqrt {b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt {x}}-\frac {2 x^{3/2} \sqrt {b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}} \]
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Rule 226
Rule 335
Rule 2046
Rule 2049
Rule 2057
Rule 2064
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}-\frac {\left (2 \left (\frac {5 b B}{2}-\frac {11 A c}{2}\right )\right ) \int \sqrt {x} \sqrt {b x^2+c x^4} \, dx}{11 c} \\ & = -\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}-\frac {(2 b (5 b B-11 A c)) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c} \\ & = -\frac {4 b (5 b B-11 A c) \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}-\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}+\frac {\left (2 b^2 (5 b B-11 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{231 c^2} \\ & = -\frac {4 b (5 b B-11 A c) \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}-\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}+\frac {\left (2 b^2 (5 b B-11 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{231 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {4 b (5 b B-11 A c) \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}-\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}+\frac {\left (4 b^2 (5 b B-11 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{231 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {4 b (5 b B-11 A c) \sqrt {b x^2+c x^4}}{231 c^2 \sqrt {x}}-\frac {2 (5 b B-11 A c) x^{3/2} \sqrt {b x^2+c x^4}}{77 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt {x}}+\frac {2 b^{7/4} (5 b B-11 A c) 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 )}{231 c^{9/4} \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 = 10.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.54 \[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (-\left (\left (b+c x^2\right ) \sqrt {1+\frac {c x^2}{b}} \left (5 b B-11 A c-7 B c x^2\right )\right )+b (5 b B-11 A c) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{b}\right )\right )}{77 c^2 \sqrt {x} \sqrt {1+\frac {c x^2}{b}}} \]
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Time = 1.83 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {2 \left (21 B \,c^{2} x^{4}+33 A \,c^{2} x^{2}+6 B b c \,x^{2}+22 A b c -10 B \,b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{231 c^{2} \sqrt {x}}-\frac {2 b^{2} \left (11 A c -5 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{231 c^{3} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(217\) |
default | \(-\frac {2 \sqrt {x^{4} c +b \,x^{2}}\, \left (-21 B \,c^{4} x^{7}+11 A \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c -33 A \,c^{4} x^{5}-5 B \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{3}-27 B b \,c^{3} x^{5}-55 A b \,c^{3} x^{3}+4 B \,b^{2} c^{2} x^{3}-22 A \,b^{2} c^{2} x +10 B \,b^{3} c x \right )}{231 x^{\frac {3}{2}} \left (c \,x^{2}+b \right ) c^{3}}\) | \(283\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.48 \[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\frac {2 \, {\left (2 \, {\left (5 \, B b^{3} - 11 \, A b^{2} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (21 \, B c^{3} x^{4} - 10 \, B b^{2} c + 22 \, A b c^{2} + 3 \, {\left (2 \, B b c^{2} + 11 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{231 \, c^{3} x} \]
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\[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\int \sqrt {x} \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]
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\[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x} \,d x } \]
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\[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x} \,d x } \]
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Timed out. \[ \int \sqrt {x} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx=\int \sqrt {x}\,\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2} \,d x \]
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