Integrand size = 21, antiderivative size = 282 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1985, 1986, 489, 545, 429, 506, 422} \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=-\frac {c^{3/2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 d} \]
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rule 1985
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \int x^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \, dx \\ & = \frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {x^2 \sqrt {b+a c+a d x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a c+a d x^2}} \\ & = \frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {c (b+a c)-(b-a c) d x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a c+a d x^2}} \\ & = \frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {\left ((b-a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 \sqrt {b+a c+a d x^2}}-\frac {\left (c (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a c+a d x^2}} \\ & = \frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {\left (c (b-a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a d \sqrt {b+a c+a d x^2}} \\ & = \frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )+i \left (-b^2+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 a d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]
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Time = 5.92 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.44
method | result | size |
default | \(\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{5}+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} x -2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {-\frac {a d}{a c +b}}\, b c x \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\) | \(406\) |
risch | \(\frac {x \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 d}-\frac {\left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d -b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 d \left (a d \,x^{2}+a c +b \right )}\) | \(525\) |
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Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.59 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {{\left (a c^{2} - b c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c^{2} - b c + {\left (a c + b\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a d^{2} x^{4} + b d x^{2} - a c^{2} + b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, a d^{2} x} \]
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\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int x^{2} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}\, dx \]
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\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int x^2\,\sqrt {a+\frac {b}{d\,x^2+c}} \,d x \]
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