Integrand size = 21, antiderivative size = 443 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {(4 b+3 a c) x \left (b+a c+a d x^2\right )}{15 a^2 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \left (b+a c+a d x^2\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \left (8 b^2+13 a b c+3 a^2 c^2\right ) \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} (4 b+3 a c) \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Time = 0.36 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 489, 596, 545, 429, 506, 422} \[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {c^{3/2} (3 a c+4 b) \left (a c+a d x^2+b\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {x (3 a c+4 b) \left (a c+a d x^2+b\right )}{15 a^2 d^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {\sqrt {c} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {x^3 \left (a c+a d x^2+b\right )}{5 a d \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \]
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rule 596
Rule 1985
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4}{\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \, dx \\ & = \frac {\sqrt {b+a c+a d x^2} \int \frac {x^4 \sqrt {c+d x^2}}{\sqrt {b+a c+a d x^2}} \, dx}{\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = \frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {b+a c+a d x^2} \int \frac {x^2 \left (3 c (b+a c)+(4 b+3 a c) d x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 a d \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {(4 b+3 a c) x \left (b+a c+a d x^2\right )}{15 a^2 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {b+a c+a d x^2} \int \frac {c (b+a c) (4 b+3 a c) d+\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^2 d^3 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {(4 b+3 a c) x \left (b+a c+a d x^2\right )}{15 a^2 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (c (b+a c) (4 b+3 a c) \sqrt {b+a c+a d x^2}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^2 d^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (\left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt {b+a c+a d x^2}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^2 d \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {(4 b+3 a c) x \left (b+a c+a d x^2\right )}{15 a^2 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \left (b+a c+a d x^2\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {c^{3/2} (4 b+3 a c) \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^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 \left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt {b+a c+a d x^2}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 a^3 d^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {(4 b+3 a c) x \left (b+a c+a d x^2\right )}{15 a^2 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {x^3 \left (b+a c+a d x^2\right )}{5 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \left (b+a c+a d x^2\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \left (8 b^2+13 a b c+3 a^2 c^2\right ) \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^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} (4 b+3 a c) \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^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 = 8.78 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (4 b^2+a b \left (7 c+d x^2\right )+3 a^2 \left (c^2-d^2 x^4\right )\right )+i \left (8 b^3+21 a b^2 c+16 a^2 b c^2+3 a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b \left (8 b^2+17 a b c+9 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{15 a^3 c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \]
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Time = 8.06 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.50
method | result | size |
default | \(-\frac {\left (-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{3} x^{7}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{2} x^{5}+\sqrt {-\frac {a d}{a c +b}}\, a b \,d^{2} x^{5}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d \,x^{3}+8 \sqrt {-\frac {a d}{a c +b}}\, a b c d \,x^{3}-3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{3}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} x +4 \sqrt {-\frac {a d}{a c +b}}\, b^{2} d \,x^{3}+6 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b \,c^{2}-13 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b \,c^{2}+7 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} x +4 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b^{2} c -8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b^{2} c +4 \sqrt {-\frac {a d}{a c +b}}\, b^{2} c x \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 a^{2} d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\) | \(664\) |
risch | \(-\frac {x \left (-3 a d \,x^{2}+3 a c +4 b \right ) \left (a d \,x^{2}+a c +b \right )}{15 d^{2} a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {\left (\frac {3 a^{2} c^{3} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {4 b^{2} c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {7 a b \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (3 a^{2} c^{2} d +13 a b c d +8 b^{2} d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{15 a^{2} d^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(686\) |
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Time = 0.11 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.54 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {{\left (3 \, a^{2} c^{3} + 13 \, a b c^{2} + 8 \, b^{2} c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (3 \, a^{2} c^{3} + 13 \, a b c^{2} + 8 \, b^{2} c + {\left (3 \, a^{2} c^{2} + 7 \, a b c + 4 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (3 \, a^{2} d^{3} x^{6} - 4 \, a b d^{2} x^{4} + 3 \, a^{2} c^{3} + 13 \, a b c^{2} + {\left (9 \, a b c + 8 \, b^{2}\right )} d x^{2} + 8 \, b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{15 \, a^{3} d^{3} x} \]
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\[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {x^{4}}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
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\[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {x^{4}}{\sqrt {a + \frac {b}{d x^{2} + c}}} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {x^{4}}{\sqrt {a + \frac {b}{d x^{2} + c}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {x^4}{\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]
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