Integrand size = 21, antiderivative size = 405 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Time = 0.44 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1985, 1986, 478, 595, 596, 545, 429, 506, 422} \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {\sqrt {c} \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 a d^2}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (7 b-a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d}-\frac {x^3 \left (a c+a d x^2+b\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d} \]
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Rule 422
Rule 429
Rule 478
Rule 506
Rule 545
Rule 595
Rule 596
Rule 1985
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \int x^4 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{3/2} \, 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^4 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a c+a d x^2}} \\ & = -\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\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} \left (3 (b+a c)+6 a d x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {b+a c+a d x^2}} \\ & = \frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\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 \left (3 (5 b-a c) (b+a c) d+3 a (7 b-a c) d^2 x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d^2 \sqrt {b+a c+a d x^2}} \\ & = \frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {3 a c (7 b-a c) (b+a c) d^2-3 a \left (b^2-14 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^4 \sqrt {b+a c+a d x^2}} \\ & = \frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\left (c (7 b-a 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}{5 d^2 \sqrt {b+a c+a d x^2}}+\frac {\left (\left (b^2-14 a b c+a^2 c^2\right ) \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}{5 d \sqrt {b+a c+a d x^2}} \\ & = \frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {c^{3/2} (7 b-a c) \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 )}{5 d^{5/2} \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 (b^2-14 a b c+a^2 c^2\right ) \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}{5 a d^2 \sqrt {b+a c+a d x^2}} \\ & = \frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \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 )}{5 a d^{5/2} \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} (7 b-a c) \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 )}{5 d^{5/2} \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 = 10.63 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.77 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (-a^2 \left (c-d x^2\right ) \left (c+d x^2\right )^2+b^2 \left (7 c+2 d x^2\right )+3 a b \left (2 c^2+3 c d x^2+d^2 x^4\right )\right )-i \left (b^3-13 a b^2 c-13 a^2 b c^2+a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b \left (b^2-6 a b c-7 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{5 a c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(441)=882\).
Time = 11.57 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.63
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1065\) |
default | \(\text {Expression too large to display}\) | \(1101\) |
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Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.58 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c + {\left (a^{2} c^{2} - 6 \, a b c - 7 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} d^{3} x^{6} + 2 \, a b d^{2} x^{4} + a^{2} c^{3} - 14 \, a b c^{2} - {\left (7 \, a b c - b^{2}\right )} d x^{2} + b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{5 \, a d^{3} x} \]
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\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4} \,d x } \]
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\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]
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