Integrand size = 25, antiderivative size = 205 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\frac {62 \sqrt {x} (2+3 x)}{21 \sqrt {2+5 x+3 x^2}}-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {62 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}+\frac {43 (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {824, 848, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\frac {43 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {62 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {3 x^2+5 x+2}}-\frac {62 \sqrt {3 x^2+5 x+2}}{21 \sqrt {x}}+\frac {62 \sqrt {x} (3 x+2)}{21 \sqrt {3 x^2+5 x+2}}-\frac {4 \sqrt {3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}+\frac {43 \sqrt {3 x^2+5 x+2}}{21 x^{3/2}} \]
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Rule 824
Rule 848
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps \begin{align*} \text {integral}& = -\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}-\frac {1}{35} \int \frac {215+255 x}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}+\frac {1}{105} \int \frac {310+\frac {645 x}{2}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {1}{105} \int \frac {-\frac {645}{2}-465 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {2}{105} \text {Subst}\left (\int \frac {-\frac {645}{2}-465 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}+\frac {43}{7} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {62}{7} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {62 \sqrt {x} (2+3 x)}{21 \sqrt {2+5 x+3 x^2}}-\frac {4 (1-3 x) \sqrt {2+5 x+3 x^2}}{7 x^{7/2}}+\frac {43 \sqrt {2+5 x+3 x^2}}{21 x^{3/2}}-\frac {62 \sqrt {2+5 x+3 x^2}}{21 \sqrt {x}}-\frac {62 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}+\frac {43 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{7 \sqrt {2} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.76 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\frac {-48+24 x+460 x^2+646 x^3+258 x^4+124 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+5 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{42 x^{7/2} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {62 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-57 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-1116 x^{5}-1086 x^{4}+1194 x^{3}+1380 x^{2}+72 x -144}{126 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {7}{2}}}\) | \(129\) |
risch | \(-\frac {186 x^{5}+181 x^{4}-199 x^{3}-230 x^{2}-12 x +24}{21 x^{\frac {7}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {43 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{42 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {31 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{21 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {4 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{4}}+\frac {12 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{3}}+\frac {43 \sqrt {3 x^{3}+5 x^{2}+2 x}}{21 x^{2}}-\frac {62 \left (3 x^{2}+5 x +2\right )}{21 \sqrt {x \left (3 x^{2}+5 x +2\right )}}+\frac {43 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{42 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {31 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{21 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(248\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.34 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\frac {77 \, \sqrt {3} x^{4} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 558 \, \sqrt {3} x^{4} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (62 \, x^{3} - 43 \, x^{2} - 36 \, x + 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{189 \, x^{4}} \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=- \int \left (- \frac {2 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {9}{2}}}\right )\, dx - \int \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2}}{x^{9/2}} \,d x \]
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