Integrand size = 25, antiderivative size = 256 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {6907 \sqrt {x} (2+3 x)}{10010 \sqrt {2+5 x+3 x^2}}+\frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {6907 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {2+5 x+3 x^2}}-\frac {3693 (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {824, 848, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {3693 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {6907 (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {6907 \sqrt {3 x^2+5 x+2}}{10010 \sqrt {x}}-\frac {6907 \sqrt {x} (3 x+2)}{10010 \sqrt {3 x^2+5 x+2}}-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{1001 x^{9/2}}-\frac {1231 \sqrt {3 x^2+5 x+2}}{2002 x^{3/2}}+\frac {204 \sqrt {3 x^2+5 x+2}}{385 x^{5/2}} \]
[In]
[Out]
Rule 824
Rule 848
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps \begin{align*} \text {integral}& = -\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {3}{143} \int \frac {(393+465 x) \sqrt {2+5 x+3 x^2}}{x^{11/2}} \, dx \\ & = \frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {\int \frac {-7956-\frac {20745 x}{2}}{x^{7/2} \sqrt {2+5 x+3 x^2}} \, dx}{3003} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {\int \frac {-\frac {55395}{2}-35802 x}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{15015} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {\int \frac {-\frac {62163}{2}-\frac {166185 x}{4}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {\int \frac {\frac {166185}{4}+\frac {186489 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {2 \text {Subst}\left (\int \frac {\frac {166185}{4}+\frac {186489 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {3693 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{2002}-\frac {20721 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{10010} \\ & = -\frac {6907 \sqrt {x} (2+3 x)}{10010 \sqrt {2+5 x+3 x^2}}+\frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {6907 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {2+5 x+3 x^2}}-\frac {3693 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.66 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\frac {-24640-67200 x+125440 x^2+654400 x^3+840316 x^4+361120 x^5-29726 x^6-36930 x^7-13814 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-4651 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{20020 x^{13/2} \sqrt {2+5 x+3 x^2}} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {2256 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}-6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}+124326 x^{8}+96420 x^{7}-6294 x^{6}+1083360 x^{5}+2520948 x^{4}+1963200 x^{3}+376320 x^{2}-201600 x -73920}{60060 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {13}{2}}}\) | \(144\) |
risch | \(\frac {20721 x^{8}+16070 x^{7}-1049 x^{6}+180560 x^{5}+420158 x^{4}+327200 x^{3}+62720 x^{2}-33600 x -12320}{10010 x^{\frac {13}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (\frac {1231 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{4004 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{20020 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(218\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{13 x^{7}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{6}}+\frac {630 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{5}}+\frac {5545 \sqrt {3 x^{3}+5 x^{2}+2 x}}{1001 x^{4}}+\frac {204 \sqrt {3 x^{3}+5 x^{2}+2 x}}{385 x^{3}}-\frac {1231 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2002 x^{2}}+\frac {\frac {20721}{10010} x^{2}+\frac {6907}{2002} x +\frac {6907}{5005}}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {1231 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{4004 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{20020 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(311\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.33 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {20860 \, \sqrt {3} x^{7} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 62163 \, \sqrt {3} x^{7} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (6907 \, x^{6} - 6155 \, x^{5} + 5304 \, x^{4} + 55450 \, x^{3} + 44100 \, x^{2} - 1400 \, x - 6160\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{90090 \, x^{7}} \]
[In]
[Out]
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{15/2}} \,d x \]
[In]
[Out]