Optimal. Leaf size=256 \[ \frac {211144 \sqrt {3 x^2+5 x+2} \sqrt {x}}{5103}-\frac {1521056 (3 x+2) \sqrt {x}}{76545 \sqrt {3 x^2+5 x+2}}-\frac {211144 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {1521056 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {3 x^2+5 x+2}}+\frac {2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {4 (1685 x+1484) x^{7/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {45820}{567} \sqrt {3 x^2+5 x+2} x^{5/2}-\frac {167336 \sqrt {3 x^2+5 x+2} x^{3/2}}{2835} \]
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Rubi [A] time = 0.19, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {818, 832, 839, 1189, 1100, 1136} \[ \frac {2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {4 (1685 x+1484) x^{7/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {45820}{567} \sqrt {3 x^2+5 x+2} x^{5/2}-\frac {167336 \sqrt {3 x^2+5 x+2} x^{3/2}}{2835}+\frac {211144 \sqrt {3 x^2+5 x+2} \sqrt {x}}{5103}-\frac {1521056 (3 x+2) \sqrt {x}}{76545 \sqrt {3 x^2+5 x+2}}-\frac {211144 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {1521056 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(-407-340 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {x^{5/2} \left (5194+\frac {11455 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {8}{567} \int \frac {\left (-\frac {57275}{2}-\frac {62751 x}{2}\right ) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {16 \int \frac {\sqrt {x} \left (\frac {188253}{2}+\frac {395895 x}{4}\right )}{\sqrt {2+5 x+3 x^2}} \, dx}{8505}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {32 \int \frac {-\frac {395895}{4}-\frac {142599 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{76545}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {64 \operatorname {Subst}\left (\int \frac {-\frac {395895}{4}-\frac {142599 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{76545}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {1521056 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{25515}-\frac {422288 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{5103}\\ &=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {1521056 \sqrt {x} (2+3 x)}{76545 \sqrt {2+5 x+3 x^2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {1521056 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {2+5 x+3 x^2}}-\frac {211144 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 187, normalized size = 0.73 \[ \frac {-1646104 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-1521056 i \sqrt {\frac {2}{x}+2} \sqrt {\frac {2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (18225 x^7-70956 x^6+262710 x^5-2106756 x^4-2967300 x^3+5504080 x^2+8876240 x+3042112\right )}{76545 \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{7} - 2 \, x^{6}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 312, normalized size = 1.22 \[ -\frac {2 \left (54675 x^{7}-212868 x^{6}+788130 x^{5}-26854524 x^{4}-77349420 x^{3}+1140792 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+1328364 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-67906368 x^{2}+1901320 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+2213940 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-19002960 x +760528 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+885576 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right ) \sqrt {3 x^{2}+5 x +2}}{229635 \left (3 x +2\right )^{2} \left (x +1\right )^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {x^{13/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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