Optimal. Leaf size=224 \[ \frac {2693 \sqrt {x} (3 x+2)}{30 \sqrt {3 x^2+5 x+2}}-\frac {2693 \sqrt {3 x^2+5 x+2}}{30 \sqrt {x}}+\frac {157 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {2693 (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {157 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.14, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {822, 834, 839, 1189, 1100, 1136} \[ \frac {2693 \sqrt {x} (3 x+2)}{30 \sqrt {3 x^2+5 x+2}}-\frac {2693 \sqrt {3 x^2+5 x+2}}{30 \sqrt {x}}+\frac {157 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}+\frac {157 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {2693 (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\int \frac {-191-225 x}{x^{7/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {1}{5} \int \frac {-785-\frac {1719 x}{2}}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {157 \sqrt {2+5 x+3 x^2}}{3 x^{3/2}}-\frac {1}{15} \int \frac {-\frac {2693}{2}-\frac {2355 x}{2}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {157 \sqrt {2+5 x+3 x^2}}{3 x^{3/2}}-\frac {2693 \sqrt {2+5 x+3 x^2}}{30 \sqrt {x}}+\frac {1}{15} \int \frac {\frac {2355}{2}+\frac {8079 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {157 \sqrt {2+5 x+3 x^2}}{3 x^{3/2}}-\frac {2693 \sqrt {2+5 x+3 x^2}}{30 \sqrt {x}}+\frac {2}{15} \operatorname {Subst}\left (\int \frac {\frac {2355}{2}+\frac {8079 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {157 \sqrt {2+5 x+3 x^2}}{3 x^{3/2}}-\frac {2693 \sqrt {2+5 x+3 x^2}}{30 \sqrt {x}}+157 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {2693}{10} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2693 \sqrt {x} (2+3 x)}{30 \sqrt {2+5 x+3 x^2}}+\frac {2 (38+45 x)}{x^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {191 \sqrt {2+5 x+3 x^2}}{5 x^{5/2}}+\frac {157 \sqrt {2+5 x+3 x^2}}{3 x^{3/2}}-\frac {2693 \sqrt {2+5 x+3 x^2}}{30 \sqrt {x}}-\frac {2693 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2} \sqrt {2+5 x+3 x^2}}+\frac {157 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 150, normalized size = 0.67 \[ \frac {-338 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+2693 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+4710 x^3+4412 x^2+110 x-12}{30 x^{5/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x}}{9 \, x^{8} + 30 \, x^{7} + 37 \, x^{6} + 20 \, x^{5} + 4 \, x^{4}}, x\right ) \]
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 \[ \int -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 124, normalized size = 0.55 \[ -\frac {48474 x^{4}+52530 x^{3}-2693 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+3369 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, x^{2} \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+5844 x^{2}-660 x +72}{180 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {5}{2}}} \]
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 \[ -\int \frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \]
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 \[ \int -\frac {5\,x-2}{x^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {5}{3 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 5 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2} + 2 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {2}{3 x^{\frac {11}{2}} \sqrt {3 x^{2} + 5 x + 2} + 5 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 2 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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