Optimal. Leaf size=223 \[ \frac {11320 \sqrt {3 x^2+5 x+2} \sqrt {x}}{5103}-\frac {68920 (3 x+2) \sqrt {x}}{15309 \sqrt {3 x^2+5 x+2}}-\frac {11320 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {68920 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15309 \sqrt {3 x^2+5 x+2}}-\frac {10}{27} \sqrt {3 x^2+5 x+2} x^{7/2}+\frac {508}{567} \sqrt {3 x^2+5 x+2} x^{5/2}-\frac {820}{567} \sqrt {3 x^2+5 x+2} x^{3/2} \]
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Rubi [A] time = 0.16, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {832, 839, 1189, 1100, 1136} \[ -\frac {10}{27} \sqrt {3 x^2+5 x+2} x^{7/2}+\frac {508}{567} \sqrt {3 x^2+5 x+2} x^{5/2}-\frac {820}{567} \sqrt {3 x^2+5 x+2} x^{3/2}+\frac {11320 \sqrt {3 x^2+5 x+2} \sqrt {x}}{5103}-\frac {68920 (3 x+2) \sqrt {x}}{15309 \sqrt {3 x^2+5 x+2}}-\frac {11320 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {3 x^2+5 x+2}}+\frac {68920 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15309 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{7/2}}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {2}{27} \int \frac {x^{5/2} (35+127 x)}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {4}{567} \int \frac {\left (-635-\frac {3075 x}{2}\right ) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {820}{567} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {8 \int \frac {\sqrt {x} \left (\frac {9225}{2}+\frac {21225 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx}{8505}\\ &=\frac {11320 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {820}{567} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {16 \int \frac {-\frac {21225}{2}-\frac {129225 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{76545}\\ &=\frac {11320 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {820}{567} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {32 \operatorname {Subst}\left (\int \frac {-\frac {21225}{2}-\frac {129225 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{76545}\\ &=\frac {11320 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {820}{567} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}-\frac {22640 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{5103}-\frac {68920 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{5103}\\ &=-\frac {68920 \sqrt {x} (2+3 x)}{15309 \sqrt {2+5 x+3 x^2}}+\frac {11320 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {820}{567} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {508}{567} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {10}{27} x^{7/2} \sqrt {2+5 x+3 x^2}+\frac {68920 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15309 \sqrt {2+5 x+3 x^2}}-\frac {11320 \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 = 0.19, size = 168, normalized size = 0.75 \[ \frac {34960 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-68920 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (8505 x^6-6399 x^5+4590 x^4-9306 x^3+40620 x^2+138340 x+68920\right )}{15309 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt {x}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}, x\right ) \]
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 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {7}{2}}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 127, normalized size = 0.57 \[ \frac {-\frac {10 x^{6}}{9}+\frac {158 x^{5}}{189}-\frac {340 x^{4}}{567}+\frac {2068 x^{3}}{1701}+\frac {41840 x^{2}}{5103}+\frac {22640 x}{5103}-\frac {34460 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{45927}+\frac {23140 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{15309}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}} \]
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 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {7}{2}}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}\,{d x} \]
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 \[ -\int \frac {x^{7/2}\,\left (5\,x-2\right )}{\sqrt {3\,x^2+5\,x+2}} \,d x \]
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 \[ - \int \left (- \frac {2 x^{\frac {7}{2}}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {5 x^{\frac {9}{2}}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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