Optimal. Leaf size=202 \[ -\frac {2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt {3 x^2+5 x+2}}-\frac {59512}{81} \sqrt {3 x^2+5 x+2} \sqrt {2 x+3}+\frac {148780 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {110516 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {818, 832, 843, 718, 424, 419} \[ -\frac {2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt {3 x^2+5 x+2}}-\frac {59512}{81} \sqrt {3 x^2+5 x+2} \sqrt {2 x+3}+\frac {148780 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {110516 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 818
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^{5/2} (4+411 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {(-18243-22317 x) \sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {8}{243} \int \frac {-34269-\frac {82887 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {55258}{81} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {74390}{81} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {\left (110516 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (148780 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 220, normalized size = 1.09 \[ -\frac {2 \left (2 \left (3 x^2+5 x+2\right ) \left (55258 \left (3 x^2+5 x+2\right )-5312 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+27629 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )+3 (2 x+3) \left (144 x^4-166566 x^3-411640 x^2-330053 x-85285\right )\right )}{243 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (16 \, x^{5} + 16 \, x^{4} - 264 \, x^{3} - 864 \, x^{2} - 999 \, x - 405\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]
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 \[ \int -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 325, normalized size = 1.61 \[ \frac {2 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-4320 x^{5}+4990500 x^{4}+19844670 x^{3}+82887 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+28698 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+28425390 x^{2}+138145 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+47830 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+17410935 x +55258 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+19132 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3837825\right )}{1215 \left (x +1\right ) \left (2 x^{2}+5 x +3\right ) \left (3 x +2\right )^{2}} \]
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 \[ -\int \frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ -\int \frac {{\left (2\,x+3\right )}^{9/2}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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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