Optimal. Leaf size=170 \[ \frac {\sqrt {3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac {49 \sqrt {3 x^2+5 x+2}}{125 \sqrt {2 x+3}}+\frac {9 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{50 \sqrt {3 x^2+5 x+2}}-\frac {49 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{250 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.11, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {810, 834, 843, 718, 424, 419} \[ \frac {\sqrt {3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac {49 \sqrt {3 x^2+5 x+2}}{125 \sqrt {2 x+3}}+\frac {9 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{50 \sqrt {3 x^2+5 x+2}}-\frac {49 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{250 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 810
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx &=\frac {(32+43 x) \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac {1}{150} \int \frac {-48-81 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {49 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}+\frac {(32+43 x) \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac {1}{375} \int \frac {-\frac {459}{2}-\frac {441 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {49 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}+\frac {(32+43 x) \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac {27}{100} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx-\frac {147}{500} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {49 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}+\frac {(32+43 x) \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac {\left (9 \sqrt {3} \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 )}{50 \sqrt {2+5 x+3 x^2}}-\frac {\left (49 \sqrt {3} \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 )}{250 \sqrt {2+5 x+3 x^2}}\\ &=\frac {49 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}+\frac {(32+43 x) \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac {49 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{250 \sqrt {2+5 x+3 x^2}}+\frac {9 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{50 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 182, normalized size = 1.07 \[ \frac {1290 x^3+3110 x^2+2460 x+22 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{7/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-49 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+640}{250 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81}, x\right ) \]
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 \[ \int -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 296, normalized size = 1.74 \[ -\frac {-11760 x^{4}-67780 x^{3}-196 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+16 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-124200 x^{2}-588 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+48 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-92220 x -441 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+36 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-24040}{2500 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {5}{2}}} \]
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 \[ -\int \frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^{7/2}} \,d x \]
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 \[ - \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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