Optimal. Leaf size=136 \[ -\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {\sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {13 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.08, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {834, 843, 718, 424, 419} \[ -\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {\sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {13 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {26 \sqrt {2+5 x+3 x^2}}{5 \sqrt {3+2 x}}-\frac {2}{5} \int \frac {-28-\frac {39 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{5 \sqrt {3+2 x}}-\frac {1}{2} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx+\frac {39}{10} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{5 \sqrt {3+2 x}}-\frac {\sqrt {-2-5 x-3 x^2} \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 )}{\sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (13 \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 )}{5 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{5 \sqrt {3+2 x}}+\frac {13 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}}-\frac {\sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 104, normalized size = 0.76 \[ \frac {(x+1) \sqrt {\frac {3 x+2}{2 x+3}} \left (13 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-12 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )}{\sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18}, x\right ) \]
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 \[ \int -\frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 131, normalized size = 0.96 \[ \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-2340 x^{2}-3900 x -39 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+34 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-1560\right )}{900 x^{3}+2850 x^{2}+2850 x +900} \]
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 \[ -\int \frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{3/2}\,\sqrt {3\,x^2+5\,x+2}} \,d x \]
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 \[ - \int \frac {x}{2 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{2 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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