Optimal. Leaf size=165 \[ -\frac {386 \sqrt {3 x^2+5 x+2}}{75 \sqrt {2 x+3}}-\frac {26 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {13 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {193 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.10, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {386 \sqrt {3 x^2+5 x+2}}{75 \sqrt {2 x+3}}-\frac {26 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {13 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {193 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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)^{5/2} \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {26 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac {2}{15} \int \frac {-19+\frac {39 x}{2}}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac {386 \sqrt {2+5 x+3 x^2}}{75 \sqrt {3+2 x}}+\frac {4}{75} \int \frac {\frac {771}{4}+\frac {579 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac {386 \sqrt {2+5 x+3 x^2}}{75 \sqrt {3+2 x}}-\frac {13}{10} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx+\frac {193}{50} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac {386 \sqrt {2+5 x+3 x^2}}{75 \sqrt {3+2 x}}-\frac {\left (13 \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 )}{5 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (193 \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 )}{25 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {26 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac {386 \sqrt {2+5 x+3 x^2}}{75 \sqrt {3+2 x}}+\frac {193 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {13 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 183, normalized size = 1.11 \[ \frac {193 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt {\frac {3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-2 \left (65 \left (3 x^2+5 x+2\right )+77 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{5/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )}{75 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54}, x\right ) \]
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 \[ \int -\frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 203, normalized size = 1.23 \[ \frac {-23160 x^{3}-77240 x^{2}-386 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+256 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-79840 x -579 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+384 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-25760}{750 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {3}{2}}} \]
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 \[ -\int \frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{5/2}\,\sqrt {3\,x^2+5\,x+2}} \,d x \]
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 \[ - \int \frac {x}{4 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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