Optimal. Leaf size=197 \[ -\frac {6 (47 x+37)}{5 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}-\frac {14876 \sqrt {3 x^2+5 x+2}}{375 \sqrt {2 x+3}}-\frac {2516 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{3/2}}-\frac {1258 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {7438 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{125 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 197, 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 = {822, 834, 843, 718, 424, 419} \[ -\frac {6 (47 x+37)}{5 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}-\frac {14876 \sqrt {3 x^2+5 x+2}}{375 \sqrt {2 x+3}}-\frac {2516 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{3/2}}-\frac {1258 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {7438 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{125 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 822
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {320+423 x}{(3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2516 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^{3/2}}+\frac {4}{75} \int \frac {-\frac {971}{2}-\frac {1887 x}{2}}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2516 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^{3/2}}-\frac {14876 \sqrt {2+5 x+3 x^2}}{375 \sqrt {3+2 x}}-\frac {8}{375} \int \frac {-\frac {6009}{2}-\frac {11157 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2516 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^{3/2}}-\frac {14876 \sqrt {2+5 x+3 x^2}}{375 \sqrt {3+2 x}}-\frac {629}{25} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx+\frac {3719}{125} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2516 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^{3/2}}-\frac {14876 \sqrt {2+5 x+3 x^2}}{375 \sqrt {3+2 x}}-\frac {\left (1258 \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 )}{25 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (7438 \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 )}{125 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {2516 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^{3/2}}-\frac {14876 \sqrt {2+5 x+3 x^2}}{375 \sqrt {3+2 x}}+\frac {7438 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{125 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {1258 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 177, normalized size = 0.90 \[ \frac {2 \left (-18870 x^2-42025 x-1832 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{5/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+3719 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{5/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-20905\right )}{375 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108}, x\right ) \]
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 \[ \int -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 203, normalized size = 1.03 \[ \frac {-446280 x^{3}-1601920 x^{2}-7438 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1148 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-1833470 x -11157 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1722 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-655330}{1875 \left (2 x +3\right )^{\frac {3}{2}} \sqrt {3 x^{2}+5 x +2}} \]
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 \[ -\int \frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{5/2}\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]
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 \[ - \int \frac {x}{12 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 69 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 18 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{12 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 69 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 18 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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