Optimal. Leaf size=202 \[ \frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac {23 \sqrt {3 x^2+5 x+2}}{11250 \sqrt {2 x+3}}+\frac {7 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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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 = {810, 834, 843, 718, 424, 419} \[ \frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac {23 \sqrt {3 x^2+5 x+2}}{11250 \sqrt {2 x+3}}+\frac {7 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx &=\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {1}{210} \int \frac {(28-21 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx\\ &=-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\int \frac {301+147 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{31500}\\ &=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {\int \frac {-546-\frac {483 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{78750}\\ &=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{15000}+\frac {7 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3000}\\ &=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\left (23 \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 )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (7 \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 )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 192, normalized size = 0.95 \[ \frac {204180 x^5+822160 x^4+1297210 x^3+998860 x^2+373610 x-44 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{11/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+23 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{11/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+53980}{22500 (2 x+3)^{9/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 482, normalized size = 2.39 \[ \frac {-22080 x^{6}+1872520 x^{5}-368 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+928 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+7688000 x^{4}-2208 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+5568 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+12088900 x^{3}-4968 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+12528 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+9181300 x^{2}-4968 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+12528 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3351080 x -1863 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+4698 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+465280}{225000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {9}{2}}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{11/2}} \,d x \]
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 \[ - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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