Optimal. Leaf size=207 \[ \frac {(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac {(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac {(7877 x+27213) \sqrt {3 x^2+5 x+2}}{2100 \sqrt {2 x+3}}-\frac {32513 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{840 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {17699 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{600 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 207, 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, 812, 843, 718, 424, 419} \[ \frac {(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac {(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac {(7877 x+27213) \sqrt {3 x^2+5 x+2}}{2100 \sqrt {2 x+3}}-\frac {32513 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{840 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {17699 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{600 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 810
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{11/2}} \, dx &=\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac {1}{126} \int \frac {(548+603 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx\\ &=\frac {(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac {\int \frac {(-60147-70893 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx}{6300}\\ &=-\frac {(27213+7877 x) \sqrt {2+5 x+3 x^2}}{2100 \sqrt {3+2 x}}+\frac {(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac {\int \frac {-941013-1115037 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{37800}\\ &=-\frac {(27213+7877 x) \sqrt {2+5 x+3 x^2}}{2100 \sqrt {3+2 x}}+\frac {(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac {17699 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{1200}-\frac {32513 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{1680}\\ &=-\frac {(27213+7877 x) \sqrt {2+5 x+3 x^2}}{2100 \sqrt {3+2 x}}+\frac {(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac {\left (17699 \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 )}{600 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (32513 \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 )}{840 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {(27213+7877 x) \sqrt {2+5 x+3 x^2}}{2100 \sqrt {3+2 x}}+\frac {(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac {(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac {17699 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{600 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {32513 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{840 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 227, normalized size = 1.10 \[ -\frac {8 \left (3 x^2+5 x+2\right ) \left (18900 x^5+1579088 x^4+8510198 x^3+17853828 x^2+16893773 x+6047963\right )-4 (2 x+3)^4 \left (247786 \left (3 x^2+5 x+2\right )-26354 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+123893 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )}{50400 (2 x+3)^{9/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 487, normalized size = 2.35 \[ -\frac {1134000 x^{7}+96635280 x^{6}+669276680 x^{5}+1982288 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+618752 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1985413000 x^{4}+11893728 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3712512 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3139417100 x^{3}+26760888 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+8353152 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+2766408200 x^{2}+26760888 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+8353152 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1280547220 x +10035333 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3132432 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+241918520}{126000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {9}{2}}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{11/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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