Optimal. Leaf size=256 \[ -\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac {215096 \sqrt {3 x^2+5 x+2}}{15625 \sqrt {2 x+3}}+\frac {258536 \sqrt {3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac {87144 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac {4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}+\frac {129268 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3125 \sqrt {3 x^2+5 x+2}}-\frac {107548 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{15625 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.19, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac {215096 \sqrt {3 x^2+5 x+2}}{15625 \sqrt {2 x+3}}+\frac {258536 \sqrt {3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac {87144 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac {4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}+\frac {129268 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3125 \sqrt {3 x^2+5 x+2}}-\frac {107548 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{15625 \sqrt {3 x^2+5 x+2}} \]
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)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {1362+1269 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {28953+30195 x}{(3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}-\frac {8 \int \frac {-147870-\frac {294111 x}{2}}{(3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{1875}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac {258536 \sqrt {2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac {16 \int \frac {\frac {1187847}{4}+\frac {872559 x}{4}}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{28125}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac {258536 \sqrt {2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac {215096 \sqrt {2+5 x+3 x^2}}{15625 \sqrt {3+2 x}}-\frac {32 \int \frac {-\frac {546237}{4}+\frac {725949 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{140625}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac {258536 \sqrt {2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac {215096 \sqrt {2+5 x+3 x^2}}{15625 \sqrt {3+2 x}}-\frac {161322 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{15625}+\frac {193902 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3125}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac {258536 \sqrt {2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac {215096 \sqrt {2+5 x+3 x^2}}{15625 \sqrt {3+2 x}}-\frac {\left (107548 \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 )}{15625 \sqrt {2+5 x+3 x^2}}+\frac {\left (129268 \sqrt {3} \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 )}{3125 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {87144 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac {258536 \sqrt {2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac {215096 \sqrt {2+5 x+3 x^2}}{15625 \sqrt {3+2 x}}-\frac {107548 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{15625 \sqrt {2+5 x+3 x^2}}+\frac {129268 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{3125 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 229, normalized size = 0.89 \[ \frac {2 \left (3871728 x^6+36155064 x^5+129381052 x^4+231620622 x^3+220795962 x^2-2 (2 x+3)^2 \left (3 x^2+5 x+2\right ) \left (53774 \left (3 x^2+5 x+2\right )+70064 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+26887 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )+106756189 x+20514383\right )}{15625 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{432 \, x^{10} + 4752 \, x^{9} + 23256 \, x^{8} + 66656 \, x^{7} + 123867 \, x^{6} + 155895 \, x^{5} + 134543 \, x^{4} + 78609 \, x^{3} + 29754 \, x^{2} + 6588 \, x + 648}, x\right ) \]
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 \[ \int -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 494, normalized size = 1.93 \[ \frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (19358640 x^{6}+180775320 x^{5}+322644 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1616376 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+646905260 x^{4}+1505672 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+7543088 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1158103110 x^{3}+2554265 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+12796310 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1103979810 x^{2}+1855203 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+9294162 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+533780945 x +483966 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+2424564 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+102571915\right )}{78125 \left (2 x +3\right )^{\frac {5}{2}} \left (3 x +2\right )^{2} \left (x +1\right )^{2}} \]
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 \[ -\int \frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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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