Optimal. Leaf size=199 \[ -\frac{3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac{717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac{19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac{51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac{153927 (8 x+7) \sqrt{3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac{153927 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{64000000 \sqrt{5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.134913, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \[ -\frac{3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac{717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac{19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac{51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac{153927 (8 x+7) \sqrt{3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac{153927 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{64000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 834
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{1}{40} \int \frac{\left (-\frac{181}{2}+117 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}+\frac{\int \frac{\left (\frac{5481}{2}-3192 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx}{1400}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{\int \frac{\left (-\frac{190323}{2}+45171 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{42000}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac{51309 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{20000}\\ &=\frac{51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}-\frac{153927 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1600000}\\ &=-\frac{153927 (7+8 x) \sqrt{2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac{51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac{153927 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{64000000}\\ &=-\frac{153927 (7+8 x) \sqrt{2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac{51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}-\frac{153927 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{32000000}\\ &=-\frac{153927 (7+8 x) \sqrt{2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac{51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac{19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac{717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac{3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac{153927 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{64000000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.131901, size = 182, normalized size = 0.91 \[ \frac{1}{40} \left (-\frac{7758 \left (3 x^2+5 x+2\right )^{5/2}}{625 (2 x+3)^5}-\frac{717 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^6}-\frac{76 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8}+\frac{51309 \left (\frac{10 \sqrt{3 x^2+5 x+2} \left (864 x^3+2068 x^2+1572 x+371\right )}{(2 x+3)^4}-3 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{8000000}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 274, normalized size = 1.4 \begin{align*} -{\frac{19}{6400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{717}{128000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{3879}{400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{51309}{3200000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{51309}{2000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1590579}{40000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{7439805+8927766\,x}{50000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1487961}{25000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{769635+923562\,x}{40000000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{153927\,\sqrt{5}}{320000000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{51309}{200000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{153927}{320000000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{10240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.55546, size = 532, normalized size = 2.67 \begin{align*} \frac{4771737}{40000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{40 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac{19 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{50 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{717 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{2000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{3879 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{12500 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{51309 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{200000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{51309 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{250000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1590579 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{10000000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{461781}{20000000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{153927}{320000000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{2924613}{160000000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1487961 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{10000000 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73183, size = 629, normalized size = 3.16 \begin{align*} \frac{153927 \, \sqrt{5}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \,{\left (5681664 \, x^{7} + 60161472 \, x^{6} + 272314944 \, x^{5} + 682163760 \, x^{4} + 1007243840 \, x^{3} + 924451956 \, x^{2} + 512781828 \, x + 131091161\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{640000000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.25654, size = 691, normalized size = 3.47 \begin{align*} \frac{153927}{320000000} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{19702656 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 443309760 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 13775440320 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 88813739520 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 1135723030560 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 3326100961968 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 20795205897360 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 31719485197440 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 108381222834920 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 93303707056820 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 182905948708404 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 90199904722080 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 98616726439110 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 25302796273485 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 12323187970155 \, \sqrt{3} x + 954490882968 \, \sqrt{3} - 12323187970155 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{32000000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]