3.1367 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=148 \[ -\frac{1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac{111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac{281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac{1017 (4-9 x) \sqrt{3 x^2+2}}{7503125 (2 x+3)^2}-\frac{6102 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

[Out]

(-1017*(4 - 9*x)*Sqrt[2 + 3*x^2])/(7503125*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))
/(210*(3 + 2*x)^6) - (281*(2 + 3*x^2)^(3/2))/(12250*(3 + 2*x)^5) - (111*(2 + 3*x
^2)^(3/2))/(17500*(3 + 2*x)^4) - (1207*(2 + 3*x^2)^(3/2))/(857500*(3 + 2*x)^3) -
(6102*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

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Rubi [A]  time = 0.261929, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac{111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac{281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac{1017 (4-9 x) \sqrt{3 x^2+2}}{7503125 (2 x+3)^2}-\frac{6102 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]

[Out]

(-1017*(4 - 9*x)*Sqrt[2 + 3*x^2])/(7503125*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))
/(210*(3 + 2*x)^6) - (281*(2 + 3*x^2)^(3/2))/(12250*(3 + 2*x)^5) - (111*(2 + 3*x
^2)^(3/2))/(17500*(3 + 2*x)^4) - (1207*(2 + 3*x^2)^(3/2))/(857500*(3 + 2*x)^3) -
(6102*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

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Rubi in Sympy [A]  time = 27.1227, size = 139, normalized size = 0.94 \[ - \frac{1017 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{15006250 \left (2 x + 3\right )^{2}} - \frac{6102 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{262609375} - \frac{1207 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{857500 \left (2 x + 3\right )^{3}} - \frac{111 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{17500 \left (2 x + 3\right )^{4}} - \frac{281 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{12250 \left (2 x + 3\right )^{5}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{210 \left (2 x + 3\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**7,x)

[Out]

-1017*(-18*x + 8)*sqrt(3*x**2 + 2)/(15006250*(2*x + 3)**2) - 6102*sqrt(35)*atanh
(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/262609375 - 1207*(3*x**2 + 2)**(3/2)
/(857500*(2*x + 3)**3) - 111*(3*x**2 + 2)**(3/2)/(17500*(2*x + 3)**4) - 281*(3*x
**2 + 2)**(3/2)/(12250*(2*x + 3)**5) - 13*(3*x**2 + 2)**(3/2)/(210*(2*x + 3)**6)

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Mathematica [A]  time = 0.16197, size = 95, normalized size = 0.64 \[ \frac{-36612 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (642132 x^5+5388660 x^4+18236055 x^3+30753930 x^2+18651300 x+22308548\right )}{(2 x+3)^6}+36612 \sqrt{35} \log (2 x+3)}{1575656250} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(22308548 + 18651300*x + 30753930*x^2 + 18236055*x^3 + 538
8660*x^4 + 642132*x^5))/(3 + 2*x)^6 + 36612*Sqrt[35]*Log[3 + 2*x] - 36612*Sqrt[3
5]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/1575656250

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Maple [A]  time = 0.022, size = 191, normalized size = 1.3 \[ -{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{281}{392000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{111}{280000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1207}{6860000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1017}{15006250} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{9153}{262609375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{6102}{262609375}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{6102\,\sqrt{35}}{262609375}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{27459\,x}{262609375}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^7,x)

[Out]

-13/13440/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(3/2)-281/392000/(x+3/2)^5*(3*(x+3/2)
^2-9*x-19/4)^(3/2)-111/280000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(3/2)-1207/686000
0/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(3/2)-1017/15006250/(x+3/2)^2*(3*(x+3/2)^2-9*
x-19/4)^(3/2)-9153/262609375/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+6102/262609375
*(12*(x+3/2)^2-36*x-19)^(1/2)-6102/262609375*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1
/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+27459/262609375*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [A]  time = 0.78013, size = 309, normalized size = 2.09 \[ \frac{6102}{262609375} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{3051}{15006250} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{281 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{12250 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{111 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{17500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1207 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{857500 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2034 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{7503125 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{9153 \, \sqrt{3 \, x^{2} + 2}}{15006250 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="maxima")

[Out]

6102/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x
+ 3)) + 3051/15006250*sqrt(3*x^2 + 2) - 13/210*(3*x^2 + 2)^(3/2)/(64*x^6 + 576*
x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 281/12250*(3*x^2 + 2)^(3/
2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 111/17500*(3*x^2 + 2)
^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1207/857500*(3*x^2 + 2)^(3/2)/
(8*x^3 + 36*x^2 + 54*x + 27) - 2034/7503125*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9)
- 9153/15006250*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 0.296431, size = 208, normalized size = 1.41 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (642132 \, x^{5} + 5388660 \, x^{4} + 18236055 \, x^{3} + 30753930 \, x^{2} + 18651300 \, x + 22308548\right )} \sqrt{3 \, x^{2} + 2} - 18306 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{1575656250 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="fricas")

[Out]

-1/1575656250*sqrt(35)*(sqrt(35)*(642132*x^5 + 5388660*x^4 + 18236055*x^3 + 3075
3930*x^2 + 18651300*x + 22308548)*sqrt(3*x^2 + 2) - 18306*(64*x^6 + 576*x^5 + 21
60*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-(sqrt(35)*(93*x^2 - 36*x + 43)
+ 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(64*x^6 + 576*x^5 + 2160*x
^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.35133, size = 490, normalized size = 3.31 \[ \frac{6102}{262609375} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{3 \,{\left (65088 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 1073952 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 20936640 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 87678735 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 199001970 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 258582989 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1280293308 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 755892540 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 1065400320 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 207134880 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 59561856 \, \sqrt{3} x + 2283136 \, \sqrt{3} + 59561856 \, \sqrt{3 \, x^{2} + 2}\right )}}{240100000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="giac")

[Out]

6102/262609375*sqrt(35)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x
^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/240100000
*(65088*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 1073952*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2))^10 + 20936640*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 87678735*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 2))^8 - 199001970*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 25858298
9*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 1280293308*(sqrt(3)*x - sqrt(3*x^2 +
2))^5 + 755892540*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 1065400320*(sqrt(3)
*x - sqrt(3*x^2 + 2))^3 + 207134880*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 59
561856*sqrt(3)*x + 2283136*sqrt(3) + 59561856*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqr
t(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^6