3.2573 \(\int \frac{5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=141 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac{9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24409}{3125 \sqrt{2 x+3}}+\frac{102697}{1875 (2 x+3)^{3/2}}+\frac{56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{806841 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{3125} \]

[Out]

56399/(625*(3 + 2*x)^(5/2)) + 102697/(1875*(3 + 2*x)^(3/2)) - 24409/(3125*Sqrt[3
+ 2*x]) - (3*(37 + 47*x))/(10*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^2) + (8852 + 99
57*x)/(50*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 266*ArcTanh[Sqrt[3 + 2*x]] - (806
841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

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Rubi [A]  time = 0.35096, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac{9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24409}{3125 \sqrt{2 x+3}}+\frac{102697}{1875 (2 x+3)^{3/2}}+\frac{56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{806841 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

56399/(625*(3 + 2*x)^(5/2)) + 102697/(1875*(3 + 2*x)^(3/2)) - 24409/(3125*Sqrt[3
+ 2*x]) - (3*(37 + 47*x))/(10*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^2) + (8852 + 99
57*x)/(50*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 266*ArcTanh[Sqrt[3 + 2*x]] - (806
841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

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Rubi in Sympy [A]  time = 58.7459, size = 124, normalized size = 0.88 \[ - \frac{806841 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15625} + 266 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{24409}{3125 \sqrt{2 x + 3}} + \frac{102697}{1875 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{141 x + 111}{10 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{9957 x + 8852}{50 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )} + \frac{56399}{625 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-806841*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/15625 + 266*atanh(sqrt(2*x + 3)
) - 24409/(3125*sqrt(2*x + 3)) + 102697/(1875*(2*x + 3)**(3/2)) - (141*x + 111)/
(10*(2*x + 3)**(5/2)*(3*x**2 + 5*x + 2)**2) + (9957*x + 8852)/(50*(2*x + 3)**(5/
2)*(3*x**2 + 5*x + 2)) + 56399/(625*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.470182, size = 151, normalized size = 1.07 \[ \frac{-\frac{225 \sqrt{2 x+3} (4209 x+2959)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{75 \sqrt{2 x+3} (67665 x+67538)}{6 x^2+10 x+4}-\frac{2057760}{\sqrt{2 x+3}}-\frac{245600}{(2 x+3)^{3/2}}-\frac{31200}{(2 x+3)^{5/2}}-6234375 \log \left (1-\sqrt{2 x+3}\right )+6234375 \log \left (\sqrt{2 x+3}+1\right )-2420523 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{46875} \]

Antiderivative was successfully verified.

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

[Out]

(-31200/(3 + 2*x)^(5/2) - 245600/(3 + 2*x)^(3/2) - 2057760/Sqrt[3 + 2*x] - (225*
Sqrt[3 + 2*x]*(2959 + 4209*x))/(2*(2 + 5*x + 3*x^2)^2) + (75*Sqrt[3 + 2*x]*(6753
8 + 67665*x))/(4 + 10*x + 6*x^2) - 2420523*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2
*x]] - 6234375*Log[1 - Sqrt[3 + 2*x]] + 6234375*Log[1 + Sqrt[3 + 2*x]])/46875

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Maple [A]  time = 0.033, size = 151, normalized size = 1.1 \[ 3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+8\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-133\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +{\frac{13122}{3125\, \left ( 4+6\,x \right ) ^{2}} \left ({\frac{775}{18} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4045}{54}\sqrt{3+2\,x}} \right ) }-{\frac{806841\,\sqrt{15}}{15625}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{416}{625} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}}-{\frac{9824}{1875} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{137184}{3125}{\frac{1}{\sqrt{3+2\,x}}}}-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+8\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+133\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/(-1+(3+2*x)^(1/2))^2+8/(-1+(3+2*x)^(1/2))-133*ln(-1+(3+2*x)^(1/2))+13122/3125*
(775/18*(3+2*x)^(3/2)-4045/54*(3+2*x)^(1/2))/(4+6*x)^2-806841/15625*arctanh(1/5*
15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-416/625/(3+2*x)^(5/2)-9824/1875/(3+2*x)^(3/2)-1
37184/3125/(3+2*x)^(1/2)-3/(1+(3+2*x)^(1/2))^2+8/(1+(3+2*x)^(1/2))+133*ln(1+(3+2
*x)^(1/2))

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Maxima [A]  time = 0.789492, size = 217, normalized size = 1.54 \[ \frac{806841}{31250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{659043 \,{\left (2 \, x + 3\right )}^{6} - 8136261 \,{\left (2 \, x + 3\right )}^{5} + 23916753 \,{\left (2 \, x + 3\right )}^{4} - 24720095 \,{\left (2 \, x + 3\right )}^{3} + 6945760 \,{\left (2 \, x + 3\right )}^{2} + 1457600 \, x + 2342400}{9375 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 25 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}\right )}} + 133 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 133 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

806841/31250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x +
3))) - 1/9375*(659043*(2*x + 3)^6 - 8136261*(2*x + 3)^5 + 23916753*(2*x + 3)^4
- 24720095*(2*x + 3)^3 + 6945760*(2*x + 3)^2 + 1457600*x + 2342400)/(9*(2*x + 3)
^(13/2) - 48*(2*x + 3)^(11/2) + 94*(2*x + 3)^(9/2) - 80*(2*x + 3)^(7/2) + 25*(2*
x + 3)^(5/2)) + 133*log(sqrt(2*x + 3) + 1) - 133*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.292056, size = 347, normalized size = 2.46 \[ \frac{\sqrt{5}{\left (2493750 \, \sqrt{5}{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 2493750 \, \sqrt{5}{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 2420523 \, \sqrt{3}{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) - \sqrt{5}{\left (5272344 \, x^{6} + 14906052 \, x^{5} - 18312714 \, x^{4} - 114099329 \, x^{3} - 160041829 \, x^{2} - 94082723 \, x - 20250051\right )}\right )}}{93750 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/93750*sqrt(5)*(2493750*sqrt(5)*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2
+ 228*x + 36)*sqrt(2*x + 3)*log(sqrt(2*x + 3) + 1) - 2493750*sqrt(5)*(36*x^6 +
228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*sqrt(2*x + 3)*log(sqrt(2*x +
3) - 1) + 2420523*sqrt(3)*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228
*x + 36)*sqrt(2*x + 3)*log((sqrt(5)*(3*x + 7) - 5*sqrt(3)*sqrt(2*x + 3))/(3*x +
2)) - sqrt(5)*(5272344*x^6 + 14906052*x^5 - 18312714*x^4 - 114099329*x^3 - 16004
1829*x^2 - 94082723*x - 20250051))/((36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*
x^2 + 228*x + 36)*sqrt(2*x + 3))

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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+2*x)**(7/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285385, size = 193, normalized size = 1.37 \[ \frac{806841}{31250} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{202995 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 745077 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 831169 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 259087 \, \sqrt{2 \, x + 3}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} - \frac{32 \,{\left (12861 \,{\left (2 \, x + 3\right )}^{2} + 3070 \, x + 4800\right )}}{9375 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 133 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 133 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

806841/31250*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sq
rt(2*x + 3))) + 1/625*(202995*(2*x + 3)^(7/2) - 745077*(2*x + 3)^(5/2) + 831169*
(2*x + 3)^(3/2) - 259087*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)^2 - 32/9375*
(12861*(2*x + 3)^2 + 3070*x + 4800)/(2*x + 3)^(5/2) + 133*ln(sqrt(2*x + 3) + 1)
- 133*ln(abs(sqrt(2*x + 3) - 1))