3.2453 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=181 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac{7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac{7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac{7 (167495-349806 x) \sqrt{3 x^2+5 x+2}}{36864}-\frac{12443893 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{73728 \sqrt{3}}+\frac{44625 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

[Out]

(7*(167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/36864 + (7*(805 - 17394*x)*(2 + 5*
x + 3*x^2)^(3/2))/4608 + (7*(584 + 121*x)*(2 + 5*x + 3*x^2)^(5/2))/(240*(3 + 2*x
)) - ((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(12*(3 + 2*x)^2) - (12443893*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(73728*Sqrt[3]) + (44625*Sqrt[5]*ArcT
anh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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Rubi [A]  time = 0.375079, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac{7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac{7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac{7 (167495-349806 x) \sqrt{3 x^2+5 x+2}}{36864}-\frac{12443893 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{73728 \sqrt{3}}+\frac{44625 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

Antiderivative was successfully verified.

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

[Out]

(7*(167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/36864 + (7*(805 - 17394*x)*(2 + 5*
x + 3*x^2)^(3/2))/4608 + (7*(584 + 121*x)*(2 + 5*x + 3*x^2)^(5/2))/(240*(3 + 2*x
)) - ((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(12*(3 + 2*x)^2) - (12443893*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(73728*Sqrt[3]) + (44625*Sqrt[5]*ArcT
anh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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Rubi in Sympy [A]  time = 48.6408, size = 168, normalized size = 0.93 \[ \frac{7 \left (- 16790688 x + 8039760\right ) \sqrt{3 x^{2} + 5 x + 2}}{1769472} + \frac{7 \left (- 417456 x + 19320\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{110592} - \frac{12443893 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{221184} - \frac{44625 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024} + \frac{7 \left (968 x + 4672\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{1920 \left (2 x + 3\right )} - \frac{\left (4 x + 84\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{48 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-16790688*x + 8039760)*sqrt(3*x**2 + 5*x + 2)/1769472 + 7*(-417456*x + 19320)
*(3*x**2 + 5*x + 2)**(3/2)/110592 - 12443893*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*
sqrt(3*x**2 + 5*x + 2)))/221184 - 44625*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqr
t(3*x**2 + 5*x + 2)))/1024 + 7*(968*x + 4672)*(3*x**2 + 5*x + 2)**(5/2)/(1920*(2
*x + 3)) - (4*x + 84)*(3*x**2 + 5*x + 2)**(7/2)/(48*(2*x + 3)**2)

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Mathematica [A]  time = 0.230124, size = 139, normalized size = 0.77 \[ \frac{-48195000 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-62219465 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{6 \sqrt{3 x^2+5 x+2} \left (414720 x^7-926208 x^6-6830784 x^5-15112992 x^4-12848072 x^3-19284852 x^2-89867034 x-91912653\right )}{(2 x+3)^2}[phone] \sqrt{5} \log (2 x+3)}{1105920} \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(-91912653 - 89867034*x - 19284852*x^2 - 12848072*x^3
- 15112992*x^4 - 6830784*x^5 - 926208*x^6 + 414720*x^7))/(3 + 2*x)^2 + 48195000
*Sqrt[5]*Log[3 + 2*x] - 48195000*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x +
3*x^2]] - 62219465*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/1105920

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Maple [A]  time = 0.016, size = 253, normalized size = 1.4 \[ -{\frac{13}{40} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{27}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{51}{8} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{5635+6762\,x}{480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{101465+121758\,x}{4608} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2040535+2448642\,x}{36864}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{12443893\,\sqrt{3}}{221184}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{357}{32} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{2975}{128} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{44625}{1024}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{44625\,\sqrt{5}}{1024}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{135+162\,x}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+27/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/
4)^(9/2)+51/8*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1127/480*(5+6*x)*(3*(x+3/2)^2-4*x-19/
4)^(5/2)-20293/4608*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-408107/36864*(5+6*x)*(3
*(x+3/2)^2-4*x-19/4)^(1/2)-12443893/221184*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2
-4*x-19/4)^(1/2))*3^(1/2)+357/32*(3*(x+3/2)^2-4*x-19/4)^(5/2)+2975/128*(3*(x+3/2
)^2-4*x-19/4)^(3/2)+44625/1024*(12*(x+3/2)^2-16*x-19)^(1/2)-44625/1024*5^(1/2)*a
rctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-27/20*(5+6*x)*(3*(x+
3/2)^2-4*x-19/4)^(7/2)

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Maxima [A]  time = 0.81357, size = 294, normalized size = 1.62 \[ \frac{39}{40} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1127}{80} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{7}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{4 \,{\left (2 \, x + 3\right )}} - \frac{20293}{768} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{5635}{4608} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{408107}{6144} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{12443893}{221184} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{44625}{1024} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{1172465}{36864} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39/40*(3*x^2 + 5*x + 2)^(7/2) - 13/10*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9)
- 1127/80*(3*x^2 + 5*x + 2)^(5/2)*x - 7/12*(3*x^2 + 5*x + 2)^(5/2) + 27/4*(3*x^
2 + 5*x + 2)^(7/2)/(2*x + 3) - 20293/768*(3*x^2 + 5*x + 2)^(3/2)*x + 5635/4608*(
3*x^2 + 5*x + 2)^(3/2) - 408107/6144*sqrt(3*x^2 + 5*x + 2)*x - 12443893/221184*s
qrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 44625/1024*sqrt(5)*log(s
qrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 1172465/3686
4*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.294805, size = 244, normalized size = 1.35 \[ \frac{\sqrt{3}{\left (16065000 \, \sqrt{5} \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, \sqrt{3}{\left (414720 \, x^{7} - 926208 \, x^{6} - 6830784 \, x^{5} - 15112992 \, x^{4} - 12848072 \, x^{3} - 19284852 \, x^{2} - 89867034 \, x - 91912653\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 62219465 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{2211840 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2211840*sqrt(3)*(16065000*sqrt(5)*sqrt(3)*(4*x^2 + 12*x + 9)*log((4*sqrt(5)*sq
rt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 4*sq
rt(3)*(414720*x^7 - 926208*x^6 - 6830784*x^5 - 15112992*x^4 - 12848072*x^3 - 192
84852*x^2 - 89867034*x - 91912653)*sqrt(3*x^2 + 5*x + 2) + 62219465*(4*x^2 + 12*
x + 9)*log(sqrt(3)*(72*x^2 + 120*x + 49) - 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))/
(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{40 \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac{292 x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac{870 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac{1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac{1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac{396 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac{27 x^{7} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integr
al(-292*x*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-
870*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-1
339*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-1
090*x**4*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-3
96*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(27*
x**7*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef