3.2436 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx\)

Optimal. Leaf size=146 \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

((51455 - 106734*x)*Sqrt[2 + 5*x + 3*x^2])/27648 + ((25 - 5586*x)*(2 + 5*x + 3*x
^2)^(3/2))/3456 + ((209 - 30*x)*(2 + 5*x + 3*x^2)^(5/2))/360 - (543811*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(55296*Sqrt[3]) + (325*Sqrt[5]*ArcT
anh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

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Rubi [A]  time = 0.298556, antiderivative size = 146, 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 \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((51455 - 106734*x)*Sqrt[2 + 5*x + 3*x^2])/27648 + ((25 - 5586*x)*(2 + 5*x + 3*x
^2)^(3/2))/3456 + ((209 - 30*x)*(2 + 5*x + 3*x^2)^(5/2))/360 - (543811*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(55296*Sqrt[3]) + (325*Sqrt[5]*ArcT
anh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

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Rubi in Sympy [A]  time = 40.8677, size = 133, normalized size = 0.91 \[ \frac{\left (- 1280808 x + 617460\right ) \sqrt{3 x^{2} + 5 x + 2}}{331776} + \frac{\left (- 33516 x + 150\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{20736} + \frac{\left (- 30 x + 209\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{360} - \frac{543811 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{165888} - \frac{325 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1280808*x + 617460)*sqrt(3*x**2 + 5*x + 2)/331776 + (-33516*x + 150)*(3*x**2 +
5*x + 2)**(3/2)/20736 + (-30*x + 209)*(3*x**2 + 5*x + 2)**(5/2)/360 - 543811*sq
rt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/165888 - 325*sqrt(5)*a
tanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/128

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Mathematica [A]  time = 0.134184, size = 122, normalized size = 0.84 \[ \frac{-2106000 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-2719055 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (103680 x^5-376704 x^4-1311120 x^3-1624872 x^2-583490 x-580299\right )+2106000 \sqrt{5} \log (2 x+3)}{829440} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-580299 - 583490*x - 1624872*x^2 - 1311120*x^3 - 3767
04*x^4 + 103680*x^5) + 2106000*Sqrt[5]*Log[3 + 2*x] - 2106000*Sqrt[5]*Log[-7 - 8
*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 2719055*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6
+ 15*x + 9*x^2]])/829440

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Maple [B]  time = 0.012, size = 239, normalized size = 1.6 \[ -{\frac{5+6\,x}{72} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{25+30\,x}{3456} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{25+30\,x}{27648}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{5\,\sqrt{3}}{165888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{65+78\,x}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1235+1482\,x}{384}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{7553\,\sqrt{3}}{2304}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{65}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{325}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{325\,\sqrt{5}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(5+6*x)*(3*x^2+5*x+2)^(5/2)+5/3456*(5+6*x)*(3*x^2+5*x+2)^(3/2)-5/27648*(5+
6*x)*(3*x^2+5*x+2)^(1/2)+5/165888*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*
3^(1/2)+13/20*(3*(x+3/2)^2-4*x-19/4)^(5/2)-13/48*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^
(3/2)-247/384*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-7553/2304*ln(1/3*(5/2+3*x)*3^
(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+65/48*(3*(x+3/2)^2-4*x-19/4)^(3/2)+3
25/128*(12*(x+3/2)^2-16*x-19)^(1/2)-325/128*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/
2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.812973, size = 212, normalized size = 1.45 \[ -\frac{1}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{209}{360} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{931}{576} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{25}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{17789}{4608} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{543811}{165888} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{325}{128} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{51455}{27648} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*x^2 + 5*x + 2)^(5/2)*x + 209/360*(3*x^2 + 5*x + 2)^(5/2) - 931/576*(3*x
^2 + 5*x + 2)^(3/2)*x + 25/3456*(3*x^2 + 5*x + 2)^(3/2) - 17789/4608*sqrt(3*x^2
+ 5*x + 2)*x - 543811/165888*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5
/2) - 325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2
*x + 3) - 2) + 51455/27648*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.288204, size = 188, normalized size = 1.29 \[ -\frac{1}{1658880} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (103680 \, x^{5} - 376704 \, x^{4} - 1311120 \, x^{3} - 1624872 \, x^{2} - 583490 \, x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 702000 \, \sqrt{5} \sqrt{3} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 2719055 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1658880*sqrt(3)*(4*sqrt(3)*(103680*x^5 - 376704*x^4 - 1311120*x^3 - 1624872*x
^2 - 583490*x - 580299)*sqrt(3*x^2 + 5*x + 2) - 702000*sqrt(5)*sqrt(3)*log((4*sq
rt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)
) - 2719055*log(sqrt(3)*(72*x^2 + 120*x + 49) - 12*sqrt(3*x^2 + 5*x + 2)*(6*x +
5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-96*x*sqrt(3*x**2
+ 5*x + 2)/(2*x + 3), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x + 3),
x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-15*x**4
*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(
2*x + 3), x)

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GIAC/XCAS [A]  time = 0.309497, size = 197, normalized size = 1.35 \[ -\frac{1}{138240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 109\right )} x - 3035\right )} x - 67703\right )} x - 291745\right )} x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{325}{128} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{543811}{165888} \, \sqrt{3}{\rm ln}\left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/138240*(2*(12*(18*(8*(30*x - 109)*x - 3035)*x - 67703)*x - 291745)*x - 580299
)*sqrt(3*x^2 + 5*x + 2) + 325/128*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sq
rt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sq
rt(3*x^2 + 5*x + 2))) + 543811/165888*sqrt(3)*ln(abs(-6*sqrt(3)*x - 5*sqrt(3) +
6*sqrt(3*x^2 + 5*x + 2)))