3.136 \(\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx\)

Optimal. Leaf size=84 \[ -\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{3226944} \]

[Out]

-(275 + 388*x)/(98*(10 - 3*x)*Sqrt[6 + 17*x + 12*x^2]) + (3137*Sqrt[6 + 17*x + 1
2*x^2])/(38416*(10 - 3*x)) + (97*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^
2])])/3226944

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Rubi [A]  time = 0.209075, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147 \[ -\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{3226944} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^2*(30 + 31*x - 12*x^2)^2),x]

[Out]

-(275 + 388*x)/(98*(10 - 3*x)*Sqrt[6 + 17*x + 12*x^2]) + (3137*Sqrt[6 + 17*x + 1
2*x^2])/(38416*(10 - 3*x)) + (97*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^
2])])/3226944

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Rubi in Sympy [A]  time = 32.0861, size = 71, normalized size = 0.85 \[ - \frac{97 \operatorname{atanh}{\left (\frac{- 291 x - 206}{84 \sqrt{12 x^{2} + 17 x + 6}} \right )}}{3226944} - \frac{3492 x + 2475}{882 \left (- 3 x + 10\right ) \sqrt{12 x^{2} + 17 x + 6}} + \frac{3137 \sqrt{12 x^{2} + 17 x + 6}}{38416 \left (- 3 x + 10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((12*x**2+17*x+6)**(1/2)/(2+3*x)**2/(-12*x**2+31*x+30)**2,x)

[Out]

-97*atanh((-291*x - 206)/(84*sqrt(12*x**2 + 17*x + 6)))/3226944 - (3492*x + 2475
)/(882*(-3*x + 10)*sqrt(12*x**2 + 17*x + 6)) + 3137*sqrt(12*x**2 + 17*x + 6)/(38
416*(-3*x + 10))

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Mathematica [A]  time = 0.23998, size = 93, normalized size = 1.11 \[ \frac{\sqrt{12 x^2+17 x+6} \left (\frac{97 \tanh ^{-1}\left (\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right )}{\sqrt{3 x+2} \sqrt{4 x+3}}-\frac{42 \left (37644 x^2-98767 x-88978\right )}{(3 x+2) \left (12 x^2-31 x-30\right )}\right )}{1613472} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^2*(30 + 31*x - 12*x^2)^2),x]

[Out]

(Sqrt[6 + 17*x + 12*x^2]*((-42*(-88978 - 98767*x + 37644*x^2))/((2 + 3*x)*(-30 -
31*x + 12*x^2)) + (97*ArcTanh[(7*Sqrt[2 + 3*x])/(6*Sqrt[3 + 4*x])])/(Sqrt[2 + 3
*x]*Sqrt[3 + 4*x])))/1613472

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Maple [B]  time = 0.027, size = 245, normalized size = 2.9 \[ -{\frac{1}{72} \left ( 12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{2}{3}} \right ) ^{-2}}+{\frac{1}{288}\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}}}+{\frac{\sqrt{12}}{6912}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}} \right ) }-{\frac{1}{67765824} \left ( 12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}} \right ) ^{{\frac{3}{2}}} \left ( x-{\frac{10}{3}} \right ) ^{-1}}-{\frac{97}{45177216}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}-{\frac{7057\,\sqrt{12}}{813189888}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}} \right ) }+{\frac{97}{3226944}{\it Artanh} \left ({\frac{1}{28} \left ({\frac{206}{3}}+97\,x \right ){\frac{1}{\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}}} \right ) }+{\frac{17+24\,x}{135531648}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}+{\frac{32}{2401} \left ( 12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{4}} \right ) ^{-2}}+{\frac{384}{117649}\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}}}-{\frac{16\,\sqrt{12}}{117649}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((12*x^2+17*x+6)^(1/2)/(2+3*x)^2/(-12*x^2+31*x+30)^2,x)

[Out]

-1/72/(x+2/3)^2*(12*(x+2/3)^2+x+2/3)^(3/2)+1/288*(12*(x+2/3)^2+x+2/3)^(1/2)+1/69
12*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x+2/3)^2+x+2/3)^(1/2))*12^(1/2)-1/67765824/
(x-10/3)*(12*(x-10/3)^2+97*x-382/3)^(3/2)-97/45177216*(12*(x-10/3)^2+97*x-382/3)
^(1/2)-7057/813189888*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x-10/3)^2+97*x-382/3)^(1
/2))*12^(1/2)+97/3226944*arctanh(1/28*(206/3+97*x)/(12*(x-10/3)^2+97*x-382/3)^(1
/2))+1/135531648*(17+24*x)*(12*(x-10/3)^2+97*x-382/3)^(1/2)+32/2401/(x+3/4)^2*(1
2*(x+3/4)^2-x-3/4)^(3/2)+384/117649*(12*(x+3/4)^2-x-3/4)^(1/2)-16/117649*ln(1/12
*(17/2+12*x)*12^(1/2)+(12*(x+3/4)^2-x-3/4)^(1/2))*12^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )}^{2}{\left (3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(12*x^2 + 17*x + 6)/((12*x^2 - 31*x - 30)^2*(3*x + 2)^2),x, algorithm="maxima")

[Out]

integrate(sqrt(12*x^2 + 17*x + 6)/((12*x^2 - 31*x - 30)^2*(3*x + 2)^2), x)

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Fricas [A]  time = 0.289709, size = 170, normalized size = 2.02 \[ \frac{97 \,{\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac{291 \, x + 84 \, \sqrt{12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 97 \,{\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac{291 \, x - 84 \, \sqrt{12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 168 \,{\left (37644 \, x^{2} - 98767 \, x - 88978\right )} \sqrt{12 \, x^{2} + 17 \, x + 6}}{6453888 \,{\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(12*x^2 + 17*x + 6)/((12*x^2 - 31*x - 30)^2*(3*x + 2)^2),x, algorithm="fricas")

[Out]

1/6453888*(97*(36*x^3 - 69*x^2 - 152*x - 60)*log((291*x + 84*sqrt(12*x^2 + 17*x
+ 6) + 206)/x) - 97*(36*x^3 - 69*x^2 - 152*x - 60)*log((291*x - 84*sqrt(12*x^2 +
17*x + 6) + 206)/x) - 168*(37644*x^2 - 98767*x - 88978)*sqrt(12*x^2 + 17*x + 6)
)/(36*x^3 - 69*x^2 - 152*x - 60)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (3 x + 2\right ) \left (4 x + 3\right )}}{\left (3 x - 10\right )^{2} \left (3 x + 2\right )^{2} \left (4 x + 3\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12*x**2+17*x+6)**(1/2)/(2+3*x)**2/(-12*x**2+31*x+30)**2,x)

[Out]

Integral(sqrt((3*x + 2)*(4*x + 3))/((3*x - 10)**2*(3*x + 2)**2*(4*x + 3)**2), x)

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GIAC/XCAS [A]  time = 0.289936, size = 215, normalized size = 2.56 \[ \frac{1}{9680832} \, \sqrt{3}{\left (\sqrt{3}{\left (175672 \, \sqrt{3} + 97 \,{\rm ln}\left (\frac{7 \, \sqrt{3} - 12}{7 \, \sqrt{3} + 12}\right )\right )}{\rm sign}\left (\frac{1}{3 \, x + 2}\right ) -{\left (97 \, \sqrt{3}{\rm ln}\left (\frac{{\left | -28 \, \sqrt{3} + 24 \, \sqrt{\frac{1}{3 \, x + 2} + 4} \right |}}{4 \,{\left (7 \, \sqrt{3} + 6 \, \sqrt{\frac{1}{3 \, x + 2} + 4}\right )}}\right ) + 134456 \, \sqrt{\frac{1}{3 \, x + 2} + 4} + \frac{28 \,{\left (\frac{221183}{3 \, x + 2} - 18436\right )}}{12 \,{\left (\frac{1}{3 \, x + 2} + 4\right )}^{\frac{3}{2}} - 49 \, \sqrt{\frac{1}{3 \, x + 2} + 4}}\right )}{\rm sign}\left (\frac{1}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(12*x^2 + 17*x + 6)/((12*x^2 - 31*x - 30)^2*(3*x + 2)^2),x, algorithm="giac")

[Out]

1/9680832*sqrt(3)*(sqrt(3)*(175672*sqrt(3) + 97*ln((7*sqrt(3) - 12)/(7*sqrt(3) +
12)))*sign(1/(3*x + 2)) - (97*sqrt(3)*ln(1/4*abs(-28*sqrt(3) + 24*sqrt(1/(3*x +
2) + 4))/(7*sqrt(3) + 6*sqrt(1/(3*x + 2) + 4))) + 134456*sqrt(1/(3*x + 2) + 4)
+ 28*(221183/(3*x + 2) - 18436)/(12*(1/(3*x + 2) + 4)^(3/2) - 49*sqrt(1/(3*x + 2
) + 4)))*sign(1/(3*x + 2)))