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

Optimal. Leaf size=139 \[ -\frac{388 x+275}{294 (10-3 x)^2 \left (12 x^2+17 x+6\right )^{3/2}}-\frac{1634466587 \sqrt{12 x^2+17 x+6}}{7589772288 (10-3 x)}-\frac{50555899 \sqrt{12 x^2+17 x+6}}{19361664 (10-3 x)^2}+\frac{1042556 x+738029}{8232 (10-3 x)^2 \sqrt{12 x^2+17 x+6}}+\frac{40325 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{637540872192} \]

[Out]

-(275 + 388*x)/(294*(10 - 3*x)^2*(6 + 17*x + 12*x^2)^(3/2)) + (738029 + 1042556*x)/(8232*(10 - 3*x)^2*Sqrt[6 +
17*x + 12*x^2]) - (50555899*Sqrt[6 + 17*x + 12*x^2])/(19361664*(10 - 3*x)^2) - (1634466587*Sqrt[6 + 17*x + 12
*x^2])/(7589772288*(10 - 3*x)) + (40325*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])])/637540872192

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Rubi [A]  time = 0.117485, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206, Rules used = {1002, 740, 822, 834, 806, 724, 206} \[ -\frac{388 x+275}{294 (10-3 x)^2 \left (12 x^2+17 x+6\right )^{3/2}}-\frac{1634466587 \sqrt{12 x^2+17 x+6}}{7589772288 (10-3 x)}-\frac{50555899 \sqrt{12 x^2+17 x+6}}{19361664 (10-3 x)^2}+\frac{1042556 x+738029}{8232 (10-3 x)^2 \sqrt{12 x^2+17 x+6}}+\frac{40325 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{637540872192} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(275 + 388*x)/(294*(10 - 3*x)^2*(6 + 17*x + 12*x^2)^(3/2)) + (738029 + 1042556*x)/(8232*(10 - 3*x)^2*Sqrt[6 +
17*x + 12*x^2]) - (50555899*Sqrt[6 + 17*x + 12*x^2])/(19361664*(10 - 3*x)^2) - (1634466587*Sqrt[6 + 17*x + 12
*x^2])/(7589772288*(10 - 3*x)) + (40325*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])])/637540872192

Rule 1002

Int[((g_) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(m_.
), x_Symbol] :> Int[((d*g)/a + (f*h*x)/c)^m*(a + b*x + c*x^2)^(m + p), x] /; FreeQ[{a, b, c, d, e, f, g, h, p}
, x] && EqQ[c*g^2 - b*g*h + a*h^2, 0] && EqQ[c^2*d*g^2 - a*c*e*g*h + a^2*f*h^2, 0] && IntegerQ[m]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx &=\int \frac{1}{(10-3 x)^3 \left (6+17 x+12 x^2\right )^{5/2}} \, dx\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}-\frac{\int \frac{\frac{109953}{2}-41904 x}{(10-3 x)^3 \left (6+17 x+12 x^2\right )^{3/2}} \, dx}{2646}\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac{738029+1042556 x}{8232 (10-3 x)^2 \sqrt{6+17 x+12 x^2}}+\frac{\int \frac{-\frac{5020024653}{4}-1773387756 x}{(10-3 x)^3 \sqrt{6+17 x+12 x^2}} \, dx}{2333772}\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac{738029+1042556 x}{8232 (10-3 x)^2 \sqrt{6+17 x+12 x^2}}-\frac{50555899 \sqrt{6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac{\int \frac{\frac{1461036257541}{8}[phone] x}{(10-3 x)^2 \sqrt{6+17 x+12 x^2}} \, dx}{8233547616}\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac{738029+1042556 x}{8232 (10-3 x)^2 \sqrt{6+17 x+12 x^2}}-\frac{50555899 \sqrt{6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac{1634466587 \sqrt{6+17 x+12 x^2}}{7589772288 (10-3 x)}+\frac{40325 \int \frac{1}{(10-3 x) \sqrt{6+17 x+12 x^2}} \, dx}{15179544576}\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac{738029+1042556 x}{8232 (10-3 x)^2 \sqrt{6+17 x+12 x^2}}-\frac{50555899 \sqrt{6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac{1634466587 \sqrt{6+17 x+12 x^2}}{7589772288 (10-3 x)}-\frac{40325 \operatorname{Subst}\left (\int \frac{1}{7056-x^2} \, dx,x,\frac{-206-291 x}{\sqrt{6+17 x+12 x^2}}\right )}{7589772288}\\ &=-\frac{275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac{738029+1042556 x}{8232 (10-3 x)^2 \sqrt{6+17 x+12 x^2}}-\frac{50555899 \sqrt{6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac{1634466587 \sqrt{6+17 x+12 x^2}}{7589772288 (10-3 x)}+\frac{40325 \tanh ^{-1}\left (\frac{206+291 x}{84 \sqrt{6+17 x+12 x^2}}\right )}{637540872192}\\ \end{align*}

Mathematica [A]  time = 0.38004, size = 131, normalized size = 0.94 \[ \frac{\sqrt{12 x^2+17 x+6} \left (42 \sqrt{3 x+2} \sqrt{4 x+3} \left (706089565584 x^5-3206824169544 x^4-1096520427663 x^3+9848047480070 x^2+10124325497244 x+2773753482408\right )+40325 \left (-36 x^3+69 x^2+152 x+60\right )^2 \tanh ^{-1}\left (\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right )\right )}{318770436096 (10-3 x)^2 (3 x+2)^{5/2} (4 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[6 + 17*x + 12*x^2]*(42*Sqrt[2 + 3*x]*Sqrt[3 + 4*x]*(2773753482408 + 10124325497244*x + 9848047480070*x^2
- 1096520427663*x^3 - 3206824169544*x^4 + 706089565584*x^5) + 40325*(60 + 152*x + 69*x^2 - 36*x^3)^2*ArcTanh[
(7*Sqrt[2 + 3*x])/(6*Sqrt[3 + 4*x])]))/(318770436096*(10 - 3*x)^2*(2 + 3*x)^(5/2)*(3 + 4*x)^(5/2))

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Maple [B]  time = 0.071, size = 306, normalized size = 2.2 \begin{align*}{\frac{1}{79692609024} \left ( 12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}} \right ) ^{{\frac{3}{2}}} \left ( x-{\frac{10}{3}} \right ) ^{-2}}+{\frac{47}{1152} \left ( 12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{2}{3}} \right ) ^{-2}}-{\frac{230400}{5764801} \left ( 12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{4}} \right ) ^{-2}}-{\frac{23\,\sqrt{12}}{110592}\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{570457\,\sqrt{12}}{31239502737408}\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{58752\,\sqrt{12}}{282475249}\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 ) }-{\frac{23}{4608}\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}}}+{\frac{40325}{637540872192}{\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{40325}{8925572210688}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}-{\frac{1410048}{282475249}\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}}}+{\frac{21437+30264\,x}{62479005474816}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}-{\frac{128}{352947} \left ( 12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{4}} \right ) ^{-3}}-{\frac{1261}{31239502737408} \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{1}{2592} \left ( 12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{2}{3}} \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/79692609024/(x-10/3)^2*(12*(x-10/3)^2+97*x-382/3)^(3/2)+47/1152/(x+2/3)^2*(12*(x+2/3)^2+x+2/3)^(3/2)-230400/
5764801/(x+3/4)^2*(12*(x+3/4)^2-x-3/4)^(3/2)-23/110592*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x+2/3)^2+x+2/3)^(1/2)
)*12^(1/2)-570457/31239502737408*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x-10/3)^2+97*x-382/3)^(1/2))*12^(1/2)+58752
/282475249*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x+3/4)^2-x-3/4)^(1/2))*12^(1/2)-23/4608*(12*(x+2/3)^2+x+2/3)^(1/2
)+40325/637540872192*arctanh(1/28*(206/3+97*x)/(12*(x-10/3)^2+97*x-382/3)^(1/2))-40325/8925572210688*(12*(x-10
/3)^2+97*x-382/3)^(1/2)-1410048/282475249*(12*(x+3/4)^2-x-3/4)^(1/2)+1261/62479005474816*(17+24*x)*(12*(x-10/3
)^2+97*x-382/3)^(1/2)-128/352947/(x+3/4)^3*(12*(x+3/4)^2-x-3/4)^(3/2)-1261/31239502737408/(x-10/3)*(12*(x-10/3
)^2+97*x-382/3)^(3/2)-1/2592/(x+2/3)^3*(12*(x+2/3)^2+x+2/3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)^3/(-12*x^2+31*x+30)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6898, size = 674, normalized size = 4.85 \begin{align*} \frac{40325 \,{\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )} \log \left (\frac{291 \, x + 84 \, \sqrt{12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 40325 \,{\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )} \log \left (\frac{291 \, x - 84 \, \sqrt{12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) + 168 \,{\left (706089565584 \, x^{5} - 3206824169544 \, x^{4} - 1096520427663 \, x^{3} + 9848047480070 \, x^{2} + 10124325497244 \, x + 2773753482408\right )} \sqrt{12 \, x^{2} + 17 \, x + 6}}{1275081744384 \,{\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)^3/(-12*x^2+31*x+30)^3,x, algorithm="fricas")

[Out]

1/1275081744384*(40325*(1296*x^6 - 4968*x^5 - 6183*x^4 + 16656*x^3 + 31384*x^2 + 18240*x + 3600)*log((291*x +
84*sqrt(12*x^2 + 17*x + 6) + 206)/x) - 40325*(1296*x^6 - 4968*x^5 - 6183*x^4 + 16656*x^3 + 31384*x^2 + 18240*x
+ 3600)*log((291*x - 84*sqrt(12*x^2 + 17*x + 6) + 206)/x) + 168*(706089565584*x^5 - 3206824169544*x^4 - 10965
20427663*x^3 + 9848047480070*x^2 + 10124325497244*x + 2773753482408)*sqrt(12*x^2 + 17*x + 6))/(1296*x^6 - 4968
*x^5 - 6183*x^4 + 16656*x^3 + 31384*x^2 + 18240*x + 3600)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.21618, size = 313, normalized size = 2.25 \begin{align*} \frac{\sqrt{3}{\left (282273 \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )}^{3} - 11460924 \,{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )}^{2} - 37551180 \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )} - 83365264\right )}}{159385218048 \,{\left (3 \,{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )}^{2} - 40 \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )} - 188\right )}^{2}} + \frac{{\left (8 \,{\left (2860316794 \, x + 6078171227\right )} x + 34383350229\right )} x + 8090114146}{2213683584 \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}}} + \frac{40325}{637540872192} \, \log \left ({\left | -6 \, \sqrt{3} x + 20 \, \sqrt{3} + 3 \, \sqrt{12 \, x^{2} + 17 \, x + 6} + 42 \right |}\right ) - \frac{40325}{637540872192} \, \log \left ({\left | -6 \, \sqrt{3} x + 20 \, \sqrt{3} + 3 \, \sqrt{12 \, x^{2} + 17 \, x + 6} - 42 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)^3/(-12*x^2+31*x+30)^3,x, algorithm="giac")

[Out]

1/159385218048*sqrt(3)*(282273*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 + 17*x + 6))^3 - 11460924*(2*sqrt(3)*x - sqr
t(12*x^2 + 17*x + 6))^2 - 37551180*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 + 17*x + 6)) - 83365264)/(3*(2*sqrt(3)*x
- sqrt(12*x^2 + 17*x + 6))^2 - 40*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 + 17*x + 6)) - 188)^2 + 1/2213683584*((8
*(2860316794*x + 6078171227)*x + 34383350229)*x + 8090114146)/(12*x^2 + 17*x + 6)^(3/2) + 40325/637540872192*l
og(abs(-6*sqrt(3)*x + 20*sqrt(3) + 3*sqrt(12*x^2 + 17*x + 6) + 42)) - 40325/637540872192*log(abs(-6*sqrt(3)*x
+ 20*sqrt(3) + 3*sqrt(12*x^2 + 17*x + 6) - 42))