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

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

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Rubi [A]  time = 0.12, antiderivative size = 139, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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])

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.37, size = 131, normalized size = 0.94 \begin {gather*} \frac {\sqrt {12 x^2+17 x+6} \left (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 )+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 )\right )}{318770436096 (10-3 x)^2 (3 x+2)^{5/2} (4 x+3)^{5/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.39, size = 95, normalized size = 0.68 \begin {gather*} \frac {40325 \tanh ^{-1}\left (\frac {6 \sqrt {12 x^2+17 x+6}}{7 (3 x+2)}\right )}{318770436096}+\frac {\sqrt {12 x^2+17 x+6} \left (706089565584 x^5-3206824169544 x^4-1096520427663 x^3+9848047480070 x^2+10124325497244 x+2773753482408\right )}{7589772288 (3 x-10)^2 (3 x+2)^2 (4 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]*(2773753482408 + 10124325497244*x + 9848047480070*x^2 - 1096520427663*x^3 - 320682416
9544*x^4 + 706089565584*x^5))/(7589772288*(-10 + 3*x)^2*(2 + 3*x)^2*(3 + 4*x)^2) + (40325*ArcTanh[(6*Sqrt[6 +
17*x + 12*x^2])/(7*(2 + 3*x))])/318770436096

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fricas [A]  time = 1.74, size = 186, normalized size = 1.34 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.26, size = 232, normalized size = 1.67 \begin {gather*} \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 {gather*}

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))

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maple [B]  time = 0.02, size = 306, normalized size = 2.20 \begin {gather*} \frac {40325 \arctanh \left (\frac {97 x +\frac {206}{3}}{28 \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}\right )}{637540872192}-\frac {570457 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}\right )}{31239502737408}-\frac {23 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}\right )}{110592}+\frac {58752 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}\right )}{282475249}-\frac {\left (x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}\right )^{\frac {3}{2}}}{2592 \left (x +\frac {2}{3}\right )^{3}}-\frac {1261 \left (97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}\right )^{\frac {3}{2}}}{31239502737408 \left (x -\frac {10}{3}\right )}+\frac {1261 \left (24 x +17\right ) \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}{62479005474816}+\frac {\left (97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}\right )^{\frac {3}{2}}}{79692609024 \left (x -\frac {10}{3}\right )^{2}}-\frac {230400 \left (-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}\right )^{\frac {3}{2}}}{5764801 \left (x +\frac {3}{4}\right )^{2}}-\frac {128 \left (-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}\right )^{\frac {3}{2}}}{352947 \left (x +\frac {3}{4}\right )^{3}}+\frac {47 \left (x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}\right )^{\frac {3}{2}}}{1152 \left (x +\frac {2}{3}\right )^{2}}-\frac {40325 \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}{8925572210688}-\frac {23 \sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}}{4608}-\frac {1410048 \sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}}{282475249} \end {gather*}

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]

-1/2592/(x+2/3)^3*(x+12*(x+2/3)^2+2/3)^(3/2)-1261/31239502737408/(x-10/3)*(97*x+12*(x-10/3)^2-382/3)^(3/2)+126
1/62479005474816*(24*x+17)*(97*x+12*(x-10/3)^2-382/3)^(1/2)+1/79692609024/(x-10/3)^2*(97*x+12*(x-10/3)^2-382/3
)^(3/2)-230400/5764801/(x+3/4)^2*(-x+12*(x+3/4)^2-3/4)^(3/2)-128/352947/(x+3/4)^3*(-x+12*(x+3/4)^2-3/4)^(3/2)-
570457/31239502737408*12^(1/2)*ln(1/12*(12*x+17/2)*12^(1/2)+(97*x+12*(x-10/3)^2-382/3)^(1/2))+47/1152/(x+2/3)^
2*(x+12*(x+2/3)^2+2/3)^(3/2)-23/110592*12^(1/2)*ln(1/12*(12*x+17/2)*12^(1/2)+(x+12*(x+2/3)^2+2/3)^(1/2))+58752
/282475249*12^(1/2)*ln(1/12*(12*x+17/2)*12^(1/2)+(-x+12*(x+3/4)^2-3/4)^(1/2))-40325/8925572210688*(97*x+12*(x-
10/3)^2-382/3)^(1/2)+40325/637540872192*arctanh(1/28*(97*x+206/3)/(97*x+12*(x-10/3)^2-382/3)^(1/2))-23/4608*(x
+12*(x+2/3)^2+2/3)^(1/2)-1410048/282475249*(-x+12*(x+3/4)^2-3/4)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {12\,x^2+17\,x+6}}{{\left (3\,x+2\right )}^3\,{\left (-12\,x^2+31\,x+30\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {12 x^{2} + 17 x + 6}}{46656 x^{9} - 268272 x^{8} - 76788 x^{7} + 1703619 x^{6} + 1218456 x^{5} - 3669588 x^{4} - 6898688 x^{3} - 4903920 x^{2} - 1641600 x - 216000}\, dx \end {gather*}

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]

-Integral(sqrt(12*x**2 + 17*x + 6)/(46656*x**9 - 268272*x**8 - 76788*x**7 + 1703619*x**6 + 1218456*x**5 - 3669
588*x**4 - 6898688*x**3 - 4903920*x**2 - 1641600*x - 216000), x)

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