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

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 + 12*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 = 0.0777517, 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, Rules used = {1002, 740, 806, 724, 206} \[ -\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{3226944} \]

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

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 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)^2 \left (30+31 x-12 x^2\right )^2} \, dx &=\int \frac{1}{(10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}} \, dx\\ &=-\frac{275+388 x}{98 (10-3 x) \sqrt{6+17 x+12 x^2}}-\frac{1}{882} \int \frac{-\frac{14859}{2}-10476 x}{(10-3 x)^2 \sqrt{6+17 x+12 x^2}} \, dx\\ &=-\frac{275+388 x}{98 (10-3 x) \sqrt{6+17 x+12 x^2}}+\frac{3137 \sqrt{6+17 x+12 x^2}}{38416 (10-3 x)}+\frac{97 \int \frac{1}{(10-3 x) \sqrt{6+17 x+12 x^2}} \, dx}{76832}\\ &=-\frac{275+388 x}{98 (10-3 x) \sqrt{6+17 x+12 x^2}}+\frac{3137 \sqrt{6+17 x+12 x^2}}{38416 (10-3 x)}-\frac{97 \operatorname{Subst}\left (\int \frac{1}{7056-x^2} \, dx,x,\frac{-206-291 x}{\sqrt{6+17 x+12 x^2}}\right )}{38416}\\ &=-\frac{275+388 x}{98 (10-3 x) \sqrt{6+17 x+12 x^2}}+\frac{3137 \sqrt{6+17 x+12 x^2}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left (\frac{206+291 x}{84 \sqrt{6+17 x+12 x^2}}\right )}{3226944}\\ \end{align*}

Mathematica [A]  time = 0.243785, size = 114, normalized size = 1.36 \[ \frac{\sqrt{12 x^2+17 x+6} \left (97 \left (36 x^3-69 x^2-152 x-60\right ) \tanh ^{-1}\left (\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right )-42 \sqrt{3 x+2} \sqrt{4 x+3} \left (37644 x^2-98767 x-88978\right )\right )}{1613472 (3 x-10) (3 x+2)^{3/2} (4 x+3)^{3/2}} \]

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*Sqrt[2 + 3*x]*Sqrt[3 + 4*x]*(-88978 - 98767*x + 37644*x^2) + 97*(-60 - 152*x - 6
9*x^2 + 36*x^3)*ArcTanh[(7*Sqrt[2 + 3*x])/(6*Sqrt[3 + 4*x])]))/(1613472*(-10 + 3*x)*(2 + 3*x)^(3/2)*(3 + 4*x)^
(3/2))

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Maple [B]  time = 0.064, size = 245, normalized size = 2.9 \begin{align*}{\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 ) }-{\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{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{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{17+24\,x}{135531648}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{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)^2/(-12*x^2+31*x+30)^2,x)

[Out]

32/2401/(x+3/4)^2*(12*(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)-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)+1/6912*ln(1/12*(17/2+12*x)*12^(1/2)+(12*(x+2/3)^2+x+2/3)^(1/2))*12^(1/2)-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)+9
7/3226944*arctanh(1/28*(206/3+97*x)/(12*(x-10/3)^2+97*x-382/3)^(1/2))-1/67765824/(x-10/3)*(12*(x-10/3)^2+97*x-
382/3)^(3/2)+1/135531648*(17+24*x)*(12*(x-10/3)^2+97*x-382/3)^(1/2)

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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 )}^{2}{\left (3 \, x + 2\right )}^{2}}\,{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)^2/(-12*x^2+31*x+30)^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 = 1.58619, size = 370, normalized size = 4.4 \begin{align*} \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 )}} \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)^2/(-12*x^2+31*x+30)^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)*sqr
t(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. \begin{align*} \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 \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)**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 [B]  time = 1.22682, size = 215, normalized size = 2.56 \begin{align*} \frac{1}{9680832} \, \sqrt{3}{\left (\sqrt{3}{\left (175672 \, \sqrt{3} + 97 \, \log \left (\frac{7 \, \sqrt{3} - 12}{7 \, \sqrt{3} + 12}\right )\right )} \mathrm{sgn}\left (\frac{1}{3 \, x + 2}\right ) -{\left (97 \, \sqrt{3} \log \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 )} \mathrm{sgn}\left (\frac{1}{3 \, x + 2}\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)^2/(-12*x^2+31*x+30)^2,x, algorithm="giac")

[Out]

1/9680832*sqrt(3)*(sqrt(3)*(175672*sqrt(3) + 97*log((7*sqrt(3) - 12)/(7*sqrt(3) + 12)))*sgn(1/(3*x + 2)) - (97
*sqrt(3)*log(1/4*abs(-28*sqrt(3) + 24*sqrt(1/(3*x + 2) + 4))/(7*sqrt(3) + 6*sqrt(1/(3*x + 2) + 4))) + 134456*s
qrt(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)))*
sgn(1/(3*x + 2)))