\(\int (2+3 x) (30+31 x-12 x^2) \sqrt {6+17 x+12 x^2} \, dx\) [134]

Optimal result

Integrand size = 30, antiderivative size = 103 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \text {arctanh}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{98304 \sqrt {3}} \]

[Out]

97/768*(17+24*x)*(12*x^2+17*x+6)^(3/2)-1/20*(12*x^2+17*x+6)^(5/2)+97/294912*arctanh(1/12*(17+24*x)*3^(1/2)/(12
*x^2+17*x+6)^(1/2))*3^(1/2)-97/24576*(17+24*x)*(12*x^2+17*x+6)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1016, 654, 626, 635, 212} \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {97 \text {arctanh}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{98304 \sqrt {3}}-\frac {1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac {97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac {97 (24 x+17) \sqrt {12 x^2+17 x+6}}{24576} \]

[In]

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

[Out]

(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/768 - (6 + 17*x +
12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])

Rule 212

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

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

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

Rule 654

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

Rule 1016

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*} \text {integral}& = \int (10-3 x) \left (6+17 x+12 x^2\right )^{3/2} \, dx \\ & = -\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97}{8} \int \left (6+17 x+12 x^2\right )^{3/2} \, dx \\ & = \frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}-\frac {97}{512} \int \sqrt {6+17 x+12 x^2} \, dx \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \int \frac {1}{\sqrt {6+17 x+12 x^2}} \, dx}{49152} \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \text {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {17+24 x}{\sqrt {6+17 x+12 x^2}}\right )}{24576} \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \tanh ^{-1}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{98304 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.72 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {6 \sqrt {6+17 x+12 x^2} \left (1353611+5455144 x+6837888 x^2+1963008 x^3-884736 x^4\right )+485 \sqrt {3} \text {arctanh}\left (\frac {2 \sqrt {2+\frac {17 x}{3}+4 x^2}}{3+4 x}\right )}{737280} \]

[In]

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

[Out]

(6*Sqrt[6 + 17*x + 12*x^2]*(1353611 + 5455144*x + 6837888*x^2 + 1963008*x^3 - 884736*x^4) + 485*Sqrt[3]*ArcTan
h[(2*Sqrt[2 + (17*x)/3 + 4*x^2])/(3 + 4*x)])/737280

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {\left (884736 x^{4}-1963008 x^{3}-6837888 x^{2}-5455144 x -1353611\right ) \sqrt {12 x^{2}+17 x +6}}{122880}+\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{589824}\)	\(65\)
trager	\(\left (-\frac {36}{5} x^{4}+\frac {639}{40} x^{3}+\frac {17807}{320} x^{2}+\frac {681893}{15360} x +\frac {1353611}{122880}\right ) \sqrt {12 x^{2}+17 x +6}-\frac {97 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+12 \sqrt {12 x^{2}+17 x +6}\right )}{294912}\)	\(76\)
default	\(-\frac {97 \left (17+24 x \right ) \sqrt {12 x^{2}+17 x +6}}{24576}+\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{589824}+\frac {7093 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{3840}-\frac {3 x^{2} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{5}+\frac {349 x \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{160}\)	\(96\)

[In]

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

[Out]

-1/122880*(884736*x^4-1963008*x^3-6837888*x^2-5455144*x-1353611)*(12*x^2+17*x+6)^(1/2)+97/589824*ln(1/12*(17/2
+12*x)*12^(1/2)+(12*x^2+17*x+6)^(1/2))*12^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {1}{122880} \, {\left (884736 \, x^{4} - 1963008 \, x^{3} - 6837888 \, x^{2} - 5455144 \, x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {97}{589824} \, \sqrt {3} \log \left (8 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} {\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \]

[In]

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

[Out]

-1/122880*(884736*x^4 - 1963008*x^3 - 6837888*x^2 - 5455144*x - 1353611)*sqrt(12*x^2 + 17*x + 6) + 97/589824*s
qrt(3)*log(8*sqrt(3)*sqrt(12*x^2 + 17*x + 6)*(24*x + 17) + 1152*x^2 + 1632*x + 577)

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\sqrt {12 x^{2} + 17 x + 6} \left (- \frac {36 x^{4}}{5} + \frac {639 x^{3}}{40} + \frac {17807 x^{2}}{320} + \frac {681893 x}{15360} + \frac {1353611}{122880}\right ) + \frac {97 \sqrt {3} \log {\left (24 x + 4 \sqrt {3} \sqrt {12 x^{2} + 17 x + 6} + 17 \right )}}{294912} \]

[In]

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

[Out]

sqrt(12*x**2 + 17*x + 6)*(-36*x**4/5 + 639*x**3/40 + 17807*x**2/320 + 681893*x/15360 + 1353611/122880) + 97*sq
rt(3)*log(24*x + 4*sqrt(3)*sqrt(12*x**2 + 17*x + 6) + 17)/294912

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {3}{5} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {349}{160} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {7093}{3840} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} - \frac {97}{1024} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x + \frac {97}{294912} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) - \frac {1649}{24576} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \]

[In]

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

[Out]

-3/5*(12*x^2 + 17*x + 6)^(3/2)*x^2 + 349/160*(12*x^2 + 17*x + 6)^(3/2)*x + 7093/3840*(12*x^2 + 17*x + 6)^(3/2)
- 97/1024*sqrt(12*x^2 + 17*x + 6)*x + 97/294912*sqrt(3)*log(4*sqrt(3)*sqrt(12*x^2 + 17*x + 6) + 24*x + 17) -
1649/24576*sqrt(12*x^2 + 17*x + 6)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {1}{122880} \, {\left (8 \, {\left (48 \, {\left (72 \, {\left (32 \, x - 71\right )} x - 17807\right )} x - 681893\right )} x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} - \frac {97}{294912} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]

[In]

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

[Out]

-1/122880*(8*(48*(72*(32*x - 71)*x - 17807)*x - 681893)*x - 1353611)*sqrt(12*x^2 + 17*x + 6) - 97/294912*sqrt(
3)*log(abs(-4*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 + 17*x + 6)) - 17))

Mupad [B] (verification not implemented)

Time = 13.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {3753\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{80}-\frac {417\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{10240}-\frac {3\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{5}+\frac {7093\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{368640}+\frac {349\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{160}+\frac {120581\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{2949120} \]

[In]

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

[Out]

(3753*(x/2 + 17/48)*(17*x + 12*x^2 + 6)^(1/2))/80 - (417*12^(1/2)*log((17*x + 12*x^2 + 6)^(1/2) + (12^(1/2)*(1
2*x + 17/2))/12))/10240 - (3*x^2*(17*x + 12*x^2 + 6)^(3/2))/5 + (7093*(17*x + 12*x^2 + 6)^(1/2)*(408*x + 1152*
x^2 - 291))/368640 + (349*x*(17*x + 12*x^2 + 6)^(3/2))/160 + (120581*12^(1/2)*log(2*(17*x + 12*x^2 + 6)^(1/2)
+ (12^(1/2)*(24*x + 17))/12))/2949120