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

Optimal. Leaf size=149 \[ -\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}-\frac {873 \left (12 x^2+17 x+6\right )^{7/2}}{1792}+\frac {25091 (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}}{24576}-\frac {125455 (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}}{4718592}+\frac {125455 (24 x+17) \sqrt {12 x^2+17 x+6}}{150994944}-\frac {125455 \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{603979776 \sqrt {3}} \]

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Rubi [A]  time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1002, 742, 640, 612, 621, 206} \begin {gather*} -\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}-\frac {873 \left (12 x^2+17 x+6\right )^{7/2}}{1792}+\frac {25091 (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}}{24576}-\frac {125455 (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}}{4718592}+\frac {125455 (24 x+17) \sqrt {12 x^2+17 x+6}}{150994944}-\frac {125455 \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{603979776 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(125455*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/150994944 - (125455*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/471859
2 + (25091*(17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/24576 - (873*(6 + 17*x + 12*x^2)^(7/2))/1792 - ((10 - 3*x)*(
6 + 17*x + 12*x^2)^(7/2))/32 - (125455*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(603979776*Sq
rt[3])

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 612

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 621

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 640

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 742

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

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 (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx &=\int (10-3 x)^2 \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {1}{96} \int \left (11331-\frac {7857 x}{2}\right ) \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {75273}{512} \int \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \int \left (6+17 x+12 x^2\right )^{3/2} \, dx}{49152}\\ &=-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {125455 \int \sqrt {6+17 x+12 x^2} \, dx}{3145728}\\ &=\frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \int \frac {1}{\sqrt {6+17 x+12 x^2}} \, dx}{301989888}\\ &=\frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \operatorname {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {17+24 x}{\sqrt {6+17 x+12 x^2}}\right )}{150994944}\\ &=\frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \tanh ^{-1}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{603979776 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 87, normalized size = 0.58 \begin {gather*} \frac {12 \sqrt {12 x^2+17 x+6} \left (171228266496 x^7-732816211968 x^6-1190083166208 x^5+3438453030912 x^4+8974844476416 x^3+7899203409792 x^2+3132157281976 x+474999091769\right )-878185 \sqrt {3} \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {36 x^2+51 x+18}}\right )}{12683575296} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*Sqrt[6 + 17*x + 12*x^2]*(474999091769 + 3132157281976*x + 7899203409792*x^2 + 8974844476416*x^3 + 34384530
30912*x^4 - 1190083166208*x^5 - 732816211968*x^6 + 171228266496*x^7) - 878185*Sqrt[3]*ArcTanh[(17 + 24*x)/(4*S
qrt[18 + 51*x + 36*x^2])])/12683575296

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IntegrateAlgebraic [A]  time = 0.73, size = 92, normalized size = 0.62 \begin {gather*} \frac {\sqrt {12 x^2+17 x+6} \left (171228266496 x^7-732816211968 x^6-1190083166208 x^5+3438453030912 x^4+8974844476416 x^3+7899203409792 x^2+3132157281976 x+474999091769\right )}{1056964608}-\frac {125455 \tanh ^{-1}\left (\frac {2 \sqrt {12 x^2+17 x+6}}{\sqrt {3} (4 x+3)}\right )}{301989888 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[6 + 17*x + 12*x^2]*(474999091769 + 3132157281976*x + 7899203409792*x^2 + 8974844476416*x^3 + 34384530309
12*x^4 - 1190083166208*x^5 - 732816211968*x^6 + 171228266496*x^7))/1056964608 - (125455*ArcTanh[(2*Sqrt[6 + 17
*x + 12*x^2])/(Sqrt[3]*(3 + 4*x))])/(301989888*Sqrt[3])

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fricas [A]  time = 0.95, size = 88, normalized size = 0.59 \begin {gather*} \frac {1}{1056964608} \, {\left (171228266496 \, x^{7} - 732816211968 \, x^{6} - 1190083166208 \, x^{5} + 3438453030912 \, x^{4} + 8974844476416 \, x^{3} + 7899203409792 \, x^{2} + 3132157281976 \, x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{3623878656} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1056964608*(171228266496*x^7 - 732816211968*x^6 - 1190083166208*x^5 + 3438453030912*x^4 + 8974844476416*x^3
+ 7899203409792*x^2 + 3132157281976*x + 474999091769)*sqrt(12*x^2 + 17*x + 6) + 125455/3623878656*sqrt(3)*log(
-8*sqrt(3)*sqrt(12*x^2 + 17*x + 6)*(24*x + 17) + 1152*x^2 + 1632*x + 577)

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giac [A]  time = 0.30, size = 85, normalized size = 0.57 \begin {gather*} \frac {1}{1056964608} \, {\left (8 \, {\left (48 \, {\left (24 \, {\left (96 \, {\left (24 \, {\left (48 \, {\left (168 \, x - 719\right )} x - 56047\right )} x + 3886417\right )} x + 973832951\right )} x + 20570842213\right )} x + 391519660247\right )} x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{1811939328} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1056964608*(8*(48*(24*(96*(24*(48*(168*x - 719)*x - 56047)*x + 3886417)*x + 973832951)*x + 20570842213)*x +
391519660247)*x + 474999091769)*sqrt(12*x^2 + 17*x + 6) + 125455/1811939328*sqrt(3)*log(abs(-4*sqrt(3)*(2*sqrt
(3)*x - sqrt(12*x^2 + 17*x + 6)) - 17))

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maple [A]  time = 0.02, size = 147, normalized size = 0.99 \begin {gather*} \frac {27 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{5}}{2}-\frac {8613 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{4}}{112}+\frac {14991 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{3}}{1792}+\frac {4267751 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{2}}{14336}+\frac {129220757 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x}{458752}-\frac {125455 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right )}{3623878656}+\frac {2473875847 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{33030144}+\frac {125455 \left (24 x +17\right ) \sqrt {12 x^{2}+17 x +6}}{150994944} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/2*x^5*(12*x^2+17*x+6)^(3/2)-8613/112*x^4*(12*x^2+17*x+6)^(3/2)+14991/1792*x^3*(12*x^2+17*x+6)^(3/2)+4267751
/14336*x^2*(12*x^2+17*x+6)^(3/2)[phone]/458752*x*(12*x^2+17*x+6)^(3/2)[phone]/33030144*(12*x^2+17*x+6)^
(3/2)-125455/3623878656*ln(1/12*(17/2+12*x)*12^(1/2)+(12*x^2+17*x+6)^(1/2))*12^(1/2)+125455/150994944*(17+24*x
)*(12*x^2+17*x+6)^(1/2)

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maxima [A]  time = 1.00, size = 155, normalized size = 1.04 \begin {gather*} \frac {27}{2} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{5} - \frac {8613}{112} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{4} + \frac {14991}{1792} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{3} + \frac {4267751}{14336} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {129220757}{458752} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {2473875847}{33030144} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} + \frac {125455}{6291456} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x - \frac {125455}{1811939328} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) + \frac {2132735}{150994944} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/2*(12*x^2 + 17*x + 6)^(3/2)*x^5 - 8613/112*(12*x^2 + 17*x + 6)^(3/2)*x^4 + 14991/1792*(12*x^2 + 17*x + 6)^(
3/2)*x^3 + 4267751/14336*(12*x^2 + 17*x + 6)^(3/2)*x^2 + 129220757/458752*(12*x^2 + 17*x + 6)^(3/2)*x + 247387
5847/33030144*(12*x^2 + 17*x + 6)^(3/2) + 125455/6291456*sqrt(12*x^2 + 17*x + 6)*x - 125455/1811939328*sqrt(3)
*log(4*sqrt(3)*sqrt(12*x^2 + 17*x + 6) + 24*x + 17) + 2132735/150994944*sqrt(12*x^2 + 17*x + 6)

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mupad [B]  time = 5.08, size = 187, normalized size = 1.26 \begin {gather*} \frac {4267751\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{14336}+\frac {14991\,x^3\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{1792}-\frac {8613\,x^4\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{112}+\frac {27\,x^5\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{2}-\frac {146030443\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{88080384}+\frac {438091329\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{229376}+\frac {2473875847\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{3170893824}+\frac {129220757\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{458752}+\frac {42055889399\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{25367150592} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4267751*x^2*(17*x + 12*x^2 + 6)^(3/2))/14336 + (14991*x^3*(17*x + 12*x^2 + 6)^(3/2))/1792 - (8613*x^4*(17*x +
12*x^2 + 6)^(3/2))/112 + (27*x^5*(17*x + 12*x^2 + 6)^(3/2))/2 - (146030443*12^(1/2)*log((17*x + 12*x^2 + 6)^(
1/2) + (12^(1/2)*(12*x + 17/2))/12))/88080384 + (438091329*(x/2 + 17/48)*(17*x + 12*x^2 + 6)^(1/2))/229376 + (
2473875847*(17*x + 12*x^2 + 6)^(1/2)*(408*x + 1152*x^2 - 291))/3170893824 + (129220757*x*(17*x + 12*x^2 + 6)^(
3/2))/458752 + (42055889399*12^(1/2)*log(2*(17*x + 12*x^2 + 6)^(1/2) + (12^(1/2)*(24*x + 17))/12))/25367150592

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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