3.22.67 \(\int (5-x) (3+2 x)^2 \sqrt {2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac {1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac {2267 (6 x+5) \sqrt {3 x^2+5 x+2}}{2592}-\frac {2267 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5184 \sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac {2267 (6 x+5) \sqrt {3 x^2+5 x+2}}{2592}-\frac {2267 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5184 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(2267*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/2592 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/15 + ((7969 + 3006*x)*(2 +
5*x + 3*x^2)^(3/2))/1620 - (2267*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5184*Sqrt[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 779

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

Rule 832

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

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^2 \sqrt {2+5 x+3 x^2} \, dx &=-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{15} \int (3+2 x) \left (\frac {511}{2}+167 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}+\frac {2267}{216} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{5184}\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{2592}\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{5184 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 72, normalized size = 0.65 \begin {gather*} \frac {-11335 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (10368 x^4-23760 x^3-229416 x^2-375250 x-168627\right )}{77760} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-168627 - 375250*x - 229416*x^2 - 23760*x^3 + 10368*x^4) - 11335*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/77760

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IntegrateAlgebraic [A]  time = 0.53, size = 74, normalized size = 0.67 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-10368 x^4+23760 x^3+229416 x^2+375250 x+168627\right )}{12960}-\frac {2267 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2592 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(168627 + 375250*x + 229416*x^2 + 23760*x^3 - 10368*x^4))/12960 - (2267*ArcTanh[Sqrt[2
+ 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(2592*Sqrt[3])

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fricas [A]  time = 0.40, size = 73, normalized size = 0.66 \begin {gather*} -\frac {1}{12960} \, {\left (10368 \, x^{4} - 23760 \, x^{3} - 229416 \, x^{2} - 375250 \, x - 168627\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2267}{31104} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

-1/12960*(10368*x^4 - 23760*x^3 - 229416*x^2 - 375250*x - 168627)*sqrt(3*x^2 + 5*x + 2) + 2267/31104*sqrt(3)*l
og(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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giac [A]  time = 0.19, size = 69, normalized size = 0.63 \begin {gather*} -\frac {1}{12960} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (24 \, x - 55\right )} x - 9559\right )} x - 187625\right )} x - 168627\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2267}{15552} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/12960*(2*(12*(18*(24*x - 55)*x - 9559)*x - 187625)*x - 168627)*sqrt(3*x^2 + 5*x + 2) + 2267/15552*sqrt(3)*l
og(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.05, size = 96, normalized size = 0.87 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {19 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{18}-\frac {2267 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{15552}+\frac {6997 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{1620}+\frac {2267 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{2592} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(2*x+3)^2*(3*x^2+5*x+2)^(1/2),x)

[Out]

-4/15*(3*x^2+5*x+2)^(3/2)*x^2+19/18*(3*x^2+5*x+2)^(3/2)*x+6997/1620*(3*x^2+5*x+2)^(3/2)+2267/2592*(6*x+5)*(3*x
^2+5*x+2)^(1/2)-2267/15552*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))

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maxima [A]  time = 1.32, size = 104, normalized size = 0.95 \begin {gather*} -\frac {4}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {19}{18} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {6997}{1620} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {2267}{432} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {2267}{15552} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {11335}{2592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-4/15*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 19/18*(3*x^2 + 5*x + 2)^(3/2)*x + 6997/1620*(3*x^2 + 5*x + 2)^(3/2) + 2267
/432*sqrt(3*x^2 + 5*x + 2)*x - 2267/15552*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 11335/2592*
sqrt(3*x^2 + 5*x + 2)

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mupad [B]  time = 3.51, size = 136, normalized size = 1.24 \begin {gather*} \frac {386\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{9}-\frac {193\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{324}-\frac {4\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{15}+\frac {6997\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{38880}+\frac {19\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{18}+\frac {6997\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{15552} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 3)^2*(x - 5)*(5*x + 3*x^2 + 2)^(1/2),x)

[Out]

(386*(x/2 + 5/12)*(5*x + 3*x^2 + 2)^(1/2))/9 - (193*3^(1/2)*log((5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2)
)/3))/324 - (4*x^2*(5*x + 3*x^2 + 2)^(3/2))/15 + (6997*(5*x + 3*x^2 + 2)^(1/2)*(30*x + 72*x^2 - 27))/38880 + (
19*x*(5*x + 3*x^2 + 2)^(3/2))/18 + (6997*3^(1/2)*log(2*(5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(6*x + 5))/3))/15552

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 51 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 8 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 4 x^{3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 45 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**2*(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-51*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(4*x**3*sqr
t(3*x**2 + 5*x + 2), x) - Integral(-45*sqrt(3*x**2 + 5*x + 2), x)

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