3.22.81 \(\int (5-x) (2+5 x+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=103 \[ -\frac {1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac {35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {35 (6 x+5) \sqrt {3 x^2+5 x+2}}{1152}+\frac {35 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2304 \sqrt {3}} \]

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Rubi [A]  time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \begin {gather*} -\frac {1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac {35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {35 (6 x+5) \sqrt {3 x^2+5 x+2}}{1152}+\frac {35 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2304 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1152 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/144 - (2 + 5*x + 3*x^2)^(5
/2)/15 + (35*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(2304*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 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]

Rubi steps

\begin {align*} \int (5-x) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}-\frac {35}{96} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{2304}\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{1152}\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{2304 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 72, normalized size = 0.70 \begin {gather*} \frac {175 \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 (3456 x^4-13680 x^3-48792 x^2-43070 x-11589\right )}{34560} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-11589 - 43070*x - 48792*x^2 - 13680*x^3 + 3456*x^4) + 175*Sqrt[3]*ArcTanh[(5 + 6*x
)/(2*Sqrt[6 + 15*x + 9*x^2])])/34560

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IntegrateAlgebraic [A]  time = 0.46, size = 74, normalized size = 0.72 \begin {gather*} \frac {35 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{1152 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-3456 x^4+13680 x^3+48792 x^2+43070 x+11589\right )}{5760} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(11589 + 43070*x + 48792*x^2 + 13680*x^3 - 3456*x^4))/5760 + (35*ArcTanh[Sqrt[2 + 5*x +
3*x^2]/(Sqrt[3]*(1 + x))])/(1152*Sqrt[3])

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fricas [A]  time = 0.40, size = 73, normalized size = 0.71 \begin {gather*} -\frac {1}{5760} \, {\left (3456 \, x^{4} - 13680 \, x^{3} - 48792 \, x^{2} - 43070 \, x - 11589\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {35}{13824} \, \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*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

-1/5760*(3456*x^4 - 13680*x^3 - 48792*x^2 - 43070*x - 11589)*sqrt(3*x^2 + 5*x + 2) + 35/13824*sqrt(3)*log(4*sq
rt(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 = 0.67 \begin {gather*} -\frac {1}{5760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (24 \, x - 95\right )} x - 2033\right )} x - 21535\right )} x - 11589\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {35}{6912} \, \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*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

-1/5760*(2*(12*(6*(24*x - 95)*x - 2033)*x - 21535)*x - 11589)*sqrt(3*x^2 + 5*x + 2) - 35/6912*sqrt(3)*log(abs(
-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.08, size = 83, normalized size = 0.81 \begin {gather*} \frac {35 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{6912}+\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{144}-\frac {35 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{1152}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/144*(6*x+5)*(3*x^2+5*x+2)^(3/2)-35/1152*(6*x+5)*(3*x^2+5*x+2)^(1/2)+35/6912*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2
)+(3*x^2+5*x+2)^(1/2))-1/15*(3*x^2+5*x+2)^(5/2)

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maxima [A]  time = 1.33, size = 101, normalized size = 0.98 \begin {gather*} -\frac {1}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {35}{24} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {175}{144} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {35}{192} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {35}{6912} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {175}{1152} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(3*x^2 + 5*x + 2)^(5/2) + 35/24*(3*x^2 + 5*x + 2)^(3/2)*x + 175/144*(3*x^2 + 5*x + 2)^(3/2) - 35/192*sqr
t(3*x^2 + 5*x + 2)*x + 35/6912*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 175/1152*sqrt(3*x^2 +
5*x + 2)

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mupad [B]  time = 2.87, size = 130, normalized size = 1.26 \begin {gather*} \frac {35\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{6912}-\frac {5\,\left (6\,x+5\right )\,\sqrt {3\,x^2+5\,x+2}}{1152}+\frac {5\,\left (3\,x+\frac {5}{2}\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{12}-\frac {5\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{16}+\frac {5\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{24}+\frac {25\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{144}-\frac {{\left (3\,x^2+5\,x+2\right )}^{5/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35*3^(1/2)*log((5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2))/3))/6912 - (5*(6*x + 5)*(5*x + 3*x^2 + 2)^(1/2
))/1152 + (5*(3*x + 5/2)*(5*x + 3*x^2 + 2)^(3/2))/12 - (5*(x/2 + 5/12)*(5*x + 3*x^2 + 2)^(1/2))/16 + (5*x*(5*x
+ 3*x^2 + 2)^(3/2))/24 + (25*(5*x + 3*x^2 + 2)^(3/2))/144 - (5*x + 3*x^2 + 2)^(5/2)/15

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 23 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 3 x^{3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 10 \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*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-23*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(3*x**3*sq
rt(3*x**2 + 5*x + 2), x) - Integral(-10*sqrt(3*x**2 + 5*x + 2), x)

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