3.2436 \(\int (5-x) (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac{35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac{175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac{175 (6 x+5) \sqrt{3 x^2+5 x+2}}{82944}-\frac{175 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{165888 \sqrt{3}} \]

[Out]

(175*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/82944 - (175*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/10368 + (35*(5 + 6*x)*(2
+ 5*x + 3*x^2)^(5/2))/216 - (2 + 5*x + 3*x^2)^(7/2)/21 - (175*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x
^2])])/(165888*Sqrt[3])

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Rubi [A]  time = 0.040626, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 621, 206} \[ -\frac{1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac{35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac{175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac{175 (6 x+5) \sqrt{3 x^2+5 x+2}}{82944}-\frac{175 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{165888 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(175*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/82944 - (175*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/10368 + (35*(5 + 6*x)*(2
+ 5*x + 3*x^2)^(5/2))/216 - (2 + 5*x + 3*x^2)^(7/2)/21 - (175*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x
^2])])/(165888*Sqrt[3])

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 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 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 (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac{35}{6} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac{175}{432} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac{175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac{35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac{175 \int \sqrt{2+5 x+3 x^2} \, dx}{6912}\\ &=\frac{175 (5+6 x) \sqrt{2+5 x+3 x^2}}{82944}-\frac{175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac{35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac{175 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{165888}\\ &=\frac{175 (5+6 x) \sqrt{2+5 x+3 x^2}}{82944}-\frac{175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac{35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac{175 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{82944}\\ &=\frac{175 (5+6 x) \sqrt{2+5 x+3 x^2}}{82944}-\frac{175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac{35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac{175 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{165888 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0689723, size = 108, normalized size = 0.86 \[ -\frac{1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac{35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac{175 \left (6 \sqrt{3 x^2+5 x+2} \left (144 x^3+360 x^2+290 x+75\right )+\sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{497664} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/216 - (2 + 5*x + 3*x^2)^(7/2)/21 - (175*(6*Sqrt[2 + 5*x + 3*x^2]*(75 +
290*x + 360*x^2 + 144*x^3) + Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]))/497664

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Maple [A]  time = 0.004, size = 102, normalized size = 0.8 \begin{align*}{\frac{175+210\,x}{216} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{875+1050\,x}{10368} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{875+1050\,x}{82944}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{175\,\sqrt{3}}{497664}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{1}{21} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/216*(5+6*x)*(3*x^2+5*x+2)^(5/2)-175/10368*(5+6*x)*(3*x^2+5*x+2)^(3/2)+175/82944*(5+6*x)*(3*x^2+5*x+2)^(1/2)
-175/497664*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-1/21*(3*x^2+5*x+2)^(7/2)

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Maxima [A]  time = 1.51076, size = 176, normalized size = 1.4 \begin{align*} -\frac{1}{21} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{35}{36} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{175}{216} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{175}{1728} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{875}{10368} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{175}{13824} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{175}{497664} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{875}{82944} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(3*x^2 + 5*x + 2)^(7/2) + 35/36*(3*x^2 + 5*x + 2)^(5/2)*x + 175/216*(3*x^2 + 5*x + 2)^(5/2) - 175/1728*(
3*x^2 + 5*x + 2)^(3/2)*x - 875/10368*(3*x^2 + 5*x + 2)^(3/2) + 175/13824*sqrt(3*x^2 + 5*x + 2)*x - 175/497664*
sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 875/82944*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.33699, size = 294, normalized size = 2.33 \begin{align*} -\frac{1}{580608} \,{\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{175}{995328} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/580608*(746496*x^6 - 1347840*x^5 - 13454208*x^4 - 26388720*x^3 - 23110872*x^2 - 9651790*x - 1568541)*sqrt(3
*x^2 + 5*x + 2) + 175/995328*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 96 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 165 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 113 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 15 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 9 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 20 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-96*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-113*x**
3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(9*x**5*sqrt(3*x**2 + 5*
x + 2), x) - Integral(-20*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.12032, size = 107, normalized size = 0.85 \begin{align*} -\frac{1}{580608} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (6 \,{\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{175}{497664} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/580608*(2*(12*(18*(8*(6*(36*x - 65)*x - 3893)*x - 61085)*x - 962953)*x - 4825895)*x - 1568541)*sqrt(3*x^2 +
5*x + 2) + 175/497664*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))