3.2451 \(\int (5-x) (2+5 x+3 x^2)^{7/2} \, dx\)

Optimal. Leaf size=149 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{9/2}+\frac{35}{288} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac{245 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{20736}+\frac{1225 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{995328}-\frac{1225 (6 x+5) \sqrt{3 x^2+5 x+2}}{7962624}+\frac{1225 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{15925248 \sqrt{3}} \]

[Out]

(-1225*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/7962624 + (1225*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/995328 - (245*(5 +
6*x)*(2 + 5*x + 3*x^2)^(5/2))/20736 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/288 - (2 + 5*x + 3*x^2)^(9/2)/27
+ (1225*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(15925248*Sqrt[3])

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Rubi [A]  time = 0.0502869, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 621, 206} \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{9/2}+\frac{35}{288} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac{245 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{20736}+\frac{1225 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{995328}-\frac{1225 (6 x+5) \sqrt{3 x^2+5 x+2}}{7962624}+\frac{1225 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{15925248 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-1225*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/7962624 + (1225*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/995328 - (245*(5 +
6*x)*(2 + 5*x + 3*x^2)^(5/2))/20736 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/288 - (2 + 5*x + 3*x^2)^(9/2)/27
+ (1225*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(15925248*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 )^{7/2} \, dx &=-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac{35}{6} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}-\frac{245}{576} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=-\frac{245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac{1225 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{41472}\\ &=\frac{1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac{245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}-\frac{1225 \int \sqrt{2+5 x+3 x^2} \, dx}{663552}\\ &=-\frac{1225 (5+6 x) \sqrt{2+5 x+3 x^2}}{7962624}+\frac{1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac{245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac{1225 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{15925248}\\ &=-\frac{1225 (5+6 x) \sqrt{2+5 x+3 x^2}}{7962624}+\frac{1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac{245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac{1225 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{7962624}\\ &=-\frac{1225 (5+6 x) \sqrt{2+5 x+3 x^2}}{7962624}+\frac{1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac{245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac{35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac{1225 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{15925248 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.133829, size = 119, normalized size = 0.8 \[ \frac{-64 \left (3 x^2+5 x+2\right )^{9/2}+210 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac{245 \left (6 \sqrt{3 x^2+5 x+2} \left (20736 x^5+86400 x^4+142128 x^3+115320 x^2+46166 x+7305\right )-5 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{27648}}{1728} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(210*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2) - 64*(2 + 5*x + 3*x^2)^(9/2) - (245*(6*Sqrt[2 + 5*x + 3*x^2]*(7305 + 46
166*x + 115320*x^2 + 142128*x^3 + 86400*x^4 + 20736*x^5) - 5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^
2])]))/27648)/1728

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Maple [A]  time = 0.004, size = 121, normalized size = 0.8 \begin{align*}{\frac{175+210\,x}{288} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{1225+1470\,x}{20736} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{6125+7350\,x}{995328} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{6125+7350\,x}{7962624}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1225\,\sqrt{3}}{47775744}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{1}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2),x)

[Out]

35/288*(5+6*x)*(3*x^2+5*x+2)^(7/2)-245/20736*(5+6*x)*(3*x^2+5*x+2)^(5/2)+1225/995328*(5+6*x)*(3*x^2+5*x+2)^(3/
2)-1225/7962624*(5+6*x)*(3*x^2+5*x+2)^(1/2)+1225/47775744*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2
)-1/27*(3*x^2+5*x+2)^(9/2)

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Maxima [A]  time = 1.74951, size = 215, normalized size = 1.44 \begin{align*} -\frac{1}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{35}{48} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{175}{288} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{245}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{1225}{20736} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1225}{165888} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{6125}{995328} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{1225}{1327104} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1225}{47775744} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{6125}{7962624} \, \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)^(7/2),x, algorithm="maxima")

[Out]

-1/27*(3*x^2 + 5*x + 2)^(9/2) + 35/48*(3*x^2 + 5*x + 2)^(7/2)*x + 175/288*(3*x^2 + 5*x + 2)^(7/2) - 245/3456*(
3*x^2 + 5*x + 2)^(5/2)*x - 1225/20736*(3*x^2 + 5*x + 2)^(5/2) + 1225/165888*(3*x^2 + 5*x + 2)^(3/2)*x + 6125/9
95328*(3*x^2 + 5*x + 2)^(3/2) - 1225/1327104*sqrt(3*x^2 + 5*x + 2)*x + 1225/47775744*sqrt(3)*log(2*sqrt(3)*sqr
t(3*x^2 + 5*x + 2) + 6*x + 5) - 6125/7962624*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.40117, size = 358, normalized size = 2.4 \begin{align*} -\frac{1}{7962624} \,{\left (23887872 \, x^{8} + 2488320 \, x^{7} - 452625408 \, x^{6} - 1507127040 \, x^{5} - 2320737408 \, x^{4} - 2013572880 \, x^{3} - 1014795048 \, x^{2} - 278256050 \, x - 32198883\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1225}{95551488} \, \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)^(7/2),x, algorithm="fricas")

[Out]

-1/7962624*(23887872*x^8 + 2488320*x^7 - 452625408*x^6 - 1507127040*x^5 - 2320737408*x^4 - 2013572880*x^3 - 10
14795048*x^2 - 278256050*x - 32198883)*sqrt(3*x^2 + 5*x + 2) + 1225/95551488*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 - 292 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 870 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 396 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 27 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 40 \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)**(7/2),x)

[Out]

-Integral(-292*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1339*x
**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-396*x**5*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2), x) - Integral(-40*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.23354, size = 120, normalized size = 0.81 \begin{align*} -\frac{1}{7962624} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \, x + 5\right )} x - 1819\right )} x - 218045\right )} x - 2014529\right )} x - 13983145\right )} x - 42283127\right )} x - 139128025\right )} x - 32198883\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1225}{47775744} \, \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)^(7/2),x, algorithm="giac")

[Out]

-1/7962624*(2*(12*(6*(8*(6*(36*(2*(48*x + 5)*x - 1819)*x - 218045)*x - 2014529)*x - 13983145)*x - 42283127)*x
- 139128025)*x - 32198883)*sqrt(3*x^2 + 5*x + 2) - 1225/47775744*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5))