3.2449 \(\int (5-x) (3+2 x)^2 (2+5 x+3 x^2)^{7/2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}}{26730}+\frac{5627 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{25920}-\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1866240}+\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{17915904}-\frac{39389 (6 x+5) \sqrt{3 x^2+5 x+2}}{143327232}+\frac{39389 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{286654464 \sqrt{3}} \]

[Out]

(-39389*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/143327232 + (39389*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/17915904 - (393
89*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/1866240 + (5627*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/25920 - ((3 + 2*x)^2*
(2 + 5*x + 3*x^2)^(9/2))/33 + ((47425 + 20358*x)*(2 + 5*x + 3*x^2)^(9/2))/26730 + (39389*ArcTanh[(5 + 6*x)/(2*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(286654464*Sqrt[3])

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Rubi [A]  time = 0.0837517, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}}{26730}+\frac{5627 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{25920}-\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1866240}+\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{17915904}-\frac{39389 (6 x+5) \sqrt{3 x^2+5 x+2}}{143327232}+\frac{39389 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{286654464 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-39389*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/143327232 + (39389*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/17915904 - (393
89*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/1866240 + (5627*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/25920 - ((3 + 2*x)^2*
(2 + 5*x + 3*x^2)^(9/2))/33 + ((47425 + 20358*x)*(2 + 5*x + 3*x^2)^(9/2))/26730 + (39389*ArcTanh[(5 + 6*x)/(2*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(286654464*Sqrt[3])

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])

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 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) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx &=-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{1}{33} \int (3+2 x) \left (\frac{1141}{2}+377 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac{5627}{540} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}-\frac{39389 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{51840}\\ &=-\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac{39389 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{746496}\\ &=\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}-\frac{39389 \int \sqrt{2+5 x+3 x^2} \, dx}{11943936}\\ &=-\frac{39389 (5+6 x) \sqrt{2+5 x+3 x^2}}{143327232}+\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac{39389 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{286654464}\\ &=-\frac{39389 (5+6 x) \sqrt{2+5 x+3 x^2}}{143327232}+\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac{39389 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{143327232}\\ &=-\frac{39389 (5+6 x) \sqrt{2+5 x+3 x^2}}{143327232}+\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac{39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac{5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac{1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac{39389 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{286654464 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.158176, size = 138, normalized size = 0.77 \[ \frac{1}{33} \left (-(2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{1}{810} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}+\frac{61897 \left (6 \sqrt{3 x^2+5 x+2} \left (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )+35 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{1433272320}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2)) + ((47425 + 20358*x)*(2 + 5*x + 3*x^2)^(9/2))/810 + (61897*(6*Sqrt[2 +
5*x + 3*x^2]*(1054785 + 9298342*x + 34858680*x^2 + 72023472*x^3 + 88560000*x^4 + 64800000*x^5 + 26127360*x^6
+ 4478976*x^7) + 35*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]))/1433272320)/33

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Maple [A]  time = 0.004, size = 153, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{33} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{197\,x}{495} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{28135+33762\,x}{25920} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{196945+236334\,x}{1866240} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{196945+236334\,x}{17915904} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{39389\,\sqrt{3}}{859963392}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{196945+236334\,x}{143327232}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{8027}{5346} \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+2*x)^2*(3*x^2+5*x+2)^(7/2),x)

[Out]

-4/33*x^2*(3*x^2+5*x+2)^(9/2)+197/495*x*(3*x^2+5*x+2)^(9/2)+5627/25920*(5+6*x)*(3*x^2+5*x+2)^(7/2)-39389/18662
40*(5+6*x)*(3*x^2+5*x+2)^(5/2)+39389/17915904*(5+6*x)*(3*x^2+5*x+2)^(3/2)+39389/859963392*ln(1/3*(5/2+3*x)*3^(
1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-39389/143327232*(5+6*x)*(3*x^2+5*x+2)^(1/2)+8027/5346*(3*x^2+5*x+2)^(9/2)

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Maxima [A]  time = 2.18057, size = 258, normalized size = 1.44 \begin{align*} -\frac{4}{33} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{197}{495} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{8027}{5346} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{5627}{4320} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{5627}{5184} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{39389}{311040} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{39389}{373248} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{39389}{2985984} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{196945}{17915904} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{39389}{23887872} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{39389}{859963392} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{196945}{143327232} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="maxima")

[Out]

-4/33*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 197/495*(3*x^2 + 5*x + 2)^(9/2)*x + 8027/5346*(3*x^2 + 5*x + 2)^(9/2) + 56
27/4320*(3*x^2 + 5*x + 2)^(7/2)*x + 5627/5184*(3*x^2 + 5*x + 2)^(7/2) - 39389/311040*(3*x^2 + 5*x + 2)^(5/2)*x
- 39389/373248*(3*x^2 + 5*x + 2)^(5/2) + 39389/2985984*(3*x^2 + 5*x + 2)^(3/2)*x + 196945/17915904*(3*x^2 + 5
*x + 2)^(3/2) - 39389/23887872*sqrt(3*x^2 + 5*x + 2)*x + 39389/859963392*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*
x + 2) + 6*x + 5) - 196945/143327232*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.49607, size = 471, normalized size = 2.63 \begin{align*} -\frac{1}{7882997760} \,{\left (77396705280 \, x^{10} + 261858852864 \, x^{9} - 1156531322880 \, x^{8} - 9116575930368 \, x^{7} - 25723491978240 \, x^{6} - 41190616509696 \, x^{5} - 41472321125760 \, x^{4} - 26847121235760 \, x^{3} - 10882383306360 \, x^{2} - 2519542755670 \, x - 254668717065\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{39389}{1719926784} \, \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+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="fricas")

[Out]

-1/7882997760*(77396705280*x^10 + 261858852864*x^9 - 1156531322880*x^8 - 9116575930368*x^7 - 25723491978240*x^
6 - 41190616509696*x^5 - 41472321125760*x^4 - 26847121235760*x^3 - 10882383306360*x^2 - 2519542755670*x - 2546
68717065)*sqrt(3*x^2 + 5*x + 2) + 39389/1719926784*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 - 3108 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 11494 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 23659 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 29358 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 22000 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 9112 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1341 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 324 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 108 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 360 \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+2*x)**2*(3*x**2+5*x+2)**(7/2),x)

[Out]

-Integral(-3108*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-11494*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-236
59*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-29358*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-22000*x**5*sq
rt(3*x**2 + 5*x + 2), x) - Integral(-9112*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1341*x**7*sqrt(3*x**2 +
5*x + 2), x) - Integral(324*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(108*x**9*sqrt(3*x**2 + 5*x + 2), x) - I
ntegral(-360*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.15644, size = 134, normalized size = 0.75 \begin{align*} -\frac{1}{7882997760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (60 \, x + 203\right )} x - 48415\right )} x - 18318737\right )} x - 103376945\right )} x - 5959290583\right )} x - 36000278755\right )} x - 186438341915\right )} x - 453432637765\right )} x - 1259771377835\right )} x - 254668717065\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{39389}{859963392} \, \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+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="giac")

[Out]

-1/7882997760*(2*(12*(6*(8*(6*(36*(2*(48*(54*(60*x + 203)*x - 48415)*x - 18318737)*x - 103376945)*x - 59592905
83)*x - 36000278755)*x - 186438341915)*x - 453432637765)*x - 1259771377835)*x - 254668717065)*sqrt(3*x^2 + 5*x
+ 2) - 39389/859963392*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))