3.2434 \(\int (5-x) (3+2 x)^2 (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=156 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac{4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac{22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac{22535 (6 x+5) \sqrt{3 x^2+5 x+2}}{5971968}-\frac{22535 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{11943936 \sqrt{3}} \]

[Out]

(22535*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5971968 - (22535*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/746496 + (4507*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(5/2))/15552 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/27 + ((10211 + 4298*x)*(2 + 5*x
+ 3*x^2)^(7/2))/4536 - (22535*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(11943936*Sqrt[3])

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Rubi [A]  time = 0.0700996, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac{4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac{22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac{22535 (6 x+5) \sqrt{3 x^2+5 x+2}}{5971968}-\frac{22535 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{11943936 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(22535*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5971968 - (22535*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/746496 + (4507*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(5/2))/15552 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/27 + ((10211 + 4298*x)*(2 + 5*x
+ 3*x^2)^(7/2))/4536 - (22535*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(11943936*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 )^{5/2} \, dx &=-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{1}{27} \int (3+2 x) \left (\frac{931}{2}+307 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac{4507}{432} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac{22535 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{31104}\\ &=-\frac{22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac{4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac{22535 \int \sqrt{2+5 x+3 x^2} \, dx}{497664}\\ &=\frac{22535 (5+6 x) \sqrt{2+5 x+3 x^2}}{5971968}-\frac{22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac{4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac{22535 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{11943936}\\ &=\frac{22535 (5+6 x) \sqrt{2+5 x+3 x^2}}{5971968}-\frac{22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac{4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac{22535 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{5971968}\\ &=\frac{22535 (5+6 x) \sqrt{2+5 x+3 x^2}}{5971968}-\frac{22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac{4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac{1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac{22535 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{11943936 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0578163, size = 92, normalized size = 0.59 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (167215104 x^8+268240896 x^7-3275873280 x^6-15455860992 x^5-30355761024 x^4-32476001904 x^3-19762157208 x^2-6434937470 x-871825317\right )-157745 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{250822656} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-871825317 - 6434937470*x - 19762157208*x^2 - 32476001904*x^3 - 30355761024*x^4 - 1
5455860992*x^5 - 3275873280*x^6 + 268240896*x^7 + 167215104*x^8) - 157745*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6
+ 15*x + 9*x^2])])/250822656

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Maple [A]  time = 0.006, size = 134, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{163\,x}{324} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8699}{4536} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{22535+27042\,x}{15552} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{112675+135210\,x}{746496} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{112675+135210\,x}{5971968}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{22535\,\sqrt{3}}{35831808}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \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)^(5/2),x)

[Out]

-4/27*x^2*(3*x^2+5*x+2)^(7/2)+163/324*x*(3*x^2+5*x+2)^(7/2)+8699/4536*(3*x^2+5*x+2)^(7/2)+4507/15552*(5+6*x)*(
3*x^2+5*x+2)^(5/2)-22535/746496*(5+6*x)*(3*x^2+5*x+2)^(3/2)+22535/5971968*(5+6*x)*(3*x^2+5*x+2)^(1/2)-22535/35
831808*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.49433, size = 219, normalized size = 1.4 \begin{align*} -\frac{4}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{163}{324} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{8699}{4536} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{4507}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{22535}{15552} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{22535}{124416} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{112675}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{22535}{995328} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{22535}{35831808} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{112675}{5971968} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-4/27*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 163/324*(3*x^2 + 5*x + 2)^(7/2)*x + 8699/4536*(3*x^2 + 5*x + 2)^(7/2) + 45
07/2592*(3*x^2 + 5*x + 2)^(5/2)*x + 22535/15552*(3*x^2 + 5*x + 2)^(5/2) - 22535/124416*(3*x^2 + 5*x + 2)^(3/2)
*x - 112675/746496*(3*x^2 + 5*x + 2)^(3/2) + 22535/995328*sqrt(3*x^2 + 5*x + 2)*x - 22535/35831808*sqrt(3)*log
(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 112675/5971968*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.37087, size = 375, normalized size = 2.4 \begin{align*} -\frac{1}{41803776} \,{\left (167215104 \, x^{8} + 268240896 \, x^{7} - 3275873280 \, x^{6} - 15455860992 \, x^{5} - 30355761024 \, x^{4} - 32476001904 \, x^{3} - 19762157208 \, x^{2} - 6434937470 \, x - 871825317\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{22535}{71663616} \, \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)^(5/2),x, algorithm="fricas")

[Out]

-1/41803776*(167215104*x^8 + 268240896*x^7 - 3275873280*x^6 - 15455860992*x^5 - 30355761024*x^4 - 32476001904*
x^3 - 19762157208*x^2 - 6434937470*x - 871825317)*sqrt(3*x^2 + 5*x + 2) + 22535/71663616*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 - 1104 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 2717 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3381 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 2151 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 551 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 48 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 180 \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)**(5/2),x)

[Out]

-Integral(-1104*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2717*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3381
*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2151*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-551*x**5*sqrt(3*
x**2 + 5*x + 2), x) - Integral(48*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(36*x**7*sqrt(3*x**2 + 5*x + 2), x
) - Integral(-180*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.14311, size = 120, normalized size = 0.77 \begin{align*} -\frac{1}{41803776} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \, x + 77\right )} x - 13165\right )} x - 2236091\right )} x - 26350487\right )} x - 225527791\right )} x - 823423217\right )} x - 3217468735\right )} x - 871825317\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{22535}{35831808} \, \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)^(5/2),x, algorithm="giac")

[Out]

-1/41803776*(2*(12*(6*(8*(6*(36*(14*(48*x + 77)*x - 13165)*x - 2236091)*x - 26350487)*x - 225527791)*x - 82342
3217)*x - 3217468735)*x - 871825317)*sqrt(3*x^2 + 5*x + 2) + 22535/35831808*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2)) - 5))