3.2433 \(\int (5-x) (3+2 x)^3 (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac{169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{182917 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{182917 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

[Out]

(182917*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (182917*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/4478976 + (1829
17*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (169*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/405 - ((3 + 2*x)^3*(2
+ 5*x + 3*x^2)^(7/2))/30 + ((477101 + 213878*x)*(2 + 5*x + 3*x^2)^(7/2))/136080 - (182917*ArcTanh[(5 + 6*x)/(
2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*Sqrt[3])

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Rubi [A]  time = 0.0975396, antiderivative size = 181, 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} \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac{169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{182917 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{182917 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(182917*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (182917*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/4478976 + (1829
17*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (169*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/405 - ((3 + 2*x)^3*(2
+ 5*x + 3*x^2)^(7/2))/30 + ((477101 + 213878*x)*(2 + 5*x + 3*x^2)^(7/2))/136080 - (182917*ArcTanh[(5 + 6*x)/(
2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*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)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{1}{30} \int (3+2 x)^2 \left (\frac{1029}{2}+338 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{1}{810} \int (3+2 x) \left (\frac{42451}{2}+15277 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac{182917 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960}\\ &=\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac{182917 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624}\\ &=-\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac{182917 \int \sqrt{2+5 x+3 x^2} \, dx}{2985984}\\ &=\frac{182917 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac{182917 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{71663616}\\ &=\frac{182917 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac{182917 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{35831808}\\ &=\frac{182917 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac{182917 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{71663616 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0693963, size = 97, normalized size = 0.54 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (9029615616 x^9+29262643200 x^8-147947046912 x^7-1086687912960 x^6-2893044950784 x^5-4253933381760 x^4-3762746217360 x^3-1995914277480 x^2-585749416130 x-73178684475\right )-6402095 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{7524679680} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-73178684475 - 585749416130*x - 1995914277480*x^2 - 3762746217360*x^3 - 42539333817
60*x^4 - 2893044950784*x^5 - 1086687912960*x^6 - 147947046912*x^7 + 29262643200*x^8 + 9029615616*x^9) - 640209
5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/7524679680

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Maple [A]  time = 0.006, size = 151, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{3}}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{38\,{x}^{2}}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{46453\,x}{9720} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{914585+1097502\,x}{466560} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{914585+1097502\,x}{4478976} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{182917\,\sqrt{3}}{214990848}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{914585+1097502\,x}{35831808}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{173137}{27216} \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+2*x)^3*(3*x^2+5*x+2)^(5/2),x)

[Out]

-4/15*x^3*(3*x^2+5*x+2)^(7/2)+38/81*x^2*(3*x^2+5*x+2)^(7/2)+46453/9720*x*(3*x^2+5*x+2)^(7/2)+182917/466560*(5+
6*x)*(3*x^2+5*x+2)^(5/2)-182917/4478976*(5+6*x)*(3*x^2+5*x+2)^(3/2)-182917/214990848*ln(1/3*(5/2+3*x)*3^(1/2)+
(3*x^2+5*x+2)^(1/2))*3^(1/2)+182917/35831808*(5+6*x)*(3*x^2+5*x+2)^(1/2)+173137/27216*(3*x^2+5*x+2)^(7/2)

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Maxima [A]  time = 1.49248, size = 242, normalized size = 1.34 \begin{align*} -\frac{4}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{38}{81} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{46453}{9720} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{173137}{27216} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{182917}{77760} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{182917}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{182917}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{914585}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{182917}{5971968} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{182917}{214990848} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{914585}{35831808} \, \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)^3*(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-4/15*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 38/81*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 46453/9720*(3*x^2 + 5*x + 2)^(7/2)*x +
173137/27216*(3*x^2 + 5*x + 2)^(7/2) + 182917/77760*(3*x^2 + 5*x + 2)^(5/2)*x + 182917/93312*(3*x^2 + 5*x + 2
)^(5/2) - 182917/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 914585/4478976*(3*x^2 + 5*x + 2)^(3/2) + 182917/5971968*sq
rt(3*x^2 + 5*x + 2)*x - 182917/214990848*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 914585/35831
808*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.45742, size = 431, normalized size = 2.38 \begin{align*} -\frac{1}{1254113280} \,{\left (9029615616 \, x^{9} + 29262643200 \, x^{8} - 147947046912 \, x^{7} - 1086687912960 \, x^{6} - 2893044950784 \, x^{5} - 4253933381760 \, x^{4} - 3762746217360 \, x^{3} - 1995914277480 \, x^{2} - 585749416130 \, x - 73178684475\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{182917}{429981696} \, \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)^3*(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/1254113280*(9029615616*x^9 + 29262643200*x^8 - 147947046912*x^7 - 1086687912960*x^6 - 2893044950784*x^5 - 4
253933381760*x^4 - 3762746217360*x^3 - 1995914277480*x^2 - 585749416130*x - 73178684475)*sqrt(3*x^2 + 5*x + 2)
+ 182917/429981696*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 - 3672 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 10359 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 15577 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 13215 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 5955 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 958 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 204 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 72 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 540 \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)**3*(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-3672*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10359*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-155
77*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-13215*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-5955*x**5*sqr
t(3*x**2 + 5*x + 2), x) - Integral(-958*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(204*x**7*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(72*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(-540*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.12036, size = 127, normalized size = 0.7 \begin{align*} -\frac{1}{1254113280} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \,{\left (54 \, x + 175\right )} x - 42469\right )} x - 4367155\right )} x - 418553957\right )} x - 3692650505\right )} x - 26130182065\right )} x - 83163094895\right )} x - 292874708065\right )} x - 73178684475\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{182917}{214990848} \, \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)^3*(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/1254113280*(2*(12*(6*(8*(6*(36*(14*(48*(54*x + 175)*x - 42469)*x - 4367155)*x - 418553957)*x - 3692650505)*
x - 26130182065)*x - 83163094895)*x - 292874708065)*x - 73178684475)*sqrt(3*x^2 + 5*x + 2) + 182917/214990848*
sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))