3.2419 \(\int (5-x) (3+2 x)^3 (2+5 x+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{1}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac{12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac{12277 (6 x+5) \sqrt{3 x^2+5 x+2}}{165888}+\frac{12277 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{331776 \sqrt{3}} \]

[Out]

(-12277*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/165888 + (12277*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/20736 + (67*(3 + 2
*x)^2*(2 + 5*x + 3*x^2)^(5/2))/126 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/24 + ((75451 + 33210*x)*(2 + 5*x +
3*x^2)^(5/2))/15120 + (12277*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(331776*Sqrt[3])

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Rubi [A]  time = 0.082797, antiderivative size = 158, 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} \[ -\frac{1}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac{12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac{12277 (6 x+5) \sqrt{3 x^2+5 x+2}}{165888}+\frac{12277 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{331776 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12277*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/165888 + (12277*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/20736 + (67*(3 + 2
*x)^2*(2 + 5*x + 3*x^2)^(5/2))/126 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/24 + ((75451 + 33210*x)*(2 + 5*x +
3*x^2)^(5/2))/15120 + (12277*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(331776*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 )^{3/2} \, dx &=-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{1}{24} \int (3+2 x)^2 \left (\frac{819}{2}+268 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{1}{504} \int (3+2 x) \left (\frac{27209}{2}+9963 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac{12277}{864} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}-\frac{12277 \int \sqrt{2+5 x+3 x^2} \, dx}{13824}\\ &=-\frac{12277 (5+6 x) \sqrt{2+5 x+3 x^2}}{165888}+\frac{12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac{12277 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{331776}\\ &=-\frac{12277 (5+6 x) \sqrt{2+5 x+3 x^2}}{165888}+\frac{12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac{12277 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{165888}\\ &=-\frac{12277 (5+6 x) \sqrt{2+5 x+3 x^2}}{165888}+\frac{12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac{67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac{12277 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{331776 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.057488, size = 87, normalized size = 0.55 \[ \frac{429695 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (17418240 x^7+25297920 x^6-368236800 x^5-1650151296 x^4-2993047920 x^3-2762417688 x^2-1276112350 x-233137461\right )}{34836480} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-233137461 - 1276112350*x - 2762417688*x^2 - 2993047920*x^3 - 1650151296*x^4 - 3682
36800*x^5 + 25297920*x^6 + 17418240*x^7) + 429695*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/34836
480

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Maple [A]  time = 0.006, size = 132, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{x}^{2}}{126} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1063\,x}{168} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{61385+73662\,x}{20736} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{12277\,\sqrt{3}}{995328}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{61385+73662\,x}{165888}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{130801}{15120} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{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)^(3/2),x)

[Out]

-1/3*x^3*(3*x^2+5*x+2)^(5/2)+79/126*x^2*(3*x^2+5*x+2)^(5/2)+1063/168*x*(3*x^2+5*x+2)^(5/2)+12277/20736*(5+6*x)
*(3*x^2+5*x+2)^(3/2)+12277/995328*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-12277/165888*(5+6*x)*(
3*x^2+5*x+2)^(1/2)+130801/15120*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 1.50472, size = 203, normalized size = 1.28 \begin{align*} -\frac{1}{3} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{79}{126} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{1063}{168} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{130801}{15120} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{12277}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{61385}{20736} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{12277}{27648} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{12277}{995328} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{61385}{165888} \, \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)^(3/2),x, algorithm="maxima")

[Out]

-1/3*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 79/126*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 1063/168*(3*x^2 + 5*x + 2)^(5/2)*x + 1
30801/15120*(3*x^2 + 5*x + 2)^(5/2) + 12277/3456*(3*x^2 + 5*x + 2)^(3/2)*x + 61385/20736*(3*x^2 + 5*x + 2)^(3/
2) - 12277/27648*sqrt(3*x^2 + 5*x + 2)*x + 12277/995328*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5)
- 61385/165888*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.3734, size = 339, normalized size = 2.15 \begin{align*} -\frac{1}{5806080} \,{\left (17418240 \, x^{7} + 25297920 \, x^{6} - 368236800 \, x^{5} - 1650151296 \, x^{4} - 2993047920 \, x^{3} - 2762417688 \, x^{2} - 1276112350 \, x - 233137461\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{12277}{1990656} \, \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)^(3/2),x, algorithm="fricas")

[Out]

-1/5806080*(17418240*x^7 + 25297920*x^6 - 368236800*x^5 - 1650151296*x^4 - 2993047920*x^3 - 2762417688*x^2 - 1
276112350*x - 233137461)*sqrt(3*x^2 + 5*x + 2) + 12277/1990656*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 - 1161 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1872 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1367 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 382 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 28 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 24 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 270 \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)**(3/2),x)

[Out]

-Integral(-1161*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1872*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1367
*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-382*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(28*x**5*sqrt(3*x**
2 + 5*x + 2), x) - Integral(24*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-270*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.20258, size = 113, normalized size = 0.72 \begin{align*} -\frac{1}{5806080} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (12 \,{\left (42 \, x + 61\right )} x - 10655\right )} x - 1432423\right )} x - 20785055\right )} x - 115100737\right )} x - 638056175\right )} x - 233137461\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{12277}{995328} \, \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)^(3/2),x, algorithm="giac")

[Out]

-1/5806080*(2*(12*(6*(8*(30*(12*(42*x + 61)*x - 10655)*x - 1432423)*x - 20785055)*x - 115100737)*x - 638056175
)*x - 233137461)*sqrt(3*x^2 + 5*x + 2) - 12277/995328*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2)) - 5))