3.2406 \(\int (5-x) (3+2 x)^4 \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{21} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^4+\frac{229}{378} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac{478}{315} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}}{68040}+\frac{25969 (6 x+5) \sqrt{3 x^2+5 x+2}}{15552}-\frac{25969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{31104 \sqrt{3}} \]

[Out]

(25969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/15552 + (478*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/315 + (229*(3 + 2*x)
^3*(2 + 5*x + 3*x^2)^(3/2))/378 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2))/21 + ((874301 + 378774*x)*(2 + 5*x + 3
*x^2)^(3/2))/68040 - (25969*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(31104*Sqrt[3])

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Rubi [A]  time = 0.0949389, antiderivative size = 160, 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} \[ -\frac{1}{21} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^4+\frac{229}{378} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac{478}{315} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}}{68040}+\frac{25969 (6 x+5) \sqrt{3 x^2+5 x+2}}{15552}-\frac{25969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{31104 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(25969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/15552 + (478*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/315 + (229*(3 + 2*x)
^3*(2 + 5*x + 3*x^2)^(3/2))/378 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2))/21 + ((874301 + 378774*x)*(2 + 5*x + 3
*x^2)^(3/2))/68040 - (25969*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(31104*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)^4 \sqrt{2+5 x+3 x^2} \, dx &=-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{1}{21} \int (3+2 x)^3 \left (\frac{707}{2}+229 x\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{1}{378} \int (3+2 x)^2 \left (\frac{22377}{2}+8604 x\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{\int (3+2 x) \left (\frac{482121}{2}+189387 x\right ) \sqrt{2+5 x+3 x^2} \, dx}{5670}\\ &=\frac{478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}+\frac{25969 \int \sqrt{2+5 x+3 x^2} \, dx}{1296}\\ &=\frac{25969 (5+6 x) \sqrt{2+5 x+3 x^2}}{15552}+\frac{478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac{25969 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{31104}\\ &=\frac{25969 (5+6 x) \sqrt{2+5 x+3 x^2}}{15552}+\frac{478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac{25969 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{15552}\\ &=\frac{25969 (5+6 x) \sqrt{2+5 x+3 x^2}}{15552}+\frac{478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac{25969 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{31104 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0623294, size = 82, normalized size = 0.51 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (1244160 x^6+1624320 x^5-28649088 x^4-123633360 x^3-208601544 x^2-161915450 x-47009103\right )-908915 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{3265920} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-47009103 - 161915450*x - 208601544*x^2 - 123633360*x^3 - 28649088*x^4 + 1624320*x^
5 + 1244160*x^6) - 908915*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/3265920

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Maple [A]  time = 0.015, size = 130, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{21} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{52\,{x}^{3}}{189} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5542\,{x}^{2}}{315} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{34931\,x}{756} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{25969\,\sqrt{3}}{93312}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{129845+155814\,x}{15552}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{2654033}{68040} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^4*(3*x^2+5*x+2)^(1/2),x)

[Out]

-16/21*x^4*(3*x^2+5*x+2)^(3/2)+52/189*x^3*(3*x^2+5*x+2)^(3/2)+5542/315*x^2*(3*x^2+5*x+2)^(3/2)+34931/756*x*(3*
x^2+5*x+2)^(3/2)-25969/93312*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+25969/15552*(5+6*x)*(3*x^2+
5*x+2)^(1/2)+2654033/68040*(3*x^2+5*x+2)^(3/2)

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Maxima [A]  time = 1.50728, size = 186, normalized size = 1.16 \begin{align*} -\frac{16}{21} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{4} + \frac{52}{189} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{3} + \frac{5542}{315} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{34931}{756} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{2654033}{68040} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{25969}{2592} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{25969}{93312} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{129845}{15552} \, \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)^4*(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-16/21*(3*x^2 + 5*x + 2)^(3/2)*x^4 + 52/189*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 5542/315*(3*x^2 + 5*x + 2)^(3/2)*x^2
+ 34931/756*(3*x^2 + 5*x + 2)^(3/2)*x + 2654033/68040*(3*x^2 + 5*x + 2)^(3/2) + 25969/2592*sqrt(3*x^2 + 5*x +
2)*x - 25969/93312*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 129845/15552*sqrt(3*x^2 + 5*x + 2
)

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Fricas [A]  time = 1.39077, size = 305, normalized size = 1.91 \begin{align*} -\frac{1}{544320} \,{\left (1244160 \, x^{6} + 1624320 \, x^{5} - 28649088 \, x^{4} - 123633360 \, x^{3} - 208601544 \, x^{2} - 161915450 \, x - 47009103\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{25969}{186624} \, \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)^4*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

-1/544320*(1244160*x^6 + 1624320*x^5 - 28649088*x^4 - 123633360*x^3 - 208601544*x^2 - 161915450*x - 47009103)*
sqrt(3*x^2 + 5*x + 2) + 25969/186624*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 - 999 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 864 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 264 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 16 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 16 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 405 \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)**4*(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-999*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-864*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-264*x*
*3*sqrt(3*x**2 + 5*x + 2), x) - Integral(16*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(16*x**5*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-405*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.19335, size = 107, normalized size = 0.67 \begin{align*} -\frac{1}{544320} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (36 \, x + 47\right )} x - 24869\right )} x - 858565\right )} x - 8691731\right )} x - 80957725\right )} x - 47009103\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{25969}{93312} \, \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)^4*(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/544320*(2*(12*(6*(8*(30*(36*x + 47)*x - 24869)*x - 858565)*x - 8691731)*x - 80957725)*x - 47009103)*sqrt(3*
x^2 + 5*x + 2) + 25969/93312*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))