3.2450 \(\int (5-x) (3+2 x) (2+5 x+3 x^2)^{7/2} \, dx\)

Optimal. Leaf size=154 \[ \frac{1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac{1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac{9793 (6 x+5) \sqrt{3 x^2+5 x+2}}{47775744}+\frac{9793 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{95551488 \sqrt{3}} \]

[Out]

(-9793*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/47775744 + (9793*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/5971968 - (9793*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(5/2))/622080 + (1399*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/8640 + ((265 - 54*x)*(2 + 5
*x + 3*x^2)^(9/2))/810 + (9793*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(95551488*Sqrt[3])

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Rubi [A]  time = 0.0583206, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {779, 612, 621, 206} \[ \frac{1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac{1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac{9793 (6 x+5) \sqrt{3 x^2+5 x+2}}{47775744}+\frac{9793 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{95551488 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9793*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/47775744 + (9793*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/5971968 - (9793*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(5/2))/622080 + (1399*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/8640 + ((265 - 54*x)*(2 + 5
*x + 3*x^2)^(9/2))/810 + (9793*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(95551488*Sqrt[3])

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) \left (2+5 x+3 x^2\right )^{7/2} \, dx &=\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac{1399}{180} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}-\frac{9793 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{17280}\\ &=-\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac{9793 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{248832}\\ &=\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}-\frac{9793 \int \sqrt{2+5 x+3 x^2} \, dx}{3981312}\\ &=-\frac{9793 (5+6 x) \sqrt{2+5 x+3 x^2}}{47775744}+\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac{9793 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{95551488}\\ &=-\frac{9793 (5+6 x) \sqrt{2+5 x+3 x^2}}{47775744}+\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac{9793 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{47775744}\\ &=-\frac{9793 (5+6 x) \sqrt{2+5 x+3 x^2}}{47775744}+\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac{9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac{1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac{1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac{9793 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{95551488 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0661322, size = 111, normalized size = 0.72 \[ \frac{1399 \left (6 \sqrt{3 x^2+5 x+2} \left (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )+35 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{1433272320}-\frac{1}{810} (54 x-265) \left (3 x^2+5 x+2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-265 + 54*x)*(2 + 5*x + 3*x^2)^(9/2))/810 + (1399*(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

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Maple [A]  time = 0.006, size = 136, normalized size = 0.9 \begin{align*} -{\frac{x}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{53}{162} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{6995+8394\,x}{8640} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{48965+58758\,x}{622080} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{48965+58758\,x}{5971968} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{48965+58758\,x}{47775744}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{9793\,\sqrt{3}}{286654464}\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)*(3*x^2+5*x+2)^(7/2),x)

[Out]

-1/15*x*(3*x^2+5*x+2)^(9/2)+53/162*(3*x^2+5*x+2)^(9/2)+1399/8640*(5+6*x)*(3*x^2+5*x+2)^(7/2)-9793/622080*(5+6*
x)*(3*x^2+5*x+2)^(5/2)+9793/5971968*(5+6*x)*(3*x^2+5*x+2)^(3/2)-9793/47775744*(5+6*x)*(3*x^2+5*x+2)^(1/2)+9793
/286654464*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 = 1.69182, size = 235, normalized size = 1.53 \begin{align*} -\frac{1}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{53}{162} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{1399}{1440} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{1399}{1728} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{9793}{103680} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{9793}{124416} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{9793}{995328} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{48965}{5971968} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{9793}{7962624} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{9793}{286654464} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{48965}{47775744} \, \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*x^2+5*x+2)^(7/2),x, algorithm="maxima")

[Out]

-1/15*(3*x^2 + 5*x + 2)^(9/2)*x + 53/162*(3*x^2 + 5*x + 2)^(9/2) + 1399/1440*(3*x^2 + 5*x + 2)^(7/2)*x + 1399/
1728*(3*x^2 + 5*x + 2)^(7/2) - 9793/103680*(3*x^2 + 5*x + 2)^(5/2)*x - 9793/124416*(3*x^2 + 5*x + 2)^(5/2) + 9
793/995328*(3*x^2 + 5*x + 2)^(3/2)*x + 48965/5971968*(3*x^2 + 5*x + 2)^(3/2) - 9793/7962624*sqrt(3*x^2 + 5*x +
2)*x + 9793/286654464*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 48965/47775744*sqrt(3*x^2 + 5*
x + 2)

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Fricas [A]  time = 1.34801, size = 413, normalized size = 2.68 \begin{align*} -\frac{1}{238878720} \,{\left (1289945088 \, x^{9} + 2269347840 \, x^{8} - 23529056256 \, x^{7} - 117850567680 \, x^{6} - 250227954432 \, x^{5} - 302902600320 \, x^{4} - 224097754320 \, x^{3} - 100612822920 \, x^{2} - 25257845290 \, x - 2726071095\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{9793}{573308928} \, \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*x^2+5*x+2)^(7/2),x, algorithm="fricas")

[Out]

-1/238878720*(1289945088*x^9 + 2269347840*x^8 - 23529056256*x^7 - 117850567680*x^6 - 250227954432*x^5 - 302902
600320*x^4 - 224097754320*x^3 - 100612822920*x^2 - 25257845290*x - 2726071095)*sqrt(3*x^2 + 5*x + 2) + 9793/57
3308928*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 - 956 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3194 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 5757 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 5948 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3368 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 792 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 81 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 54 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 120 \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*x**2+5*x+2)**(7/2),x)

[Out]

-Integral(-956*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3194*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-5757*
x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-5948*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3368*x**5*sqrt(3*
x**2 + 5*x + 2), x) - Integral(-792*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(81*x**7*sqrt(3*x**2 + 5*x + 2),
x) - Integral(54*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(-120*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.20975, size = 127, normalized size = 0.82 \begin{align*} -\frac{1}{238878720} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \, x + 95\right )} x - 47279\right )} x - 473615\right )} x - 36201961\right )} x - 262936285\right )} x - 1556234405\right )} x - 4192200955\right )} x - 12628922645\right )} x - 2726071095\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{9793}{286654464} \, \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*x^2+5*x+2)^(7/2),x, algorithm="giac")

[Out]

-1/238878720*(2*(12*(6*(8*(6*(36*(2*(48*(54*x + 95)*x - 47279)*x - 473615)*x - 36201961)*x - 262936285)*x - 15
56234405)*x - 4192200955)*x - 12628922645)*x - 2726071095)*sqrt(3*x^2 + 5*x + 2) - 9793/286654464*sqrt(3)*log(
abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))