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

Optimal. Leaf size=108 \[ \frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {839 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {839 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {839 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {839 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-839*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (839*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/2592 + ((161 - 30*x)*(2
+ 5*x + 3*x^2)^(5/2))/270 + (839*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*Sqrt[3])

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])

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 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]

Rubi steps

\begin {align*} \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {839 \int \sqrt {2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 0.71 \begin {gather*} \frac {4195 \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 (103680 x^5-210816 x^4-2032560 x^3-3567288 x^2-2406950 x-561921\right )}{622080} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-561921 - 2406950*x - 3567288*x^2 - 2032560*x^3 - 210816*x^4 + 103680*x^5) + 4195*S
qrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/622080

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IntegrateAlgebraic [A]  time = 0.61, size = 79, normalized size = 0.73 \begin {gather*} \frac {839 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{20736 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-103680 x^5+210816 x^4+2032560 x^3+3567288 x^2+2406950 x+561921\right )}{103680} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(561921 + 2406950*x + 3567288*x^2 + 2032560*x^3 + 210816*x^4 - 103680*x^5))/103680 + (8
39*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(20736*Sqrt[3])

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fricas [A]  time = 0.40, size = 78, normalized size = 0.72 \begin {gather*} -\frac {1}{103680} \, {\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {839}{248832} \, \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)*(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

-1/103680*(103680*x^5 - 210816*x^4 - 2032560*x^3 - 3567288*x^2 - 2406950*x - 561921)*sqrt(3*x^2 + 5*x + 2) + 8
39/248832*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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giac [A]  time = 0.24, size = 74, normalized size = 0.69 \begin {gather*} -\frac {1}{103680} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {839}{124416} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)*(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

-1/103680*(2*(12*(18*(8*(30*x - 61)*x - 4705)*x - 148637)*x - 1203475)*x - 561921)*sqrt(3*x^2 + 5*x + 2) - 839
/124416*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.05, size = 98, normalized size = 0.91 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{9}+\frac {839 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{124416}+\frac {161 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{270}+\frac {839 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {839 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{20736} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(3*x^2+5*x+2)^(5/2)*x+161/270*(3*x^2+5*x+2)^(5/2)+839/2592*(6*x+5)*(3*x^2+5*x+2)^(3/2)-839/20736*(6*x+5)*
(3*x^2+5*x+2)^(1/2)+839/124416*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))

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maxima [A]  time = 1.35, size = 116, normalized size = 1.07 \begin {gather*} -\frac {1}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {161}{270} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {839}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {4195}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {839}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {839}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {4195}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)*(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-1/9*(3*x^2 + 5*x + 2)^(5/2)*x + 161/270*(3*x^2 + 5*x + 2)^(5/2) + 839/432*(3*x^2 + 5*x + 2)^(3/2)*x + 4195/25
92*(3*x^2 + 5*x + 2)^(3/2) - 839/3456*sqrt(3*x^2 + 5*x + 2)*x + 839/124416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 6*x + 5) - 4195/20736*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 89 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 30 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)*(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-89*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-76*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-11*x**3*
sqrt(3*x**2 + 5*x + 2), x) - Integral(6*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-30*sqrt(3*x**2 + 5*x + 2),
x)

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