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

Optimal. Leaf size=85 \[ \frac {1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {559}{864} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {559 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1728 \sqrt {3}} \]

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Rubi [A]  time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {559}{864} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {559 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1728 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(559*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/864 + ((109 - 18*x)*(2 + 5*x + 3*x^2)^(3/2))/108 - (559*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1728*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) \sqrt {2+5 x+3 x^2} \, dx &=\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac {559}{72} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1728}\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559}{864} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1728 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 0.79 \begin {gather*} \frac {-559 \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 (432 x^3-1896 x^2-7426 x-4539\right )}{5184} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-4539 - 7426*x - 1896*x^2 + 432*x^3) - 559*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15
*x + 9*x^2])])/5184

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IntegrateAlgebraic [A]  time = 0.41, size = 69, normalized size = 0.81 \begin {gather*} \frac {1}{864} \sqrt {3 x^2+5 x+2} \left (-432 x^3+1896 x^2+7426 x+4539\right )-\frac {559 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{864 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(4539 + 7426*x + 1896*x^2 - 432*x^3))/864 - (559*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]
*(1 + x))])/(864*Sqrt[3])

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fricas [A]  time = 0.40, size = 68, normalized size = 0.80 \begin {gather*} -\frac {1}{864} \, {\left (432 \, x^{3} - 1896 \, x^{2} - 7426 \, x - 4539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {559}{10368} \, \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)^(1/2),x, algorithm="fricas")

[Out]

-1/864*(432*x^3 - 1896*x^2 - 7426*x - 4539)*sqrt(3*x^2 + 5*x + 2) + 559/10368*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 = 0.25, size = 64, normalized size = 0.75 \begin {gather*} -\frac {1}{864} \, {\left (2 \, {\left (12 \, {\left (18 \, x - 79\right )} x - 3713\right )} x - 4539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {559}{5184} \, \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)^(1/2),x, algorithm="giac")

[Out]

-1/864*(2*(12*(18*x - 79)*x - 3713)*x - 4539)*sqrt(3*x^2 + 5*x + 2) + 559/5184*sqrt(3)*log(abs(-2*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.05, size = 79, normalized size = 0.93 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{6}-\frac {559 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{5184}+\frac {109 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{108}+\frac {559 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{864} \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)^(1/2),x)

[Out]

-1/6*(3*x^2+5*x+2)^(3/2)*x+109/108*(3*x^2+5*x+2)^(3/2)+559/864*(6*x+5)*(3*x^2+5*x+2)^(1/2)-559/5184*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 = 1.33, size = 87, normalized size = 1.02 \begin {gather*} -\frac {1}{6} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {109}{108} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {559}{144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {559}{5184} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {2795}{864} \, \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)^(1/2),x, algorithm="maxima")

[Out]

-1/6*(3*x^2 + 5*x + 2)^(3/2)*x + 109/108*(3*x^2 + 5*x + 2)^(3/2) + 559/144*sqrt(3*x^2 + 5*x + 2)*x - 559/5184*
sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 2795/864*sqrt(3*x^2 + 5*x + 2)

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mupad [B]  time = 0.70, size = 119, normalized size = 1.40 \begin {gather*} \frac {46\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{3}-\frac {23\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{108}+\frac {109\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{2592}-\frac {x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{6}+\frac {545\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{5184} \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)^(1/2),x)

[Out]

(46*(x/2 + 5/12)*(5*x + 3*x^2 + 2)^(1/2))/3 - (23*3^(1/2)*log((5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2))/
3))/108 + (109*(5*x + 3*x^2 + 2)^(1/2)*(30*x + 72*x^2 - 27))/2592 - (x*(5*x + 3*x^2 + 2)^(3/2))/6 + (545*3^(1/
2)*log(2*(5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(6*x + 5))/3))/5184

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 7 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 2 x^{2} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 15 \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)**(1/2),x)

[Out]

-Integral(-7*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(2*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-15*sqrt(3*x
**2 + 5*x + 2), x)

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