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

Optimal. Leaf size=100 \[ -\frac {1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac {511}{9} x \sqrt {3 x^2+2}+\frac {1022 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \begin {gather*} -\frac {1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac {511}{9} x \sqrt {3 x^2+2}+\frac {1022 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(511*x*Sqrt[2 + 3*x^2])/9 + (17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/30 - ((3 + 2*x)^3*(2 + 3*x^2)^(3/2))/18 + (7*(8
98 + 267*x)*(2 + 3*x^2)^(3/2))/270 + (1022*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + 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 + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + 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])

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx &=-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {1}{18} \int (3+2 x)^2 (282+153 x) \sqrt {2+3 x^2} \, dx\\ &=\frac {17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {1}{270} \int (3+2 x) (11466+11214 x) \sqrt {2+3 x^2} \, dx\\ &=\frac {17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac {1022}{9} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {511}{9} x \sqrt {2+3 x^2}+\frac {17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac {1022}{9} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {511}{9} x \sqrt {2+3 x^2}+\frac {17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac {1022 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 60, normalized size = 0.60 \begin {gather*} \frac {1}{270} \left (10220 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (360 x^5-216 x^4-8445 x^3-21918 x^2-21120 x-14516\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-14516 - 21120*x - 21918*x^2 - 8445*x^3 - 216*x^4 + 360*x^5)) + 10220*Sqrt[3]*ArcSinh[Sqrt
[3/2]*x])/270

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IntegrateAlgebraic [A]  time = 0.30, size = 71, normalized size = 0.71 \begin {gather*} \frac {1}{270} \sqrt {3 x^2+2} \left (-360 x^5+216 x^4+8445 x^3+21918 x^2+21120 x+14516\right )-\frac {1022 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(14516 + 21120*x + 21918*x^2 + 8445*x^3 + 216*x^4 - 360*x^5))/270 - (1022*Log[-(Sqrt[3]*x) +
Sqrt[2 + 3*x^2]])/(9*Sqrt[3])

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fricas [A]  time = 0.42, size = 65, normalized size = 0.65 \begin {gather*} -\frac {1}{270} \, {\left (360 \, x^{5} - 216 \, x^{4} - 8445 \, x^{3} - 21918 \, x^{2} - 21120 \, x - 14516\right )} \sqrt {3 \, x^{2} + 2} + \frac {511}{27} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/270*(360*x^5 - 216*x^4 - 8445*x^3 - 21918*x^2 - 21120*x - 14516)*sqrt(3*x^2 + 2) + 511/27*sqrt(3)*log(-sqrt
(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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giac [A]  time = 0.17, size = 57, normalized size = 0.57 \begin {gather*} -\frac {1}{270} \, {\left (3 \, {\left ({\left ({\left (24 \, {\left (5 \, x - 3\right )} x - 2815\right )} x - 7306\right )} x - 7040\right )} x - 14516\right )} \sqrt {3 \, x^{2} + 2} - \frac {1022}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-1/270*(3*(((24*(5*x - 3)*x - 2815)*x - 7306)*x - 7040)*x - 14516)*sqrt(3*x^2 + 2) - 1022/27*sqrt(3)*log(-sqrt
(3)*x + sqrt(3*x^2 + 2))

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maple [A]  time = 0.06, size = 77, normalized size = 0.77 \begin {gather*} -\frac {4 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{3}}{9}+\frac {4 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {193 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{18}+\frac {511 \sqrt {3 x^{2}+2}\, x}{9}+\frac {1022 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{27}+\frac {3629 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{135} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*(3*x^2+2)^(3/2)*x^3+193/18*(3*x^2+2)^(3/2)*x+511/9*(3*x^2+2)^(1/2)*x+1022/27*arcsinh(1/2*6^(1/2)*x)*3^(1/
2)+4/15*(3*x^2+2)^(3/2)*x^2+3629/135*(3*x^2+2)^(3/2)

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maxima [A]  time = 1.10, size = 76, normalized size = 0.76 \begin {gather*} -\frac {4}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + \frac {4}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {193}{18} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3629}{135} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {511}{9} \, \sqrt {3 \, x^{2} + 2} x + \frac {1022}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-4/9*(3*x^2 + 2)^(3/2)*x^3 + 4/15*(3*x^2 + 2)^(3/2)*x^2 + 193/18*(3*x^2 + 2)^(3/2)*x + 3629/135*(3*x^2 + 2)^(3
/2) + 511/9*sqrt(3*x^2 + 2)*x + 1022/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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mupad [B]  time = 1.69, size = 50, normalized size = 0.50 \begin {gather*} \frac {1022\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-4\,x^5+\frac {12\,x^4}{5}+\frac {563\,x^3}{6}+\frac {3653\,x^2}{15}+\frac {704\,x}{3}+\frac {7258}{45}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1022*3^(1/2)*asinh((6^(1/2)*x)/2))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((704*x)/3 + (3653*x^2)/15 + (563*x^3)/6 +
(12*x^4)/5 - 4*x^5 + 7258/45))/3

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sympy [A]  time = 20.72, size = 150, normalized size = 1.50 \begin {gather*} - \frac {4 x^{7}}{\sqrt {3 x^{2} + 2}} + \frac {547 x^{5}}{6 \sqrt {3 x^{2} + 2}} + \frac {1705 x^{3}}{18 \sqrt {3 x^{2} + 2}} + \frac {135 x \sqrt {3 x^{2} + 2}}{2} + \frac {193 x}{9 \sqrt {3 x^{2} + 2}} + \frac {16 \sqrt {2} \left (\frac {3 x^{2}}{2} + 1\right )^{\frac {5}{2}}}{45} - \frac {16 \sqrt {2} \left (\frac {3 x^{2}}{2} + 1\right )^{\frac {3}{2}}}{27} + 27 \left (3 x^{2} + 2\right )^{\frac {3}{2}} + \frac {1022 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**3*(3*x**2+2)**(1/2),x)

[Out]

-4*x**7/sqrt(3*x**2 + 2) + 547*x**5/(6*sqrt(3*x**2 + 2)) + 1705*x**3/(18*sqrt(3*x**2 + 2)) + 135*x*sqrt(3*x**2
+ 2)/2 + 193*x/(9*sqrt(3*x**2 + 2)) + 16*sqrt(2)*(3*x**2/2 + 1)**(5/2)/45 - 16*sqrt(2)*(3*x**2/2 + 1)**(3/2)/
27 + 27*(3*x**2 + 2)**(3/2) + 1022*sqrt(3)*asinh(sqrt(6)*x/2)/27

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