3.14.57 \(\int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx\) [1357]

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

[Out]

17/30*(3+2*x)^2*(3*x^2+2)^(3/2)-1/18*(3+2*x)^3*(3*x^2+2)^(3/2)+7/270*(898+267*x)*(3*x^2+2)^(3/2)+1022/27*arcsi
nh(1/2*x*6^(1/2))*3^(1/2)+511/9*x*(3*x^2+2)^(1/2)

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Rubi [A]
time = 0.03, 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 = {847, 794, 201, 221} \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 201

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 221

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 794

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 847

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 = 0.17, size = 71, normalized size = 0.71 \begin {gather*} -\frac {1}{270} \sqrt {2+3 x^2} \left (-14516-21120 x-21918 x^2-8445 x^3-216 x^4+360 x^5\right )-\frac {1022 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/270*(Sqrt[2 + 3*x^2]*(-14516 - 21120*x - 21918*x^2 - 8445*x^3 - 216*x^4 + 360*x^5)) - (1022*Log[-(Sqrt[3]*x
) + Sqrt[2 + 3*x^2]])/(9*Sqrt[3])

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Maple [A]
time = 0.73, size = 77, normalized size = 0.77

method	result	size
risch	\(-\frac {\left (360 x^{5}-216 x^{4}-8445 x^{3}-21918 x^{2}-21120 x -14516\right ) \sqrt {3 x^{2}+2}}{270}+\frac {1022 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}\)	\(50\)
trager	\(\left (-\frac {4}{3} x^{5}+\frac {4}{5} x^{4}+\frac {563}{18} x^{3}+\frac {3653}{45} x^{2}+\frac {704}{9} x +\frac {7258}{135}\right ) \sqrt {3 x^{2}+2}+\frac {1022 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\)	\(66\)
default	\(-\frac {4 x^{3} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{9}+\frac {193 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{18}+\frac {511 x \sqrt {3 x^{2}+2}}{9}+\frac {1022 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}+\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{15}+\frac {3629 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{135}\)	\(77\)
meijerg	\(-\frac {45 \sqrt {3}\, \left (-\sqrt {6}\, \sqrt {\pi }\, x \sqrt {\frac {3 x^{2}}{2}+1}-2 \sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{2 \sqrt {\pi }}-\frac {4 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}} \left (-\frac {9 x^{2}}{2}+2\right )}{15}\right )}{9 \sqrt {\pi }}-\frac {14 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (9 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{12}+\frac {\sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{\sqrt {\pi }}-\frac {81 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (3 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}\right )}{2 \sqrt {\pi }}+\frac {16 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-90 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{120}-\frac {\sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}\)	\(217\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, 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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Fricas [A]
time = 2.83, 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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Sympy [A]
time = 13.07, 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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Giac [A]
time = 1.32, 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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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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