3.13.2 \(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=106 \[ \frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac {3 (111 x+385) \sqrt {3 x^2+2}}{280 (2 x+3)}+\frac {11727 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{560 \sqrt {35}}+\frac {33}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {811, 813, 844, 215, 725, 206} \begin {gather*} \frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac {3 (111 x+385) \sqrt {3 x^2+2}}{280 (2 x+3)}+\frac {11727 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{560 \sqrt {35}}+\frac {33}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

(-3*(385 + 111*x)*Sqrt[2 + 3*x^2])/(280*(3 + 2*x)) + ((229 + 456*x)*(2 + 3*x^2)^(3/2))/(420*(3 + 2*x)^3) + (33
*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (11727*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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 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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx &=\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}-\frac {1}{560} \int \frac {(-624+1332 x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac {3 (385+111 x) \sqrt {2+3 x^2}}{280 (3+2 x)}+\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac {\int \frac {-10656+55440 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{4480}\\ &=-\frac {3 (385+111 x) \sqrt {2+3 x^2}}{280 (3+2 x)}+\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac {99}{16} \int \frac {1}{\sqrt {2+3 x^2}} \, dx-\frac {11727}{560} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {3 (385+111 x) \sqrt {2+3 x^2}}{280 (3+2 x)}+\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac {33}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {11727}{560} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {3 (385+111 x) \sqrt {2+3 x^2}}{280 (3+2 x)}+\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac {33}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {11727 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{560 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 89, normalized size = 0.84 \begin {gather*} \frac {11727 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{560 \sqrt {35}}-\frac {\sqrt {3 x^2+2} \left (1260 x^3+24474 x^2+48747 x+30269\right )}{840 (2 x+3)^3}+\frac {33}{16} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

-1/840*(Sqrt[2 + 3*x^2]*(30269 + 48747*x + 24474*x^2 + 1260*x^3))/(3 + 2*x)^3 + (33*Sqrt[3]*ArcSinh[Sqrt[3/2]*
x])/16 + (11727*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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IntegrateAlgebraic [A]  time = 0.93, size = 116, normalized size = 1.09 \begin {gather*} -\frac {33}{16} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )-\frac {11727 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{280 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-1260 x^3-24474 x^2-48747 x-30269\right )}{840 (2 x+3)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

(Sqrt[2 + 3*x^2]*(-30269 - 48747*x - 24474*x^2 - 1260*x^3))/(840*(3 + 2*x)^3) - (11727*ArcTanh[3*Sqrt[3/35] +
2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(280*Sqrt[35]) - (33*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2
]])/16

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fricas [A]  time = 0.44, size = 151, normalized size = 1.42 \begin {gather*} \frac {121275 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 35181 \, \sqrt {35} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (1260 \, x^{3} + 24474 \, x^{2} + 48747 \, x + 30269\right )} \sqrt {3 \, x^{2} + 2}}{117600 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/117600*(121275*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 35181*sqrt
(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x
+ 9)) - 140*(1260*x^3 + 24474*x^2 + 48747*x + 30269)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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giac [B]  time = 0.28, size = 265, normalized size = 2.50 \begin {gather*} -\frac {33}{16} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {11727}{19600} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3}{16} \, \sqrt {3 \, x^{2} + 2} - \frac {\sqrt {3} {\left (14792 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 189285 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 141030 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 561630 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 166480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 50144\right )}}{1120 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-33/16*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 11727/19600*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*s
qrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/16*sqrt(3*x^2 + 2) -
1/1120*sqrt(3)*(14792*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 189285*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1410
30*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 561630*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 166480*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2)) - 50144)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^
3

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maple [B]  time = 0.06, size = 173, normalized size = 1.63 \begin {gather*} \frac {3933 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{9800}+\frac {1338 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{42875}+\frac {33 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{16}+\frac {11727 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{19600}-\frac {\left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{2450 \left (x +\frac {3}{2}\right )^{2}}-\frac {446 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{42875 \left (x +\frac {3}{2}\right )}-\frac {3909 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{85750}-\frac {11727 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{19600}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{840 \left (x +\frac {3}{2}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^4,x)

[Out]

-1/2450/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-446/42875/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-3909/85750*(-9
*x+3*(x+3/2)^2-19/4)^(3/2)+3933/9800*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+33/16*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-1172
7/19600*(-36*x+12*(x+3/2)^2-19)^(1/2)+11727/19600*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-
19)^(1/2))+1338/42875*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x-13/840/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(5/2)

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maxima [A]  time = 1.43, size = 150, normalized size = 1.42 \begin {gather*} \frac {3}{2450} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{1225 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {3933}{9800} \, \sqrt {3 \, x^{2} + 2} x + \frac {33}{16} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {11727}{19600} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {11727}{9800} \, \sqrt {3 \, x^{2} + 2} - \frac {223 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{1225 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

3/2450*(3*x^2 + 2)^(3/2) - 13/105*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2/1225*(3*x^2 + 2)^(5/2)/(4
*x^2 + 12*x + 9) + 3933/9800*sqrt(3*x^2 + 2)*x + 33/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 11727/19600*sqrt(35)*a
rcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 11727/9800*sqrt(3*x^2 + 2) - 223/1225*(3*x^2 +
2)^(3/2)/(2*x + 3)

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mupad [B]  time = 0.12, size = 133, normalized size = 1.25 \begin {gather*} \frac {33\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{16}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {11727\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{19600}+\frac {11727\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{19600}-\frac {1567\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{560\,\left (x+\frac {3}{2}\right )}+\frac {77\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{384\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^4,x)

[Out]

(33*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/16 - (3*3^(1/2)*(x^2 + 2/3)^(1/2))/16 - (11727*35^(1/2)*log(x + 3/2)
)/19600 + (11727*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/19600 - (1567*3^(1/2)*(x^2 +
2/3)^(1/2))/(560*(x + 3/2)) + (77*3^(1/2)*(x^2 + 2/3)^(1/2))/(32*(3*x + x^2 + 9/4)) - (455*3^(1/2)*(x^2 + 2/3)
^(1/2))/(384*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**4,x)

[Out]

Timed out

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