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

Optimal. Leaf size=106 \[ \frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac {3 (4097 x+2943) \sqrt {3 x^2+2}}{19600 (2 x+3)^2}-\frac {39663 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{39200 \sqrt {35}}-\frac {3}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

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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 = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {811, 844, 215, 725, 206} \begin {gather*} \frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac {3 (4097 x+2943) \sqrt {3 x^2+2}}{19600 (2 x+3)^2}-\frac {39663 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{39200 \sqrt {35}}-\frac {3}{32} \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)^5,x]

[Out]

(3*(2943 + 4097*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)^2) + ((54 + 491*x)*(2 + 3*x^2)^(3/2))/(840*(3 + 2*x)^4) -
(3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (39663*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*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 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)^5} \, dx &=\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac {\int \frac {(-936+840 x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{1120}\\ &=\frac {3 (2943+4097 x) \sqrt {2+3 x^2}}{19600 (3+2 x)^2}+\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}+\frac {\int \frac {105408-352800 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{627200}\\ &=\frac {3 (2943+4097 x) \sqrt {2+3 x^2}}{19600 (3+2 x)^2}+\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac {9}{32} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {39663 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{39200}\\ &=\frac {3 (2943+4097 x) \sqrt {2+3 x^2}}{19600 (3+2 x)^2}+\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac {3}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {39663 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{39200}\\ &=\frac {3 (2943+4097 x) \sqrt {2+3 x^2}}{19600 (3+2 x)^2}+\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac {3}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {39663 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{39200 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 90, normalized size = 0.85 \begin {gather*} \frac {\frac {70 \sqrt {3 x^2+2} \left (250602 x^3+559764 x^2+718441 x+245943\right )}{(2 x+3)^4}-118989 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{4116000}-\frac {3}{32} \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)^5,x]

[Out]

(-3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + ((70*Sqrt[2 + 3*x^2]*(245943 + 718441*x + 559764*x^2 + 250602*x^3))/(3
+ 2*x)^4 - 118989*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/4116000

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IntegrateAlgebraic [A]  time = 1.17, size = 116, normalized size = 1.09 \begin {gather*} \frac {3}{32} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )+\frac {39663 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{19600 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (250602 x^3+559764 x^2+718441 x+245943\right )}{58800 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(245943 + 718441*x + 559764*x^2 + 250602*x^3))/(58800*(3 + 2*x)^4) + (39663*ArcTanh[3*Sqrt[3/
35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(19600*Sqrt[35]) + (3*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2
+ 3*x^2]])/32

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fricas [A]  time = 0.44, size = 166, normalized size = 1.57 \begin {gather*} \frac {385875 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 118989 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (250602 \, x^{3} + 559764 \, x^{2} + 718441 \, x + 245943\right )} \sqrt {3 \, x^{2} + 2}}{8232000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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)^5,x, algorithm="fricas")

[Out]

1/8232000*(385875*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
+ 118989*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 -
36*x + 43)/(4*x^2 + 12*x + 9)) + 140*(250602*x^3 + 559764*x^2 + 718441*x + 245943)*sqrt(3*x^2 + 2))/(16*x^4 +
96*x^3 + 216*x^2 + 216*x + 81)

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giac [A]  time = 0.24, size = 106, normalized size = 1.00 \begin {gather*} -\frac {1}{470400} \, {\left (\frac {35 \, {\left (\frac {35 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 1193 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 16227 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 125301 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {41767}{156800} \, \sqrt {3} \mathrm {sgn}\left (\frac {1}{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)^5,x, algorithm="giac")

[Out]

-1/470400*(35*(35*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 1193*sgn(1/(2*x + 3)))/(2*x + 3) + 16227*sgn(1/(2*x + 3))
)/(2*x + 3) - 125301*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 41767/156800*sqrt(3)*sgn(1/(
2*x + 3))

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maple [B]  time = 0.07, size = 194, normalized size = 1.83 \begin {gather*} -\frac {7227 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{686000}+\frac {17337 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{12005000}-\frac {3 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{32}-\frac {39663 \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 )}{1372000}-\frac {211 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{117600 \left (x +\frac {3}{2}\right )^{3}}-\frac {999 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{686000 \left (x +\frac {3}{2}\right )^{2}}-\frac {5779 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12005000 \left (x +\frac {3}{2}\right )}+\frac {13221 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6002500}+\frac {39663 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1372000}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}} \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)^5,x)

[Out]

-211/117600/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-999/686000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-5779/12
005000/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+13221/6002500*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-7227/686000*(-9*x+3*(
x+3/2)^2-19/4)^(1/2)*x-3/32*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+39663/1372000*(-36*x+12*(x+3/2)^2-19)^(1/2)-39663/1
372000*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+17337/12005000*(-9*x+3*(x+3/2)^2
-19/4)^(3/2)*x-13/2240/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(5/2)

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maxima [B]  time = 1.45, size = 183, normalized size = 1.73 \begin {gather*} \frac {2997}{686000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {211 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{14700 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {999 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{171500 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {7227}{686000} \, \sqrt {3 \, x^{2} + 2} x - \frac {3}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {39663}{1372000} \, \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 {39663}{686000} \, \sqrt {3 \, x^{2} + 2} - \frac {5779 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{686000 \, {\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)^5,x, algorithm="maxima")

[Out]

2997/686000*(3*x^2 + 2)^(3/2) - 13/140*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 211/14700*
(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 999/171500*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 7227/686000
*sqrt(3*x^2 + 2)*x - 3/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 39663/1372000*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*
x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 39663/686000*sqrt(3*x^2 + 2) - 5779/686000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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mupad [B]  time = 0.12, size = 155, normalized size = 1.46 \begin {gather*} \frac {39663\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1372000}-\frac {3\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {39663\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1372000}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1024\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {41767\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{156800\,\left (x+\frac {3}{2}\right )}-\frac {5409\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8960\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {1193\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536\,\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)^5,x)

[Out]

(39663*35^(1/2)*log(x + 3/2))/1372000 - (3*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 - (39663*35^(1/2)*log(x -
(3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1372000 - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(1024*((27*x)/2 + (27
*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (41767*3^(1/2)*(x^2 + 2/3)^(1/2))/(156800*(x + 3/2)) - (5409*3^(1/2)*(x^2 +
2/3)^(1/2))/(8960*(3*x + x^2 + 9/4)) + (1193*3^(1/2)*(x^2 + 2/3)^(1/2))/(1536*((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)**5,x)

[Out]

Timed out

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