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

Optimal. Leaf size=104 \[ -\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac {3 (12 x+37) \sqrt {3 x^2+2}}{4 (2 x+3)}-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8 \sqrt {35}}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 104, 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 = {813, 844, 215, 725, 206} \begin {gather*} -\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac {3 (12 x+37) \sqrt {3 x^2+2}}{4 (2 x+3)}-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8 \sqrt {35}}-\frac {111}{8} \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)^3,x]

[Out]

(3*(37 + 12*x)*Sqrt[2 + 3*x^2])/(4*(3 + 2*x)) - ((8 + x)*(2 + 3*x^2)^(3/2))/(4*(3 + 2*x)^2) - (111*Sqrt[3]*Arc
Sinh[Sqrt[3/2]*x])/8 - (1143*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8*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 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)^3} \, dx &=-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {3}{32} \int \frac {(16-192 x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}+\frac {3}{256} \int \frac {1536-7104 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {333}{8} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {1143}{8} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143}{8} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8 \sqrt {35}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 89, normalized size = 0.86 \begin {gather*} -\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8 \sqrt {35}}-\frac {\sqrt {3 x^2+2} \left (3 x^3-48 x^2-328 x-317\right )}{4 (2 x+3)^2}-\frac {111}{8} \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)^3,x]

[Out]

-1/4*(Sqrt[2 + 3*x^2]*(-317 - 328*x - 48*x^2 + 3*x^3))/(3 + 2*x)^2 - (111*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (1
143*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.75, size = 116, normalized size = 1.12 \begin {gather*} \frac {111}{8} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )+\frac {1143 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{4 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-3 x^3+48 x^2+328 x+317\right )}{4 (2 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(317 + 328*x + 48*x^2 - 3*x^3))/(4*(3 + 2*x)^2) + (1143*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x
- (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(4*Sqrt[35]) + (111*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/8

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 136, normalized size = 1.31 \begin {gather*} \frac {3885 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 1143 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\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 (3 \, x^{3} - 48 \, x^{2} - 328 \, x - 317\right )} \sqrt {3 \, x^{2} + 2}}{560 \, {\left (4 \, x^{2} + 12 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/560*(3885*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 1143*sqrt(35)*(4*x^2 + 12*
x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3*x^3 - 48*x^
2 - 328*x - 317)*sqrt(3*x^2 + 2))/(4*x^2 + 12*x + 9)

________________________________________________________________________________________

giac [B]  time = 0.28, size = 219, normalized size = 2.11 \begin {gather*} -\frac {3}{16} \, \sqrt {3 \, x^{2} + 2} {\left (x - 19\right )} + \frac {111}{8} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {1143}{280} \, \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 {5 \, {\left (1452 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 3013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6528 \, \sqrt {3} x + 1048 \, \sqrt {3} + 6528 \, \sqrt {3 \, x^{2} + 2}\right )}}{64 \, {\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 )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-3/16*sqrt(3*x^2 + 2)*(x - 19) + 111/8*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1143/280*sqrt(35)*log(-abs(
-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 +
2))) + 5/64*(1452*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 3013*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6528*sqrt(3
)*x + 1048*sqrt(3) + 6528*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^
2 + 2)) - 2)^2

________________________________________________________________________________________

maple [A]  time = 0.05, size = 152, normalized size = 1.46 \begin {gather*} -\frac {171 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{70}-\frac {561 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{4900}-\frac {111 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{8}-\frac {1143 \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 )}{280}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}+\frac {187 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {381 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1225}+\frac {1143 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{280} \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)^3,x)

[Out]

-13/280/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+187/4900/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+381/1225*(-9*x+
3*(x+3/2)^2-19/4)^(3/2)-171/70*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-111/8*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+1143/280*(
-36*x+12*(x+3/2)^2-19)^(1/2)-1143/280*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))-5
61/4900*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 122, normalized size = 1.17 \begin {gather*} \frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {171}{70} \, \sqrt {3 \, x^{2} + 2} x - \frac {111}{8} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1143}{280} \, \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 {1143}{140} \, \sqrt {3 \, x^{2} + 2} + \frac {187 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{280 \, {\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)^3,x, algorithm="maxima")

[Out]

39/280*(3*x^2 + 2)^(3/2) - 13/70*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 171/70*sqrt(3*x^2 + 2)*x - 111/8*sqrt(
3)*arcsinh(1/2*sqrt(6)*x) + 1143/280*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) +
1143/140*sqrt(3*x^2 + 2) + 187/280*(3*x^2 + 2)^(3/2)/(2*x + 3)

________________________________________________________________________________________

mupad [B]  time = 1.82, size = 117, normalized size = 1.12 \begin {gather*} \frac {1143\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{280}+\frac {57\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {111\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{8}-\frac {1143\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{280}+\frac {655\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64\,\left (x+\frac {3}{2}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {3\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{16} \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)^3,x)

[Out]

(1143*35^(1/2)*log(x + 3/2))/280 + (57*3^(1/2)*(x^2 + 2/3)^(1/2))/16 - (111*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/
2))/8 - (1143*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/280 + (655*3^(1/2)*(x^2 + 2/3)^(
1/2))/(64*(x + 3/2)) - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(128*(3*x + x^2 + 9/4)) - (3*3^(1/2)*x*(x^2 + 2/3)^(1/2
))/16

________________________________________________________________________________________

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)**3,x)

[Out]

Timed out

________________________________________________________________________________________