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

Optimal. Leaf size=73 \[ -\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}+\frac {19 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{\sqrt {35}}+2 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}+\frac {19 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{\sqrt {35}}+2 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((8 + x)*Sqrt[2 + 3*x^2])/(2*(3 + 2*x)) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/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) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx &=-\frac {(8+x) \sqrt {2+3 x^2}}{2 (3+2 x)}-\frac {1}{8} \int \frac {8-96 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {(8+x) \sqrt {2+3 x^2}}{2 (3+2 x)}+6 \int \frac {1}{\sqrt {2+3 x^2}} \, dx-19 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {(8+x) \sqrt {2+3 x^2}}{2 (3+2 x)}+2 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+19 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {(8+x) \sqrt {2+3 x^2}}{2 (3+2 x)}+2 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {19 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{\sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 71, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {3 x^2+2} (x+8)}{4 x+6}+\frac {19 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{\sqrt {35}}+2 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((8 + x)*Sqrt[2 + 3*x^2])/(6 + 4*x)) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt
[2 + 3*x^2])])/Sqrt[35]

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IntegrateAlgebraic [A]  time = 0.59, size = 102, normalized size = 1.40 \begin {gather*} \frac {\sqrt {3 x^2+2} (-x-8)}{2 (2 x+3)}-2 \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )-\frac {38 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{\sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8 - x)*Sqrt[2 + 3*x^2])/(2*(3 + 2*x)) - (38*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqr
t[35]])/Sqrt[35] - 2*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]]

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fricas [A]  time = 0.43, size = 109, normalized size = 1.49 \begin {gather*} \frac {70 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 19 \, \sqrt {35} {\left (2 \, x + 3\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 ) - 35 \, \sqrt {3 \, x^{2} + 2} {\left (x + 8\right )}}{70 \, {\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)^(1/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/70*(70*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 19*sqrt(35)*(2*x + 3)*log((sqrt(35)*s
qrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 35*sqrt(3*x^2 + 2)*(x + 8))/(2*x + 3)

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giac [B]  time = 0.57, size = 285, normalized size = 3.90 \begin {gather*} \frac {19}{35} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2 \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {13}{8} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {3 \, {\left (3 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{4 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

19/35*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x +
3)) - 2*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(s
qrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 13/8*sqrt(-18/(2*x
+ 3) + 35/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 3/4*(3*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*
x + 3))*sgn(1/(2*x + 3)) - sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2
*x + 3))^2 - 3)

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maple [A]  time = 0.05, size = 98, normalized size = 1.34 \begin {gather*} \frac {39 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{70}+2 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )+\frac {19 \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 )}{35}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{70 \left (x +\frac {3}{2}\right )}-\frac {19 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/70/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)-19/35*(-36*x+12*(x+3/2)^2-19)^(1/2)+2*arcsinh(1/2*6^(1/2)*x)*3^(1/
2)+19/35*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+39/70*x*(3*(x+3/2)^2-9*x-19/4)
^(1/2)

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maxima [A]  time = 1.25, size = 76, normalized size = 1.04 \begin {gather*} 2 \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {19}{35} \, \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 {1}{4} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{4 \, {\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)^(1/2)/(3+2*x)^2,x, algorithm="maxima")

[Out]

2*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 19/35*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3
)) - 1/4*sqrt(3*x^2 + 2) - 13/4*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 0.11, size = 80, normalized size = 1.10 \begin {gather*} 2\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4}-\frac {19\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{35}+\frac {19\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{35}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8\,\left (x+\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2) - (3^(1/2)*(x^2 + 2/3)^(1/2))/4 - (19*35^(1/2)*log(x + 3/2))/35 + (19*3
5^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/35 - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(8*(x + 3/2
))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {5 \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-5*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(x*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x)

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