3.1361 \(\int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx\)

Optimal. Leaf size=72 \[ \frac {1}{4} \sqrt {3 x^2+2} (13-x)-\frac {13}{8} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {121 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{8 \sqrt {3}} \]

[Out]

-121/24*arcsinh(1/2*x*6^(1/2))*3^(1/2)-13/8*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/4*(13-x)
*(3*x^2+2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 72, 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 = {815, 844, 215, 725, 206} \[ \frac {1}{4} \sqrt {3 x^2+2} (13-x)-\frac {13}{8} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {121 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{8 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((13 - x)*Sqrt[2 + 3*x^2])/4 - (121*ArcSinh[Sqrt[3/2]*x])/(8*Sqrt[3]) - (13*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/8

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 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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} \, dx &=\frac {1}{4} (13-x) \sqrt {2+3 x^2}+\frac {1}{24} \int \frac {276-726 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {1}{4} (13-x) \sqrt {2+3 x^2}-\frac {121}{8} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {455}{8} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {1}{4} (13-x) \sqrt {2+3 x^2}-\frac {121 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{8 \sqrt {3}}-\frac {455}{8} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {1}{4} (13-x) \sqrt {2+3 x^2}-\frac {121 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{8 \sqrt {3}}-\frac {13}{8} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.94 \[ \frac {1}{24} \left (-6 \sqrt {3 x^2+2} (x-13)-39 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-121 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-13 + x)*Sqrt[2 + 3*x^2] - 121*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 39*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqr
t[2 + 3*x^2])])/24

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fricas [A]  time = 0.63, size = 90, normalized size = 1.25 \[ -\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} {\left (x - 13\right )} + \frac {121}{48} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {13}{16} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/48*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 13/16*sqrt(35)*log
(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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giac [A]  time = 0.25, size = 104, normalized size = 1.44 \[ -\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} {\left (x - 13\right )} + \frac {121}{24} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {13}{8} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/24*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 13/8*sqrt(35)*log(-abs(-2*s
qrt(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)))

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maple [A]  time = 0.06, size = 72, normalized size = 1.00 \[ -\frac {\sqrt {3 x^{2}+2}\, x}{4}-\frac {121 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{24}-\frac {13 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{8}+\frac {13 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*x^2+2)^(1/2)*x-121/24*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+13/8*(12*(x+3/2)^2-36*x-19)^(1/2)-13/8*35^(1/2)*a
rctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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maxima [A]  time = 1.08, size = 70, normalized size = 0.97 \[ -\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} x - \frac {121}{24} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {13}{8} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {13}{4} \, \sqrt {3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3*x^2 + 2)*x - 121/24*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 13/8*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x +
3) - 2/3*sqrt(6)/abs(2*x + 3)) + 13/4*sqrt(3*x^2 + 2)

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mupad [B]  time = 0.17, size = 66, normalized size = 0.92 \[ \frac {\sqrt {35}\,\left (910\,\ln \left (x+\frac {3}{2}\right )-910\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{560}-\frac {121\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{24}-\frac {\sqrt {3}\,\left (\frac {3\,x}{4}-\frac {39}{4}\right )\,\sqrt {x^2+\frac {2}{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35^(1/2)*(910*log(x + 3/2) - 910*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)))/560 - (121*3^(1/2)*a
sinh((2^(1/2)*3^(1/2)*x)/2))/24 - (3^(1/2)*((3*x)/4 - 39/4)*(x^2 + 2/3)^(1/2))/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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