3.1405 \(\int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {10 \sqrt {3 x^2+2}}{343 (2 x+3)}-\frac {16 \sqrt {3 x^2+2}}{245 (2 x+3)^2}-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}-\frac {57 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1715 \sqrt {35}} \]

[Out]

-57/60025*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-13/105*(3*x^2+2)^(1/2)/(3+2*x)^3-16/245*(3*x
^2+2)^(1/2)/(3+2*x)^2-10/343*(3*x^2+2)^(1/2)/(3+2*x)

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Rubi [A]  time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {835, 807, 725, 206} \[ -\frac {10 \sqrt {3 x^2+2}}{343 (2 x+3)}-\frac {16 \sqrt {3 x^2+2}}{245 (2 x+3)^2}-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}-\frac {57 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1715 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) - (16*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (10*Sqrt[2 + 3*x^2])/(343*(
3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*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 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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 835

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

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx &=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {1}{105} \int \frac {-123+78 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {16 \sqrt {2+3 x^2}}{245 (3+2 x)^2}+\frac {\int \frac {1590-1440 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{7350}\\ &=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {16 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {10 \sqrt {2+3 x^2}}{343 (3+2 x)}+\frac {57 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1715}\\ &=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {16 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {10 \sqrt {2+3 x^2}}{343 (3+2 x)}-\frac {57 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1715}\\ &=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {16 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {10 \sqrt {2+3 x^2}}{343 (3+2 x)}-\frac {57 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1715 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 65, normalized size = 0.66 \[ -\frac {\sqrt {3 x^2+2} \left (600 x^2+2472 x+2995\right )}{5145 (2 x+3)^3}-\frac {57 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1715 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5145*(Sqrt[2 + 3*x^2]*(2995 + 2472*x + 600*x^2))/(3 + 2*x)^3 - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x
^2])])/(1715*Sqrt[35])

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fricas [A]  time = 0.55, size = 104, normalized size = 1.05 \[ \frac {171 \, \sqrt {35} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 70 \, {\left (600 \, x^{2} + 2472 \, x + 2995\right )} \sqrt {3 \, x^{2} + 2}}{360150 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/360150*(171*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)) - 70*(600*x^2 + 2472*x + 2995)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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giac [B]  time = 0.32, size = 232, normalized size = 2.34 \[ \frac {57}{60025} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {\sqrt {3} {\left (38 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 855 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 2250 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 13290 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3448 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 800\right )}}{3430 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

57/60025*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))) - 1/3430*sqrt(3)*(38*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 855*(sqrt(3)*x
- sqrt(3*x^2 + 2))^4 + 2250*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 13290*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 +
3448*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 800)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sq
rt(3*x^2 + 2)) - 2)^3

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maple [A]  time = 0.06, size = 95, normalized size = 0.96 \[ -\frac {57 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{60025}-\frac {4 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{245 \left (x +\frac {3}{2}\right )^{2}}-\frac {5 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{343 \left (x +\frac {3}{2}\right )}-\frac {13 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{840 \left (x +\frac {3}{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^4/(3*x^2+2)^(1/2),x)

[Out]

-4/245/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-5/343/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-57/60025*35^(1/2)*a
rctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))-13/840/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(1/2)

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maxima [A]  time = 1.38, size = 104, normalized size = 1.05 \[ \frac {57}{60025} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {16 \, \sqrt {3 \, x^{2} + 2}}{245 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {10 \, \sqrt {3 \, x^{2} + 2}}{343 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

57/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/105*sqrt(3*x^2 + 2)/(8*x
^3 + 36*x^2 + 54*x + 27) - 16/245*sqrt(3*x^2 + 2)/(4*x^2 + 12*x + 9) - 10/343*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 0.11, size = 106, normalized size = 1.07 \[ \frac {57\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{60025}-\frac {57\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{60025}-\frac {5\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{343\,\left (x+\frac {3}{2}\right )}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{245\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{840\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^4*(3*x^2 + 2)^(1/2)),x)

[Out]

(57*35^(1/2)*log(x + 3/2))/60025 - (57*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/60025 -
(5*3^(1/2)*(x^2 + 2/3)^(1/2))/(343*(x + 3/2)) - (4*3^(1/2)*(x^2 + 2/3)^(1/2))/(245*(3*x + x^2 + 9/4)) - (13*3
^(1/2)*(x^2 + 2/3)^(1/2))/(840*((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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)**4/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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