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

Optimal. Leaf size=82 \[ -\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac {41 (4-9 x) \sqrt {3 x^2+2}}{2450 (2 x+3)^2}-\frac {123 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \]

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Rubi [A]  time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {807, 721, 725, 206} \begin {gather*} -\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac {41 (4-9 x) \sqrt {3 x^2+2}}{2450 (2 x+3)^2}-\frac {123 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41*(4 - 9*x)*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(105*(3 + 2*x)^3) - (123*ArcTanh[(
4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*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 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 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]

Rubi steps

\begin {align*} \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac {41}{35} \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac {41 (4-9 x) \sqrt {2+3 x^2}}{2450 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac {123 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1225}\\ &=-\frac {41 (4-9 x) \sqrt {2+3 x^2}}{2450 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac {123 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1225}\\ &=-\frac {41 (4-9 x) \sqrt {2+3 x^2}}{2450 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac {123 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 65, normalized size = 0.79 \begin {gather*} \frac {-\frac {35 \sqrt {3 x^2+2} \left (516 x^2-2337 x+3296\right )}{(2 x+3)^3}-738 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{257250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(3296 - 2337*x + 516*x^2))/(3 + 2*x)^3 - 738*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/257250

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IntegrateAlgebraic [A]  time = 0.84, size = 81, normalized size = 0.99 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-516 x^2+2337 x-3296\right )}{7350 (2 x+3)^3}+\frac {246 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{1225 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3296 + 2337*x - 516*x^2)*Sqrt[2 + 3*x^2])/(7350*(3 + 2*x)^3) + (246*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x -
(2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(1225*Sqrt[35])

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fricas [A]  time = 0.42, size = 104, normalized size = 1.27 \begin {gather*} \frac {369 \, \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 ) - 35 \, {\left (516 \, x^{2} - 2337 \, x + 3296\right )} \sqrt {3 \, x^{2} + 2}}{257250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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)^4,x, algorithm="fricas")

[Out]

1/257250*(369*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)) - 35*(516*x^2 - 2337*x + 3296)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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giac [B]  time = 0.31, size = 232, normalized size = 2.83 \begin {gather*} \frac {123}{42875} \, \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 (1553 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 30 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 3870 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 25740 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 20 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 1376\right )}}{9800 \, {\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}} \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)^4,x, algorithm="giac")

[Out]

123/42875*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))) - 1/9800*sqrt(3)*(1553*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 30*(sqrt(3)*
x - sqrt(3*x^2 + 2))^4 + 3870*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 25740*(sqrt(3)*x - sqrt(3*x^2 + 2))^2
- 20*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 1376)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2)) - 2)^3

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maple [A]  time = 0.06, size = 128, normalized size = 1.56 \begin {gather*} \frac {1107 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{85750}-\frac {123 \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 )}{42875}-\frac {41 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{4900 \left (x +\frac {3}{2}\right )^{2}}-\frac {369 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{85750 \left (x +\frac {3}{2}\right )}+\frac {123 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{42875}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{840 \left (x +\frac {3}{2}\right )^{3}} \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)^4,x)

[Out]

-41/4900/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-369/85750/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+123/42875*(-3
6*x+12*(x+3/2)^2-19)^(1/2)-123/42875*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+11
07/85750*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-13/840/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(3/2)

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maxima [A]  time = 1.35, size = 115, normalized size = 1.40 \begin {gather*} \frac {123}{42875} \, \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 {123}{4900} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{1225 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {369 \, \sqrt {3 \, x^{2} + 2}}{4900 \, {\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)^4,x, algorithm="maxima")

[Out]

123/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 123/4900*sqrt(3*x^2 + 2) -
13/105*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 41/1225*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 369/49
00*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 1.81, size = 106, normalized size = 1.29 \begin {gather*} \frac {123\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {123\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}-\frac {43\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {3}{2}\right )}+\frac {37\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{560\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{96\,\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)^(1/2)*(x - 5))/(2*x + 3)^4,x)

[Out]

(123*35^(1/2)*log(x + 3/2))/42875 - (123*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875
- (43*3^(1/2)*(x^2 + 2/3)^(1/2))/(4900*(x + 3/2)) + (37*3^(1/2)*(x^2 + 2/3)^(1/2))/(560*(3*x + x^2 + 9/4)) -
(13*3^(1/2)*(x^2 + 2/3)^(1/2))/(96*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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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}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, 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)**4,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(x*sqrt(3*x**2 + 2)/(1
6*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

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