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

Optimal. Leaf size=104 \[ \frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {331 \sqrt {3 x^2+2}}{8575 (2 x+3)}+\frac {9 \sqrt {3 x^2+2}}{245 (2 x+3)^2}-\frac {1962 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8575 \sqrt {35}} \]

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Rubi [A]  time = 0.06, antiderivative size = 104, 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 = {823, 835, 807, 725, 206} \begin {gather*} \frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {331 \sqrt {3 x^2+2}}{8575 (2 x+3)}+\frac {9 \sqrt {3 x^2+2}}{245 (2 x+3)^2}-\frac {1962 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8575 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (9*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (331*Sqrt[2 + 3*x^2])/(
8575*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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)^3 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}-\frac {1}{210} \int \frac {-468-492 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx\\ &=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {9 \sqrt {2+3 x^2}}{245 (3+2 x)^2}+\frac {\int \frac {12360+1620 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{14700}\\ &=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {9 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {331 \sqrt {2+3 x^2}}{8575 (3+2 x)}+\frac {1962 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{8575}\\ &=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {9 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {331 \sqrt {2+3 x^2}}{8575 (3+2 x)}-\frac {1962 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{8575}\\ &=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {9 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {331 \sqrt {2+3 x^2}}{8575 (3+2 x)}-\frac {1962 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8575 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 70, normalized size = 0.67 \begin {gather*} \frac {-3924 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \left (3972 x^3+4068 x^2-7397 x-3658\right )}{(2 x+3)^2 \sqrt {3 x^2+2}}}{600250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(-3658 - 7397*x + 4068*x^2 + 3972*x^3))/((3 + 2*x)^2*Sqrt[2 + 3*x^2]) - 3924*Sqrt[35]*ArcTanh[(4 - 9*x)/
(Sqrt[35]*Sqrt[2 + 3*x^2])])/600250

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IntegrateAlgebraic [A]  time = 0.70, size = 86, normalized size = 0.83 \begin {gather*} \frac {3924 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{8575 \sqrt {35}}+\frac {-3972 x^3-4068 x^2+7397 x+3658}{17150 (2 x+3)^2 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3658 + 7397*x - 4068*x^2 - 3972*x^3)/(17150*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (3924*ArcTanh[3*Sqrt[3/35] + 2*Sqr
t[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(8575*Sqrt[35])

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fricas [A]  time = 0.44, size = 119, normalized size = 1.14 \begin {gather*} \frac {1962 \, \sqrt {35} {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\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 (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt {3 \, x^{2} + 2}}{600250 \, {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/600250*(1962*sqrt(35)*(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x
^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(3972*x^3 + 4068*x^2 - 7397*x - 3658)*sqrt(3*x^2 + 2))/(12*x^4 + 36*x
^3 + 35*x^2 + 24*x + 18)

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giac [B]  time = 0.25, size = 199, normalized size = 1.91 \begin {gather*} \frac {1962}{300125} \, \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 {3 \, {\left (157 \, x - 1478\right )}}{85750 \, \sqrt {3 \, x^{2} + 2}} - \frac {768 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt {3} x + 856 \, \sqrt {3} + 6168 \, \sqrt {3 \, x^{2} + 2}}{6125 \, {\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+2*x)^3/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

1962/300125*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))) - 3/85750*(157*x - 1478)/sqrt(3*x^2 + 2) - 1/6125*(768*(sqrt(3)*x - sqrt(3*
x^2 + 2))^3 + 2461*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6168*sqrt(3)*x + 856*sqrt(3) + 6168*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

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maple [A]  time = 0.08, size = 107, normalized size = 1.03 \begin {gather*} -\frac {993 x}{17150 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {1962 \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 )}{300125}-\frac {103}{980 \left (x +\frac {3}{2}\right ) \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {981}{8575 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{280 \left (x +\frac {3}{2}\right )^{2} \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-103/980/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(1/2)+981/8575/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-993/17150/(-9*x+3*(x+3/2
)^2-19/4)^(1/2)*x-1962/300125*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))-13/280/(x
+3/2)^2/(-9*x+3*(x+3/2)^2-19/4)^(1/2)

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maxima [A]  time = 1.42, size = 128, normalized size = 1.23 \begin {gather*} \frac {1962}{300125} \, \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 {993 \, x}{17150 \, \sqrt {3 \, x^{2} + 2}} + \frac {981}{8575 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{70 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {103}{490 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1962/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 993/17150*x/sqrt(3*x^2 +
2) + 981/8575/sqrt(3*x^2 + 2) - 13/70/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 10
3/490/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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mupad [B]  time = 1.78, size = 181, normalized size = 1.74 \begin {gather*} \frac {1962\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{300125}-\frac {1962\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{300125}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {107\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2450\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1962*35^(1/2)*log(x + 3/2))/300125 - (1962*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/30
0125 - (157*3^(1/2)*(x^2 + 2/3)^(1/2))/(171500*(x - (6^(1/2)*1i)/3)) - (157*3^(1/2)*(x^2 + 2/3)^(1/2))/(171500
*(x + (6^(1/2)*1i)/3)) - (107*3^(1/2)*(x^2 + 2/3)^(1/2))/(6125*(x + 3/2)) - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(24
50*(3*x + x^2 + 9/4)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*739i)/(171500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1
/2)*(x^2 + 2/3)^(1/2)*739i)/(171500*(x + (6^(1/2)*1i)/3))

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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+2*x)**3/(3*x**2+2)**(3/2),x)

[Out]

Timed out

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