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

Optimal. Leaf size=126 \[ \frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}-\frac {1051 \sqrt {3 x^2+2}}{42875 (2 x+3)}-\frac {27 \sqrt {3 x^2+2}}{1225 (2 x+3)^2}+\frac {23 \sqrt {3 x^2+2}}{525 (2 x+3)^3}-\frac {3312 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42875 \sqrt {35}} \]

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Rubi [A]  time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, 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)^3 \sqrt {3 x^2+2}}-\frac {1051 \sqrt {3 x^2+2}}{42875 (2 x+3)}-\frac {27 \sqrt {3 x^2+2}}{1225 (2 x+3)^2}+\frac {23 \sqrt {3 x^2+2}}{525 (2 x+3)^3}-\frac {3312 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42875 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (23*Sqrt[2 + 3*x^2])/(525*(3 + 2*x)^3) - (27*Sqrt[2 + 3*x^2])/(
1225*(3 + 2*x)^2) - (1051*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])])/(42875*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)^4 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}-\frac {1}{210} \int \frac {-624-738 x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx\\ &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}+\frac {\int \frac {25704+5796 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx}{22050}\\ &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}-\frac {27 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {\int \frac {-509040+102060 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{1543500}\\ &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}-\frac {27 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {1051 \sqrt {2+3 x^2}}{42875 (3+2 x)}+\frac {3312 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{42875}\\ &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}-\frac {27 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {1051 \sqrt {2+3 x^2}}{42875 (3+2 x)}-\frac {3312 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{42875}\\ &=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}-\frac {27 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {1051 \sqrt {2+3 x^2}}{42875 (3+2 x)}-\frac {3312 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42875 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 75, normalized size = 0.60 \begin {gather*} \frac {-19872 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \left (75672 x^4+261036 x^3+237930 x^2+23349 x+29438\right )}{(2 x+3)^3 \sqrt {3 x^2+2}}}{9003750} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(29438 + 23349*x + 237930*x^2 + 261036*x^3 + 75672*x^4))/((3 + 2*x)^3*Sqrt[2 + 3*x^2]) - 19872*Sqrt[35]*
ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/9003750

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IntegrateAlgebraic [A]  time = 0.91, size = 91, normalized size = 0.72 \begin {gather*} \frac {6624 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{42875 \sqrt {35}}+\frac {-75672 x^4-261036 x^3-237930 x^2-23349 x-29438}{257250 (2 x+3)^3 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-29438 - 23349*x - 237930*x^2 - 261036*x^3 - 75672*x^4)/(257250*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (6624*ArcTanh[
3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(42875*Sqrt[35])

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fricas [A]  time = 0.42, size = 134, normalized size = 1.06 \begin {gather*} \frac {9936 \, \sqrt {35} {\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\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 (75672 \, x^{4} + 261036 \, x^{3} + 237930 \, x^{2} + 23349 \, x + 29438\right )} \sqrt {3 \, x^{2} + 2}}{9003750 \, {\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9003750*(9936*sqrt(35)*(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9
*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(75672*x^4 + 261036*x^3 + 237930*x^2 + 23349*x + 29438)
*sqrt(3*x^2 + 2))/(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)

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giac [B]  time = 0.29, size = 248, normalized size = 1.97 \begin {gather*} \frac {3312}{1500625} \, \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 (10281 \, x - 12674\right )}}{3001250 \, \sqrt {3 \, x^{2} + 2}} - \frac {2 \, \sqrt {3} {\left (12983 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 253320 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 298170 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1481160 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 425140 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 106016\right )}}{1500625 \, {\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+2*x)^4/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

3312/1500625*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))) - 3/3001250*(10281*x - 12674)/sqrt(3*x^2 + 2) - 2/1500625*sqrt(3)*(12983*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 253320*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 298170*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 2))^3 - 1481160*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 425140*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) -
106016)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^3

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maple [A]  time = 0.06, size = 128, normalized size = 1.02 \begin {gather*} -\frac {3153 x}{85750 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {3312 \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 )}{1500625}-\frac {17}{700 \left (x +\frac {3}{2}\right )^{2} \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {101}{2450 \left (x +\frac {3}{2}\right ) \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {1656}{42875 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{840 \left (x +\frac {3}{2}\right )^{3} \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)^4/(3*x^2+2)^(3/2),x)

[Out]

-17/700/(x+3/2)^2/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-101/2450/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(1/2)+1656/42875/(-9*
x+3*(x+3/2)^2-19/4)^(1/2)-3153/85750/(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-3312/1500625*35^(1/2)*arctanh(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 = 1.30, size = 184, normalized size = 1.46 \begin {gather*} \frac {3312}{1500625} \, \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 {3153 \, x}{85750 \, \sqrt {3 \, x^{2} + 2}} + \frac {1656}{42875 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{105 \, {\left (8 \, \sqrt {3 \, x^{2} + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 2} x + 27 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {17}{175 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {101}{1225 \, {\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)^4/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

3312/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 3153/85750*x/sqrt(3*x^2
+ 2) + 1656/42875/sqrt(3*x^2 + 2) - 13/105/(8*sqrt(3*x^2 + 2)*x^3 + 36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 +
2)*x + 27*sqrt(3*x^2 + 2)) - 17/175/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 101/1
225/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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mupad [B]  time = 1.79, size = 210, normalized size = 1.67 \begin {gather*} \frac {3312\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1500625}-\frac {3312\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1500625}-\frac {10281\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6002500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {10281\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6002500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {13252\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x+\frac {3}{2}\right )}-\frac {197\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7350\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,6337{}\mathrm {i}}{6002500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,6337{}\mathrm {i}}{6002500\,\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)^4*(3*x^2 + 2)^(3/2)),x)

[Out]

(3312*35^(1/2)*log(x + 3/2))/1500625 - (3312*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1
500625 - (10281*3^(1/2)*(x^2 + 2/3)^(1/2))/(6002500*(x - (6^(1/2)*1i)/3)) - (10281*3^(1/2)*(x^2 + 2/3)^(1/2))/
(6002500*(x + (6^(1/2)*1i)/3)) - (13252*3^(1/2)*(x^2 + 2/3)^(1/2))/(1500625*(x + 3/2)) - (197*3^(1/2)*(x^2 + 2
/3)^(1/2))/(42875*(3*x + x^2 + 9/4)) - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(7350*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8
)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*6337i)/(6002500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(
1/2)*6337i)/(6002500*(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)**4/(3*x**2+2)**(3/2),x)

[Out]

Timed out

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