3.1427 \(\int \frac {5-x}{(3+2 x)^3 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=131 \[ \frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}+\frac {857 \sqrt {3 x^2+2}}{128625 (2 x+3)}+\frac {83 \sqrt {3 x^2+2}}{1225 (2 x+3)^2}+\frac {419 x+4}{1050 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {3072 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42875 \sqrt {35}} \]

[Out]

1/210*(26+41*x)/(3+2*x)^2/(3*x^2+2)^(3/2)-3072/1500625*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)
+1/1050*(4+419*x)/(3+2*x)^2/(3*x^2+2)^(1/2)+83/1225*(3*x^2+2)^(1/2)/(3+2*x)^2+857/128625*(3*x^2+2)^(1/2)/(3+2*
x)

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Rubi [A]  time = 0.08, antiderivative size = 131, 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} \[ \frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}+\frac {857 \sqrt {3 x^2+2}}{128625 (2 x+3)}+\frac {83 \sqrt {3 x^2+2}}{1225 (2 x+3)^2}+\frac {419 x+4}{1050 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {3072 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42875 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(210*(3 + 2*x)^2*(2 + 3*x^2)^(3/2)) + (4 + 419*x)/(1050*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (83*Sqrt[2
+ 3*x^2])/(1225*(3 + 2*x)^2) + (857*Sqrt[2 + 3*x^2])/(128625*(3 + 2*x)) - (3072*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[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)^3 \left (2+3 x^2\right )^{5/2}} \, dx &=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}-\frac {1}{630} \int \frac {-1518-984 x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {\int \frac {3024+211176 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx}{132300}\\ &=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {83 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {\int \frac {-1743840-1882440 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{9261000}\\ &=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {83 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}+\frac {857 \sqrt {2+3 x^2}}{128625 (3+2 x)}+\frac {3072 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{42875}\\ &=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {83 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}+\frac {857 \sqrt {2+3 x^2}}{128625 (3+2 x)}-\frac {3072 \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}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {83 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}+\frac {857 \sqrt {2+3 x^2}}{128625 (3+2 x)}-\frac {3072 \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 = 0.11, size = 80, normalized size = 0.61 \[ \frac {\frac {35 \left (10284 x^5+67716 x^4+116367 x^3+91268 x^2+89749 x+41366\right )}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}-6144 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{3001250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(41366 + 89749*x + 91268*x^2 + 116367*x^3 + 67716*x^4 + 10284*x^5))/((3 + 2*x)^2*(2 + 3*x^2)^(3/2)) - 614
4*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/3001250

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fricas [A]  time = 0.61, size = 149, normalized size = 1.14 \[ \frac {3072 \, \sqrt {35} {\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (10284 \, x^{5} + 67716 \, x^{4} + 116367 \, x^{3} + 91268 \, x^{2} + 89749 \, x + 41366\right )} \sqrt {3 \, x^{2} + 2}}{3001250 \, {\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3001250*(3072*sqrt(35)*(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)*log(-(sqrt(35)*sqrt(3*x^
2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(10284*x^5 + 67716*x^4 + 116367*x^3 + 91268*x^
2 + 89749*x + 41366)*sqrt(3*x^2 + 2))/(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)

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giac [A]  time = 0.27, size = 208, normalized size = 1.59 \[ \frac {3072}{1500625} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, {\left ({\left (59203 \, x + 69168\right )} x + 37637\right )} x + 190066}{3001250 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (9588 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 27991 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 68448 \, \sqrt {3} x + 9736 \, \sqrt {3} + 68448 \, \sqrt {3 \, x^{2} + 2}\right )}}{1500625 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3072/1500625*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))) + 1/3001250*(3*((59203*x + 69168)*x + 37637)*x + 190066)/(3*x^2 + 2)^(3/2)
- 4/1500625*(9588*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 27991*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 68448*sqr
t(3)*x + 9736*sqrt(3) + 68448*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 = 0.05, size = 140, normalized size = 1.07 \[ -\frac {173 x}{2450 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {857 x}{85750 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {3072 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1500625}-\frac {107}{700 \left (x +\frac {3}{2}\right ) \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {128}{1225 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {1536}{42875 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{280 \left (x +\frac {3}{2}\right )^{2} \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-107/700/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(3/2)+128/1225/(-9*x+3*(x+3/2)^2-19/4)^(3/2)-173/2450/(-9*x+3*(x+3/2)
^2-19/4)^(3/2)*x+857/85750/(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+1536/42875/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-3072/15006
25*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)^2-1
9/4)^(3/2)

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maxima [A]  time = 1.23, size = 151, normalized size = 1.15 \[ \frac {3072}{1500625} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {857 \, x}{85750 \, \sqrt {3 \, x^{2} + 2}} + \frac {1536}{42875 \, \sqrt {3 \, x^{2} + 2}} - \frac {173 \, x}{2450 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{70 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {107}{350 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {128}{1225 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3072/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 857/85750*x/sqrt(3*x^2
+ 2) + 1536/42875/sqrt(3*x^2 + 2) - 173/2450*x/(3*x^2 + 2)^(3/2) - 13/70/(4*(3*x^2 + 2)^(3/2)*x^2 + 12*(3*x^2
+ 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 107/350/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) + 128/1225/(3*x^2
+ 2)^(3/2)

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mupad [B]  time = 1.83, size = 301, normalized size = 2.30 \[ \frac {3072\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1500625}-\frac {3072\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1500625}-\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1029000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {59203\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{18007500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {59203\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{18007500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1029000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {4868\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x+\frac {3}{2}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,157{}\mathrm {i}}{6174000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,164201{}\mathrm {i}}{72030000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,164201{}\mathrm {i}}{72030000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,157{}\mathrm {i}}{6174000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3072*35^(1/2)*log(x + 3/2))/1500625 - (3072*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1
500625 - (739*3^(1/2)*(x^2 + 2/3)^(1/2))/(1029000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (59203*3^(1/2)*(x^2 + 2/3)
^(1/2))/(18007500*(x - (6^(1/2)*1i)/3)) + (59203*3^(1/2)*(x^2 + 2/3)^(1/2))/(18007500*(x + (6^(1/2)*1i)/3)) +
(739*3^(1/2)*(x^2 + 2/3)^(1/2))/(1029000*((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - (4868*3^(1/2)*(x^2 + 2/3)^(1/2))/(1
500625*(x + 3/2)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(42875*(3*x + x^2 + 9/4)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1
/2)*157i)/(6174000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*164201i)/(72030000*(x
- (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*164201i)/(72030000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^
(1/2)*(x^2 + 2/3)^(1/2)*157i)/(6174000*((6^(1/2)*x*2i)/3 - x^2 + 2/3))

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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)**3/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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