3.13.33 \(\int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac {10023 \sqrt {3 x^2+2}}{15006250 (2 x+3)}-\frac {1611 \sqrt {3 x^2+2}}{428750 (2 x+3)^2}-\frac {797 \sqrt {3 x^2+2}}{61250 (2 x+3)^3}-\frac {439 \sqrt {3 x^2+2}}{12250 (2 x+3)^4}-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}+\frac {19737 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \]

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Rubi [A]  time = 0.10, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {835, 807, 725, 206} \begin {gather*} -\frac {10023 \sqrt {3 x^2+2}}{15006250 (2 x+3)}-\frac {1611 \sqrt {3 x^2+2}}{428750 (2 x+3)^2}-\frac {797 \sqrt {3 x^2+2}}{61250 (2 x+3)^3}-\frac {439 \sqrt {3 x^2+2}}{12250 (2 x+3)^4}-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}+\frac {19737 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 3*x^2]),x]

[Out]

(-13*Sqrt[2 + 3*x^2])/(175*(3 + 2*x)^5) - (439*Sqrt[2 + 3*x^2])/(12250*(3 + 2*x)^4) - (797*Sqrt[2 + 3*x^2])/(6
1250*(3 + 2*x)^3) - (1611*Sqrt[2 + 3*x^2])/(428750*(3 + 2*x)^2) - (10023*Sqrt[2 + 3*x^2])/(15006250*(3 + 2*x))
+ (19737*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*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)^6 \sqrt {2+3 x^2}} \, dx &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {1}{175} \int \frac {-205+156 x}{(3+2 x)^5 \sqrt {2+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}+\frac {\int \frac {4884-7902 x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx}{24500}\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {\int \frac {-37044+200844 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx}{2572500}\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {1611 \sqrt {2+3 x^2}}{428750 (3+2 x)^2}+\frac {\int \frac {-939960-2029860 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{180075000}\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {1611 \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {10023 \sqrt {2+3 x^2}}{15006250 (3+2 x)}-\frac {19737 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{7503125}\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {1611 \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {10023 \sqrt {2+3 x^2}}{15006250 (3+2 x)}+\frac {19737 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{7503125}\\ &=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {1611 \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {10023 \sqrt {2+3 x^2}}{15006250 (3+2 x)}+\frac {19737 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{7503125 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 75, normalized size = 0.52 \begin {gather*} \frac {19737 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \sqrt {3 x^2+2} \left (80184 x^4+706644 x^3+2487944 x^2+4314244 x+3409859\right )}{(2 x+3)^5}}{262609375} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 3*x^2]),x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(3409859 + 4314244*x + 2487944*x^2 + 706644*x^3 + 80184*x^4))/(3 + 2*x)^5 + 19737*Sqrt[3
5]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/262609375

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IntegrateAlgebraic [A]  time = 1.26, size = 91, normalized size = 0.64 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-80184 x^4-706644 x^3-2487944 x^2-4314244 x-3409859\right )}{7503125 (2 x+3)^5}-\frac {39474 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{7503125 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 3*x^2]),x]

[Out]

(Sqrt[2 + 3*x^2]*(-3409859 - 4314244*x - 2487944*x^2 - 706644*x^3 - 80184*x^4))/(7503125*(3 + 2*x)^5) - (39474
*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(7503125*Sqrt[35])

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fricas [A]  time = 0.45, size = 133, normalized size = 0.93 \begin {gather*} \frac {19737 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (80184 \, x^{4} + 706644 \, x^{3} + 2487944 \, x^{2} + 4314244 \, x + 3409859\right )} \sqrt {3 \, x^{2} + 2}}{525218750 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/525218750*(19737*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((sqrt(35)*sqrt(3*x^2 + 2
)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 70*(80184*x^4 + 706644*x^3 + 2487944*x^2 + 4314244*x +
3409859)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.27, size = 322, normalized size = 2.25 \begin {gather*} -\frac {19737}{262609375} \, \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 \, \sqrt {3} {\left (8772 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 355266 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 1773406 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 11098773 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 2315313 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 49794206 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 25535944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 16740688 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 1744032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 213824\right )}}{30012500 \, {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-19737/262609375*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqr
t(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 3/30012500*sqrt(3)*(8772*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 3
55266*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 1773406*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 11098773*(sqrt(3)*x
- sqrt(3*x^2 + 2))^6 + 2315313*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 49794206*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^4 - 25535944*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 16740688*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 1744032*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 213824)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 2)) - 2)^5

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maple [A]  time = 0.07, size = 137, normalized size = 0.96 \begin {gather*} \frac {19737 \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 )}{262609375}-\frac {13 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {439 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{196000 \left (x +\frac {3}{2}\right )^{4}}-\frac {797 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{490000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1611 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1715000 \left (x +\frac {3}{2}\right )^{2}}-\frac {10023 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{30012500 \left (x +\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^6/(3*x^2+2)^(1/2),x)

[Out]

-13/5600/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-439/196000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-797/490000
/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-1611/1715000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-10023/30012500/(
x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(1/2)+19737/262609375*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)
^2-19)^(1/2))

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maxima [A]  time = 1.16, size = 175, normalized size = 1.22 \begin {gather*} -\frac {19737}{262609375} \, \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}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {439 \, \sqrt {3 \, x^{2} + 2}}{12250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {797 \, \sqrt {3 \, x^{2} + 2}}{61250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1611 \, \sqrt {3 \, x^{2} + 2}}{428750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {10023 \, \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-19737/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/175*sqrt(3*x^2 +
2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 439/12250*sqrt(3*x^2 + 2)/(16*x^4 + 96*x^3 + 216*x
^2 + 216*x + 81) - 797/61250*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1611/428750*sqrt(3*x^2 + 2)/(4*x^2
+ 12*x + 9) - 10023/15006250*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 1.93, size = 160, normalized size = 1.12 \begin {gather*} \frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {1404}{42875\,\left (x+\frac {3}{2}\right )}+\frac {54}{1225\,{\left (x+\frac {3}{2}\right )}^2}+\frac {9}{175\,{\left (x+\frac {3}{2}\right )}^3}+\frac {3}{70\,{\left (x+\frac {3}{2}\right )}^4}\right )}{96}-\frac {\sqrt {35}\,\left (\frac {555984\,\ln \left (x+\frac {3}{2}\right )}{7503125}-\frac {555984\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7503125}\right )}{1120}-\frac {\sqrt {35}\,\left (\frac {216\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {216\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{560}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {972504}{7503125\,\left (x+\frac {3}{2}\right )}+\frac {57564}{214375\,{\left (x+\frac {3}{2}\right )}^2}+\frac {12714}{30625\,{\left (x+\frac {3}{2}\right )}^3}+\frac {3159}{6125\,{\left (x+\frac {3}{2}\right )}^4}+\frac {78}{175\,{\left (x+\frac {3}{2}\right )}^5}\right )}{192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^6*(3*x^2 + 2)^(1/2)),x)

[Out]

(3^(1/2)*(x^2 + 2/3)^(1/2)*(1404/(42875*(x + 3/2)) + 54/(1225*(x + 3/2)^2) + 9/(175*(x + 3/2)^3) + 3/(70*(x +
3/2)^4)))/96 - (35^(1/2)*((555984*log(x + 3/2))/7503125 - (555984*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))
/9 - 4/9))/7503125))/1120 - (35^(1/2)*((216*log(x + 3/2))/42875 - (216*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(
1/2))/9 - 4/9))/42875))/560 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(972504/(7503125*(x + 3/2)) + 57564/(214375*(x + 3/2)
^2) + 12714/(30625*(x + 3/2)^3) + 3159/(6125*(x + 3/2)^4) + 78/(175*(x + 3/2)^5)))/192

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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)**6/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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