3.1366 \(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=126 \[ -\frac {43 \left (3 x^2+2\right )^{3/2}}{6125 (2 x+3)^3}-\frac {23 \left (3 x^2+2\right )^{3/2}}{875 (2 x+3)^4}-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}-\frac {339 (4-9 x) \sqrt {3 x^2+2}}{428750 (2 x+3)^2}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{214375 \sqrt {35}} \]

[Out]

-13/175*(3*x^2+2)^(3/2)/(3+2*x)^5-23/875*(3*x^2+2)^(3/2)/(3+2*x)^4-43/6125*(3*x^2+2)^(3/2)/(3+2*x)^3-1017/7503
125*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-339/428750*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2

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Rubi [A]  time = 0.07, 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 = {835, 807, 721, 725, 206} \[ -\frac {43 \left (3 x^2+2\right )^{3/2}}{6125 (2 x+3)^3}-\frac {23 \left (3 x^2+2\right )^{3/2}}{875 (2 x+3)^4}-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}-\frac {339 (4-9 x) \sqrt {3 x^2+2}}{428750 (2 x+3)^2}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{214375 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-339*(4 - 9*x)*Sqrt[2 + 3*x^2])/(428750*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(175*(3 + 2*x)^5) - (23*(2 + 3*
x^2)^(3/2))/(875*(3 + 2*x)^4) - (43*(2 + 3*x^2)^(3/2))/(6125*(3 + 2*x)^3) - (1017*ArcTanh[(4 - 9*x)/(Sqrt[35]*
Sqrt[2 + 3*x^2])])/(214375*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]

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) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {1}{175} \int \frac {(-205+78 x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}+\frac {\int \frac {(6132-1932 x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx}{24500}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}+\frac {339 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{6125}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}+\frac {1017 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{214375}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}-\frac {1017 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{214375}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{214375 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 75, normalized size = 0.60 \[ \frac {-2034 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \sqrt {3 x^2+2} \left (11712 x^4+76992 x^3+186392 x^2+108167 x+222112\right )}{(2 x+3)^5}}{15006250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(222112 + 108167*x + 186392*x^2 + 76992*x^3 + 11712*x^4))/(3 + 2*x)^5 - 2034*Sqrt[35]*Ar
cTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/15006250

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fricas [A]  time = 0.70, size = 134, normalized size = 1.06 \[ \frac {1017 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (11712 \, x^{4} + 76992 \, x^{3} + 186392 \, x^{2} + 108167 \, x + 222112\right )} \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15006250*(1017*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)) - 35*(11712*x^4 + 76992*x^3 + 186392*x^2 + 108167*x + 222
112)*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 = 0.26, size = 322, normalized size = 2.56 \[ \frac {1017}{7503125} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, \sqrt {3} {\left (904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 36612 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 254217 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 142464 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 338184 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 4315808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 1676892 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1737184 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 219776 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 31232\right )}}{1715000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1017/7503125*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))) - 3/1715000*sqrt(3)*(904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 36612*(
sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 254217*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 142464*(sqrt(3)*x - sqrt(3*x
^2 + 2))^6 - 338184*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4315808*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 167689
2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1737184*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 219776*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2)) - 31232)/((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 = 170, normalized size = 1.35 \[ \frac {9153 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{15006250}-\frac {1017 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{7503125}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {23 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{14000 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{49000 \left (x +\frac {3}{2}\right )^{3}}-\frac {339 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{857500 \left (x +\frac {3}{2}\right )^{2}}-\frac {3051 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )}+\frac {1017 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{7503125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/5600/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-23/14000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-43/49000/(x+
3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-339/857500/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-3051/15006250/(x+3/2)*
(-9*x+3*(x+3/2)^2-19/4)^(3/2)+1017/7503125*(-36*x+12*(x+3/2)^2-19)^(1/2)-1017/7503125*35^(1/2)*arctanh(2/35*(-
9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+9153/15006250*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x

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maxima [A]  time = 1.23, size = 186, normalized size = 1.48 \[ \frac {1017}{7503125} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1017}{857500} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {23 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{875 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {43 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{6125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {339 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{214375 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3051 \, \sqrt {3 \, x^{2} + 2}}{857500 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1017/7503125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1017/857500*sqrt(3*x^2
+ 2) - 13/175*(3*x^2 + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 23/875*(3*x^2 + 2)^(3/
2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 43/6125*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 339/214
375*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 3051/857500*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 1.80, size = 178, normalized size = 1.41 \[ \frac {1017\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{7503125}-\frac {1017\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7503125}+\frac {73\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{11200\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{640\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {183\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{214375\,\left (x+\frac {3}{2}\right )}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1017*35^(1/2)*log(x + 3/2))/7503125 - (1017*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/7
503125 + (73*3^(1/2)*(x^2 + 2/3)^(1/2))/(11200*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (13*3^(1/2)*(x
^2 + 2/3)^(1/2))/(640*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (183*3^(1/2)*(x^2
+ 2/3)^(1/2))/(214375*(x + 3/2)) - (3*3^(1/2)*(x^2 + 2/3)^(1/2))/(6125*(3*x + x^2 + 9/4)) + (3^(1/2)*(x^2 + 2
/3)^(1/2))/(7000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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

[Out]

Timed out

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