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]

(-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])

________________________________________________________________________________________

Rubi [A]  time = 0.0713158, antiderivative size = 126, normalized size of antiderivative = 1., 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 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])

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 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 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])

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*}

Mathematica [A]  time = 0.0854206, size = 75, normalized size = 0.6 \[ \frac{-\frac{35 \sqrt{3 x^2+2} \left (11712 x^4+76992 x^3+186392 x^2+108167 x+222112\right )}{(2 x+3)^5}-2034 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{15006250} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 170, normalized size = 1.4 \begin{align*} -{\frac{13}{5600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{23}{14000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{43}{49000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{339}{857500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{3051}{15006250} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{1017}{7503125}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{1017\,\sqrt{35}}{7503125}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{9153\,x}{15006250}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.51316, size = 251, normalized size = 1.99 \begin{align*} \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 )}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.24201, size = 401, normalized size = 3.18 \begin{align*} \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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.26823, size = 429, normalized size = 3.4 \begin{align*} \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 \,{\left (2712 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 36612 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 762651 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 142464 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1014552 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 4315808 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 5030676 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1737184 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 659328 \, \sqrt{3} x - 31232 \, \sqrt{3} - 659328 \, \sqrt{3 \, x^{2} + 2}\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}} \end{align*}

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*(2712*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 36612*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 2))^8 + 762651*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 142464*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^6 - 1014552*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4315808*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 5030676*(sqr
t(3)*x - sqrt(3*x^2 + 2))^3 - 1737184*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 659328*sqrt(3)*x - 31232*sqrt(
3) - 659328*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)^5