3.1407 \(\int \frac{5-x}{(3+2 x)^6 \sqrt{2+3 x^2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.095708, antiderivative size = 143, normalized size of antiderivative = 1., 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} \[ -\frac{10023 \sqrt{3 x^2+2}}{15006250 (2 x+3)}-\frac{1611 \sqrt{3 x^2+2}}{428750 (2 x+3)^2}-\frac{797 \sqrt{3 x^2+2}}{61250 (2 x+3)^3}-\frac{439 \sqrt{3 x^2+2}}{12250 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+2}}{175 (2 x+3)^5}+\frac{19737 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

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 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 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}{(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*}

Mathematica [A]  time = 0.0938616, size = 75, normalized size = 0.52 \[ \frac{19737 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{35 \sqrt{3 x^2+2} \left (80184 x^4+706644 x^3+2487944 x^2+4314244 x+3409859\right )}{(2 x+3)^5}}{262609375} \]

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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Maple [A]  time = 0.01, size = 137, normalized size = 1. \begin{align*} -{\frac{13}{5600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{439}{196000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{797}{490000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1611}{1715000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{10023}{30012500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{19737\,\sqrt{35}}{262609375}{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53151, size = 236, normalized size = 1.65 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.89105, size = 408, normalized size = 2.85 \begin{align*} \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{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.23883, size = 429, normalized size = 3. \begin{align*} -\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 \,{\left (26316 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 355266 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 5320218 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 11098773 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} + 6945939 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 49794206 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 76607832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 16740688 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 5232096 \, \sqrt{3} x + 213824 \, \sqrt{3} + 5232096 \, \sqrt{3 \, x^{2} + 2}\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{align*}

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*(26316*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 355266*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 5320218*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 11098773*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 2))^6 + 6945939*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 49794206*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 7
6607832*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 16740688*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 5232096*sqrt(3)*x
+ 213824*sqrt(3) + 5232096*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