3.1417 \(\int \frac{5-x}{(3+2 x)^5 (2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{41 x+26}{70 (2 x+3)^4 \sqrt{3 x^2+2}}-\frac{14944 \sqrt{3 x^2+2}}{1500625 (2 x+3)}-\frac{708 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}-\frac{298 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{58 \sqrt{3 x^2+2}}{1225 (2 x+3)^4}-\frac{30078 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^4*Sqrt[2 + 3*x^2]) + (58*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)^4) - (298*Sqrt[2 + 3*x^2])
/(18375*(3 + 2*x)^3) - (708*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)^2) - (14944*Sqrt[2 + 3*x^2])/(1500625*(3 + 2*x))
- (30078*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*Sqrt[35])

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Rubi [A]  time = 0.0943078, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{41 x+26}{70 (2 x+3)^4 \sqrt{3 x^2+2}}-\frac{14944 \sqrt{3 x^2+2}}{1500625 (2 x+3)}-\frac{708 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}-\frac{298 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{58 \sqrt{3 x^2+2}}{1225 (2 x+3)^4}-\frac{30078 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^4*Sqrt[2 + 3*x^2]) + (58*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)^4) - (298*Sqrt[2 + 3*x^2])
/(18375*(3 + 2*x)^3) - (708*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)^2) - (14944*Sqrt[2 + 3*x^2])/(1500625*(3 + 2*x))
- (30078*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*Sqrt[35])

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])

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)^5 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}-\frac{1}{210} \int \frac{-780-984 x}{(3+2 x)^5 \sqrt{2+3 x^2}} \, dx\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}+\frac{\int \frac{43824+12528 x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx}{29400}\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}-\frac{298 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{\int \frac{-1333584+300384 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{3087000}\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}-\frac{298 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{708 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}+\frac{\int \frac{21601440-10704960 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{216090000}\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}-\frac{298 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{708 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{14944 \sqrt{2+3 x^2}}{1500625 (3+2 x)}+\frac{30078 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1500625}\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}-\frac{298 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{708 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{14944 \sqrt{2+3 x^2}}{1500625 (3+2 x)}-\frac{30078 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1500625}\\ &=\frac{26+41 x}{70 (3+2 x)^4 \sqrt{2+3 x^2}}+\frac{58 \sqrt{2+3 x^2}}{1225 (3+2 x)^4}-\frac{298 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{708 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{14944 \sqrt{2+3 x^2}}{1500625 (3+2 x)}-\frac{30078 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1500625 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0970315, size = 80, normalized size = 0.54 \[ \frac{-\frac{35 \left (2151936 x^5+11467872 x^4+22188792 x^3+18957672 x^2+8562487 x+4197366\right )}{(2 x+3)^4 \sqrt{3 x^2+2}}-180468 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{315131250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(4197366 + 8562487*x + 18957672*x^2 + 22188792*x^3 + 11467872*x^4 + 2151936*x^5))/((3 + 2*x)^4*Sqrt[2 +
3*x^2]) - 180468*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/315131250

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Maple [A]  time = 0.012, size = 149, normalized size = 1. \begin{align*} -{\frac{913}{117600} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{9}{1000} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{2143}{171500} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{15039}{1500625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{22416\,x}{1500625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{30078\,\sqrt{35}}{52521875}{\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{13}{2240} \left ( x+{\frac{3}{2}} \right ) ^{-4}{\frac{1}{\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+2*x)^5/(3*x^2+2)^(3/2),x)

[Out]

-913/117600/(x+3/2)^3/(3*(x+3/2)^2-9*x-19/4)^(1/2)-9/1000/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(1/2)-2143/171500/(
x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)+15039/1500625/(3*(x+3/2)^2-9*x-19/4)^(1/2)-22416/1500625*x/(3*(x+3/2)^2-9*
x-19/4)^(1/2)-30078/52521875*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-13/2240/(x+3
/2)^4/(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [B]  time = 1.50337, size = 343, normalized size = 2.32 \begin{align*} \frac{30078}{52521875} \, \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{22416 \, x}{1500625 \, \sqrt{3 \, x^{2} + 2}} + \frac{15039}{1500625 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{140 \,{\left (16 \, \sqrt{3 \, x^{2} + 2} x^{4} + 96 \, \sqrt{3 \, x^{2} + 2} x^{3} + 216 \, \sqrt{3 \, x^{2} + 2} x^{2} + 216 \, \sqrt{3 \, x^{2} + 2} x + 81 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{913}{14700 \,{\left (8 \, \sqrt{3 \, x^{2} + 2} x^{3} + 36 \, \sqrt{3 \, x^{2} + 2} x^{2} + 54 \, \sqrt{3 \, x^{2} + 2} x + 27 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{9}{250 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{2143}{85750 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30078/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 22416/1500625*x/sqrt(
3*x^2 + 2) + 15039/1500625/sqrt(3*x^2 + 2) - 13/140/(16*sqrt(3*x^2 + 2)*x^4 + 96*sqrt(3*x^2 + 2)*x^3 + 216*sqr
t(3*x^2 + 2)*x^2 + 216*sqrt(3*x^2 + 2)*x + 81*sqrt(3*x^2 + 2)) - 913/14700/(8*sqrt(3*x^2 + 2)*x^3 + 36*sqrt(3*
x^2 + 2)*x^2 + 54*sqrt(3*x^2 + 2)*x + 27*sqrt(3*x^2 + 2)) - 9/250/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*
x + 9*sqrt(3*x^2 + 2)) - 2143/85750/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 1.57269, size = 460, normalized size = 3.11 \begin{align*} \frac{90234 \, \sqrt{35}{\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\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 (2151936 \, x^{5} + 11467872 \, x^{4} + 22188792 \, x^{3} + 18957672 \, x^{2} + 8562487 \, x + 4197366\right )} \sqrt{3 \, x^{2} + 2}}{315131250 \,{\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315131250*(90234*sqrt(35)*(48*x^6 + 288*x^5 + 680*x^4 + 840*x^3 + 675*x^2 + 432*x + 162)*log(-(sqrt(35)*sqrt
(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(2151936*x^5 + 11467872*x^4 + 22188792*x^
3 + 18957672*x^2 + 8562487*x + 4197366)*sqrt(3*x^2 + 2))/(48*x^6 + 288*x^5 + 680*x^4 + 840*x^3 + 675*x^2 + 432
*x + 162)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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