3.1428 \(\int \frac{5-x}{(3+2 x)^4 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=153 \[ -\frac{114-3331 x}{7350 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{5987 \sqrt{3 x^2+2}}{1500625 (2 x+3)}+\frac{541 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}+\frac{1471 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-\frac{55344 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

[Out]

(26 + 41*x)/(210*(3 + 2*x)^3*(2 + 3*x^2)^(3/2)) - (114 - 3331*x)/(7350*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (1471*Sq
rt[2 + 3*x^2])/(18375*(3 + 2*x)^3) + (541*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)^2) - (5987*Sqrt[2 + 3*x^2])/(15006
25*(3 + 2*x)) - (55344*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*Sqrt[35])

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Rubi [A]  time = 0.0981771, antiderivative size = 153, 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} \[ -\frac{114-3331 x}{7350 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{5987 \sqrt{3 x^2+2}}{1500625 (2 x+3)}+\frac{541 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}+\frac{1471 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-\frac{55344 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*(2 + 3*x^2)^(5/2)),x]

[Out]

(26 + 41*x)/(210*(3 + 2*x)^3*(2 + 3*x^2)^(3/2)) - (114 - 3331*x)/(7350*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (1471*Sq
rt[2 + 3*x^2])/(18375*(3 + 2*x)^3) + (541*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)^2) - (5987*Sqrt[2 + 3*x^2])/(15006
25*(3 + 2*x)) - (55344*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)^4 \left (2+3 x^2\right )^{5/2}} \, dx &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{1}{630} \int \frac{-1674-1230 x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{\int \frac{-16416+359748 x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx}{132300}\\ &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{1471 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}-\frac{\int \frac{-3873744-6672456 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{13891500}\\ &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{1471 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}+\frac{541 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}+\frac{\int \frac{123107040+36809640 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{972405000}\\ &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{1471 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}+\frac{541 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{5987 \sqrt{2+3 x^2}}{1500625 (3+2 x)}+\frac{55344 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1500625}\\ &=\frac{26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{1471 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}+\frac{541 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{5987 \sqrt{2+3 x^2}}{1500625 (3+2 x)}-\frac{55344 \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}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac{114-3331 x}{7350 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{1471 \sqrt{2+3 x^2}}{18375 (3+2 x)^3}+\frac{541 \sqrt{2+3 x^2}}{42875 (3+2 x)^2}-\frac{5987 \sqrt{2+3 x^2}}{1500625 (3+2 x)}-\frac{55344 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1500625 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.121697, size = 85, normalized size = 0.56 \[ \frac{-\frac{35 \left (1293192 x^6+1834596 x^5-4920642 x^4-9795297 x^3-7866162 x^2-9103449 x-3788738\right )}{(2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-332064 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{315131250} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*(2 + 3*x^2)^(5/2)),x]

[Out]

((-35*(-3788738 - 9103449*x - 7866162*x^2 - 9795297*x^3 - 4920642*x^4 + 1834596*x^5 + 1293192*x^6))/((3 + 2*x)
^3*(2 + 3*x^2)^(3/2)) - 332064*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/315131250

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Maple [A]  time = 0.011, size = 161, normalized size = 1.1 \begin{align*} -{\frac{13}{840} \left ( x+{\frac{3}{2}} \right ) ^{-3} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{79}{2450} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{516}{6125} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{2306}{42875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{4071\,x}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{17961\,x}{3001250}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{27672}{1500625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{55344\,\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x)

[Out]

-13/840/(x+3/2)^3/(3*(x+3/2)^2-9*x-19/4)^(3/2)-79/2450/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(3/2)-516/6125/(x+3/2)
/(3*(x+3/2)^2-9*x-19/4)^(3/2)+2306/42875/(3*(x+3/2)^2-9*x-19/4)^(3/2)-4071/85750*x/(3*(x+3/2)^2-9*x-19/4)^(3/2
)-17961/3001250*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)+27672/1500625/(3*(x+3/2)^2-9*x-19/4)^(1/2)-55344/52521875*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 = 1.5285, size = 279, normalized size = 1.82 \begin{align*} \frac{55344}{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{17961 \, x}{3001250 \, \sqrt{3 \, x^{2} + 2}} + \frac{27672}{1500625 \, \sqrt{3 \, x^{2} + 2}} - \frac{4071 \, x}{85750 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{13}{105 \,{\left (8 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + 36 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + 54 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} - \frac{158}{1225 \,{\left (4 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} - \frac{1032}{6125 \,{\left (2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{2306}{42875 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

55344/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 17961/3001250*x/sqrt(
3*x^2 + 2) + 27672/1500625/sqrt(3*x^2 + 2) - 4071/85750*x/(3*x^2 + 2)^(3/2) - 13/105/(8*(3*x^2 + 2)^(3/2)*x^3
+ 36*(3*x^2 + 2)^(3/2)*x^2 + 54*(3*x^2 + 2)^(3/2)*x + 27*(3*x^2 + 2)^(3/2)) - 158/1225/(4*(3*x^2 + 2)^(3/2)*x^
2 + 12*(3*x^2 + 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 1032/6125/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) +
2306/42875/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.544, size = 504, normalized size = 3.29 \begin{align*} \frac{166032 \, \sqrt{35}{\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\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 (1293192 \, x^{6} + 1834596 \, x^{5} - 4920642 \, x^{4} - 9795297 \, x^{3} - 7866162 \, x^{2} - 9103449 \, x - 3788738\right )} \sqrt{3 \, x^{2} + 2}}{315131250 \,{\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/315131250*(166032*sqrt(35)*(72*x^7 + 324*x^6 + 582*x^5 + 675*x^4 + 680*x^3 + 468*x^2 + 216*x + 108)*log(-(sq
rt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(1293192*x^6 + 1834596*x^5 - 4
920642*x^4 - 9795297*x^3 - 7866162*x^2 - 9103449*x - 3788738)*sqrt(3*x^2 + 2))/(72*x^7 + 324*x^6 + 582*x^5 + 6
75*x^4 + 680*x^3 + 468*x^2 + 216*x + 108)

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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)**4/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.30592, size = 342, normalized size = 2.24 \begin{align*} \frac{55344}{52521875} \, \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{9 \,{\left ({\left (49879 \, x + 344464\right )} x - 6729\right )} x + 2510374}{105043750 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (112956 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 695865 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 2188890 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 3472470 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3050400 \, \sqrt{3} x - 259424 \, \sqrt{3} - 3050400 \, \sqrt{3 \, x^{2} + 2}\right )}}{52521875 \,{\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

55344/52521875*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))) + 1/105043750*(9*((49879*x + 344464)*x - 6729)*x + 2510374)/(3*x^2 + 2)^
(3/2) - 8/52521875*(112956*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 695865*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 +
2188890*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 3472470*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3050400*sqrt(3)*x
- 259424*sqrt(3) - 3050400*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)^3