3.1406 \(\int \frac{5-x}{(3+2 x)^5 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^4) - (97*Sqrt[2 + 3*x^2])/(2100*(3 + 2*x)^3) - (87*Sqrt[2 + 3*x^2])/(4900
*(3 + 2*x)^2) - (991*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)) + (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/
(42875*Sqrt[35])

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Rubi [A]  time = 0.075237, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {835, 807, 725, 206} \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^4) - (97*Sqrt[2 + 3*x^2])/(2100*(3 + 2*x)^3) - (87*Sqrt[2 + 3*x^2])/(4900
*(3 + 2*x)^2) - (991*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)) + (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/
(42875*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)^5 \sqrt{2+3 x^2}} \, dx &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{1}{140} \int \frac{-164+117 x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}+\frac{\int \frac{3024-4074 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{14700}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{\int \frac{-21840+54810 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{1029000}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}-\frac{27 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{42875}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}+\frac{27 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{42875}\\ &=-\frac{13 \sqrt{2+3 x^2}}{140 (3+2 x)^4}-\frac{97 \sqrt{2+3 x^2}}{2100 (3+2 x)^3}-\frac{87 \sqrt{2+3 x^2}}{4900 (3+2 x)^2}-\frac{991 \sqrt{2+3 x^2}}{171500 (3+2 x)}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{42875 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0813856, size = 70, normalized size = 0.58 \[ \frac{81 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{35 \sqrt{3 x^2+2} \left (5946 x^3+35892 x^2+79423 x+70389\right )}{(2 x+3)^4}}{4501875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(70389 + 79423*x + 35892*x^2 + 5946*x^3))/(3 + 2*x)^4 + 81*Sqrt[35]*ArcTanh[(4 - 9*x)/(S
qrt[35]*Sqrt[2 + 3*x^2])])/4501875

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Maple [A]  time = 0.011, size = 116, normalized size = 1. \begin{align*} -{\frac{97}{16800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{87}{19600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{991}{343000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{27\,\sqrt{35}}{1500625}{\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}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-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)^(1/2),x)

[Out]

-97/16800/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(1/2)-87/19600/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-991/343000/(x
+3/2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)+27/1500625*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 [A]  time = 1.61529, size = 185, normalized size = 1.53 \begin{align*} -\frac{27}{1500625} \, \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}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{97 \, \sqrt{3 \, x^{2} + 2}}{2100 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{87 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{991 \, \sqrt{3 \, x^{2} + 2}}{171500 \,{\left (2 \, x + 3\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)^(1/2),x, algorithm="maxima")

[Out]

-27/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/140*sqrt(3*x^2 + 2)/(
16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 97/2100*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 87/4900*sqrt(
3*x^2 + 2)/(4*x^2 + 12*x + 9) - 991/171500*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 1.77651, size = 339, normalized size = 2.8 \begin{align*} \frac{81 \, \sqrt{35}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (5946 \, x^{3} + 35892 \, x^{2} + 79423 \, x + 70389\right )} \sqrt{3 \, x^{2} + 2}}{9003750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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)^(1/2),x, algorithm="fricas")

[Out]

1/9003750*(81*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x
^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 70*(5946*x^3 + 35892*x^2 + 79423*x + 70389)*sqrt(3*x^2 + 2))/(16*x^4 + 9
6*x^3 + 216*x^2 + 216*x + 81)

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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)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 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)^(1/2),x, algorithm="giac")

[Out]

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