3.1378 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=109 \[ -\frac{13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{171500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{85750 \sqrt{35}} \]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(4900*(3 + 2*x)^4) -
(13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) - (1107*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(85750*Sqrt[35
])

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Rubi [A]  time = 0.048901, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {807, 721, 725, 206} \[ -\frac{13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{171500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{85750 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(4900*(3 + 2*x)^4) -
(13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) - (1107*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(85750*Sqrt[35
])

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) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac{41}{35} \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac{369 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{2450}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{171500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac{1107 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{85750}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{171500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}-\frac{1107 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{85750}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{171500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{85750 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.10019, size = 112, normalized size = 1.03 \[ \frac{1}{350} \left (-\frac{26 \left (3 x^2+2\right )^{5/2}}{(2 x+3)^5}+\frac{41 (9 x-4) \left (3 x^2+2\right )^{3/2}}{14 (2 x+3)^4}-\frac{369 \left (6 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{35 (9 x-4) \sqrt{3 x^2+2}}{(2 x+3)^2}\right )}{17150}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

((41*(-4 + 9*x)*(2 + 3*x^2)^(3/2))/(14*(3 + 2*x)^4) - (26*(2 + 3*x^2)^(5/2))/(3 + 2*x)^5 - (369*((-35*(-4 + 9*
x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 + 6*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/17150)/350

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Maple [B]  time = 0.013, size = 203, normalized size = 1.9 \begin{align*} -{\frac{13}{5600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{41}{39200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{369}{686000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{3813}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{43173}{210087500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{1476}{52521875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{9963\,x}{6002500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{1107}{3001250}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{1107\,\sqrt{35}}{3001250}{\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{129519\,x}{210087500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/5600/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(5/2)-41/39200/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)-369/686000/(x+
3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-3813/12005000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-43173/210087500/(x+3/
2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+1476/52521875*(3*(x+3/2)^2-9*x-19/4)^(3/2)+9963/6002500*x*(3*(x+3/2)^2-9*x-19/
4)^(1/2)+1107/3001250*(12*(x+3/2)^2-36*x-19)^(1/2)-1107/3001250*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+
3/2)^2-36*x-19)^(1/2))+129519/210087500*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [B]  time = 1.52518, size = 282, normalized size = 2.59 \begin{align*} \frac{11439}{12005000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{175 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{41 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{2450 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{369 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{85750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{3813 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{3001250 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{9963}{6002500} \, \sqrt{3 \, x^{2} + 2} x + \frac{1107}{3001250} \, \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{1107}{1500625} \, \sqrt{3 \, x^{2} + 2} - \frac{43173 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{12005000 \,{\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)^(3/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

11439/12005000*(3*x^2 + 2)^(3/2) - 13/175*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 2
43) - 41/2450*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 369/85750*(3*x^2 + 2)^(5/2)/(8*x^3
+ 36*x^2 + 54*x + 27) - 3813/3001250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 9963/6002500*sqrt(3*x^2 + 2)*x + 1
107/3001250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1107/1500625*sqrt(3*x^2
+ 2) - 43173/12005000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 2.29368, size = 398, normalized size = 3.65 \begin{align*} \frac{1107 \, \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 (10602 \, x^{4} - 189543 \, x^{3} + 26682 \, x^{2} - 64493 \, x + 125252\right )} \sqrt{3 \, x^{2} + 2}}{6002500 \,{\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)^(3/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/6002500*(1107*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*(10602*x^4 - 189543*x^3 + 26682*x^2 - 64493*x + 12525
2)*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*x**2+2)**(3/2)/(3+2*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.32622, size = 429, normalized size = 3.94 \begin{align*} \frac{1107}{3001250} \, \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 (89686 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 138886 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 1224478 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 245133 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1224531 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 4374874 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 4855928 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1339152 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 586816 \, \sqrt{3} x - 37696 \, \sqrt{3} + 586816 \, \sqrt{3 \, x^{2} + 2}\right )}}{2744000 \,{\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)^(3/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

1107/3001250*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))) - 9/2744000*(89686*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 138886*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 2))^8 + 1224478*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 245133*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
2))^6 - 1224531*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4374874*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 4855928*(
sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1339152*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 586816*sqrt(3)*x - 37696*sq
rt(3) + 586816*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