3.1377 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=106 \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(2943 + 4097*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)^2) + ((54 + 491*x)*(2 + 3*x^2)^(3/2))/(840*(3 + 2*x)^4) -
 (3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (39663*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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Rubi [A]  time = 0.0618945, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {811, 844, 215, 725, 206} \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2943 + 4097*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)^2) + ((54 + 491*x)*(2 + 3*x^2)^(3/2))/(840*(3 + 2*x)^4) -
 (3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (39663*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
 x] && GtQ[a, 0] && PosQ[b]

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)^5} \, dx &=\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{\int \frac{(-936+840 x) \sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{1120}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}+\frac{\int \frac{105408-352800 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{627200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{9}{32} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{39663 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{39200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{39663 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{39200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{39200 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.151156, size = 90, normalized size = 0.85 \[ \frac{\frac{70 \sqrt{3 x^2+2} \left (250602 x^3+559764 x^2+718441 x+245943\right )}{(2 x+3)^4}-118989 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{4116000}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + ((70*Sqrt[2 + 3*x^2]*(245943 + 718441*x + 559764*x^2 + 250602*x^3))/(3
+ 2*x)^4 - 118989*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/4116000

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Maple [B]  time = 0.013, size = 194, normalized size = 1.8 \begin{align*} -{\frac{211}{117600} \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{999}{686000} \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{5779}{12005000} \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{13221}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{7227\,x}{686000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{3\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{39663}{1372000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{39663\,\sqrt{35}}{1372000}{\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{17337\,x}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}} \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)^5,x)

[Out]

-211/117600/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-999/686000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-5779/1200
5000/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+13221/6002500*(3*(x+3/2)^2-9*x-19/4)^(3/2)-7227/686000*x*(3*(x+3/2)^
2-9*x-19/4)^(1/2)-3/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)+39663/1372000*(12*(x+3/2)^2-36*x-19)^(1/2)-39663/1372000
*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+17337/12005000*x*(3*(x+3/2)^2-9*x-19/4)^
(3/2)-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [B]  time = 1.5419, size = 247, normalized size = 2.33 \begin{align*} \frac{2997}{686000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{211 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{14700 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{999 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{171500 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{7227}{686000} \, \sqrt{3 \, x^{2} + 2} x - \frac{3}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{39663}{1372000} \, \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{39663}{686000} \, \sqrt{3 \, x^{2} + 2} - \frac{5779 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{686000 \,{\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)^5,x, algorithm="maxima")

[Out]

2997/686000*(3*x^2 + 2)^(3/2) - 13/140*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 211/14700*
(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 999/171500*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 7227/686000
*sqrt(3*x^2 + 2)*x - 3/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 39663/1372000*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*
x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 39663/686000*sqrt(3*x^2 + 2) - 5779/686000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 2.24925, size = 490, normalized size = 4.62 \begin{align*} \frac{385875 \, \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 118989 \, \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 ) + 140 \,{\left (250602 \, x^{3} + 559764 \, x^{2} + 718441 \, x + 245943\right )} \sqrt{3 \, x^{2} + 2}}{8232000 \,{\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*x^2+2)^(3/2)/(3+2*x)^5,x, algorithm="fricas")

[Out]

1/8232000*(385875*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
+ 118989*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)) + 140*(250602*x^3 + 559764*x^2 + 718441*x + 245943)*sqrt(3*x^2 + 2))/(16*x^4 +
 96*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*x**2+2)**(3/2)/(3+2*x)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.37664, size = 333, normalized size = 3.14 \begin{align*} -\frac{39663}{1372000} \, \sqrt{35} \log \left (-9 \, \sqrt{35} + 35 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{35 \, \sqrt{35}}{2 \, x + 3}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{3}{32} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{470400} \,{\left (\frac{35 \,{\left (\frac{35 \,{\left (\frac{1365 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 1193 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 16227 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 125301 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} \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)^5,x, algorithm="giac")

[Out]

-39663/1372000*sqrt(35)*log(-9*sqrt(35) + 35*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 35*sqrt(35)/(2*x + 3))
*sgn(1/(2*x + 3)) + 3/32*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(
35)/(2*x + 3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 1
/470400*(35*(35*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 1193*sgn(1/(2*x + 3)))/(2*x + 3) + 16227*sgn(1/(2*x + 3)))/
(2*x + 3) - 125301*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)