3.1379 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=131 \[ -\frac{29 \left (3 x^2+2\right )^{5/2}}{1750 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}-\frac{(4-9 x) \left (3 x^2+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{9 (4-9 x) \sqrt{3 x^2+2}}{17500 (2 x+3)^2}-\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8750 \sqrt{35}} \]

[Out]

(-9*(4 - 9*x)*Sqrt[2 + 3*x^2])/(17500*(3 + 2*x)^2) - ((4 - 9*x)*(2 + 3*x^2)^(3/2))/(500*(3 + 2*x)^4) - (13*(2
+ 3*x^2)^(5/2))/(210*(3 + 2*x)^6) - (29*(2 + 3*x^2)^(5/2))/(1750*(3 + 2*x)^5) - (27*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(8750*Sqrt[35])

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Rubi [A]  time = 0.0691624, antiderivative size = 131, 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 = {835, 807, 721, 725, 206} \[ -\frac{29 \left (3 x^2+2\right )^{5/2}}{1750 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}-\frac{(4-9 x) \left (3 x^2+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{9 (4-9 x) \sqrt{3 x^2+2}}{17500 (2 x+3)^2}-\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8750 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^7,x]

[Out]

(-9*(4 - 9*x)*Sqrt[2 + 3*x^2])/(17500*(3 + 2*x)^2) - ((4 - 9*x)*(2 + 3*x^2)^(3/2))/(500*(3 + 2*x)^4) - (13*(2
+ 3*x^2)^(5/2))/(210*(3 + 2*x)^6) - (29*(2 + 3*x^2)^(5/2))/(1750*(3 + 2*x)^5) - (27*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(8750*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 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)^7} \, dx &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{1}{210} \int \frac{(-246+39 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx\\ &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}+\frac{7}{25} \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=-\frac{(4-9 x) \left (2+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}+\frac{9}{250} \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac{9 (4-9 x) \sqrt{2+3 x^2}}{17500 (3+2 x)^2}-\frac{(4-9 x) \left (2+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}+\frac{27 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{8750}\\ &=-\frac{9 (4-9 x) \sqrt{2+3 x^2}}{17500 (3+2 x)^2}-\frac{(4-9 x) \left (2+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}-\frac{27 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{8750}\\ &=-\frac{9 (4-9 x) \sqrt{2+3 x^2}}{17500 (3+2 x)^2}-\frac{(4-9 x) \left (2+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac{29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}-\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{8750 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.207752, size = 137, normalized size = 1.05 \[ \frac{1}{210} \left (-\frac{13 \left (3 x^2+2\right )^{5/2}}{(2 x+3)^6}-\frac{3 \left (10150 \left (3 x^2+2\right )^{5/2}+(2 x+3) \left (-315 (9 x-4) \sqrt{3 x^2+2} (2 x+3)^2-1225 (9 x-4) \left (3 x^2+2\right )^{3/2}+54 \sqrt{35} (2 x+3)^4 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )\right )\right )}{8750 (2 x+3)^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^7,x]

[Out]

((-13*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6 - (3*(10150*(2 + 3*x^2)^(5/2) + (3 + 2*x)*(-315*(3 + 2*x)^2*(-4 + 9*x)*Sq
rt[2 + 3*x^2] - 1225*(-4 + 9*x)*(2 + 3*x^2)^(3/2) + 54*Sqrt[35]*(3 + 2*x)^4*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])))/(8750*(3 + 2*x)^5))/210

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Maple [B]  time = 0.015, size = 224, normalized size = 1.7 \begin{align*} -{\frac{29}{56000} \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{1}{4000} \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{9}{70000} \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{93}{1225000} \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{1053}{21437500} \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{36}{5359375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{243\,x}{612500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{27}{306250}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{27\,\sqrt{35}}{306250}{\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{3159\,x}{21437500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}} \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)^7,x)

[Out]

-29/56000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1/4000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)-9/70000/(x+3/2)
^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-93/1225000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1053/21437500/(x+3/2)*(3*(x+
3/2)^2-9*x-19/4)^(5/2)+36/5359375*(3*(x+3/2)^2-9*x-19/4)^(3/2)+243/612500*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+27/30
6250*(12*(x+3/2)^2-36*x-19)^(1/2)-27/306250*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2
))+3159/21437500*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-13/13440/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [B]  time = 1.51367, size = 340, normalized size = 2.6 \begin{align*} \frac{279}{1225000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{29 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{1750 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{250 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{8750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{93 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{306250 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{243}{612500} \, \sqrt{3 \, x^{2} + 2} x + \frac{27}{306250} \, \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{27}{153125} \, \sqrt{3 \, x^{2} + 2} - \frac{1053 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225000 \,{\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)^7,x, algorithm="maxima")

[Out]

279/1225000*(3*x^2 + 2)^(3/2) - 13/210*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 +
2916*x + 729) - 29/1750*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1/250*(3*x^2
+ 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 9/8750*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) -
93/306250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 243/612500*sqrt(3*x^2 + 2)*x + 27/306250*sqrt(35)*arcsinh(3/
2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 27/153125*sqrt(3*x^2 + 2) - 1053/1225000*(3*x^2 + 2)^(3
/2)/(2*x + 3)

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Fricas [A]  time = 2.21559, size = 439, normalized size = 3.35 \begin{align*} \frac{81 \, \sqrt{35}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\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 (432 \, x^{5} + 2160 \, x^{4} - 39195 \, x^{3} + 33180 \, x^{2} + 3675 \, x + 39748\right )} \sqrt{3 \, x^{2} + 2}}{1837500 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\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)^7,x, algorithm="fricas")

[Out]

1/1837500*(81*sqrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-(sqrt(35)*sqrt(
3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(432*x^5 + 2160*x^4 - 39195*x^3 + 33180*x^
2 + 3675*x + 39748)*sqrt(3*x^2 + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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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)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.28711, size = 490, normalized size = 3.74 \begin{align*} \frac{27}{306250} \, \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 (96 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 5959 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 4120 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 8620 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 225240 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 57988 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 648336 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 213680 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 309440 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 45040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 10752 \, \sqrt{3} x + 512 \, \sqrt{3} + 10752 \, \sqrt{3 \, x^{2} + 2}\right )}}{280000 \,{\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 )}^{6}} \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)^7,x, algorithm="giac")

[Out]

27/306250*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))) - 9/280000*(96*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 5959*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2))^10 - 4120*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 8620*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 225
240*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 57988*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 648336*(sqrt(3)*x - sqrt
(3*x^2 + 2))^5 + 213680*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 309440*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 450
40*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 10752*sqrt(3)*x + 512*sqrt(3) + 10752*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)^6