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

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

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Rubi [A]  time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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 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])

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*}

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Mathematica [A]  time = 0.20, size = 137, normalized size = 1.05 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 1.69, size = 96, normalized size = 0.73 \begin {gather*} \frac {27 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{4375 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-432 x^5-2160 x^4+39195 x^3-33180 x^2-3675 x-39748\right )}{52500 (2 x+3)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-39748 - 3675*x - 33180*x^2 + 39195*x^3 - 2160*x^4 - 432*x^5))/(52500*(3 + 2*x)^6) + (27*Arc
Tanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(4375*Sqrt[35])

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fricas [A]  time = 0.44, size = 149, normalized size = 1.14 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.31, size = 367, normalized size = 2.80 \begin {gather*} \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 {3 \, \sqrt {3} {\left (96 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 17877 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 4120 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 25860 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 225240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 173964 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 648336 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 641040 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 309440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 135120 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 10752 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 1536\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 {gather*}

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))) - 3/280000*sqrt(3)*(96*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 17877*(sqrt
(3)*x - sqrt(3*x^2 + 2))^10 - 4120*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 25860*(sqrt(3)*x - sqrt(3*x^2 + 2
))^8 - 225240*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 173964*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 648336*sqrt(3
)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 641040*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 309440*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 2))^3 - 135120*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 10752*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 1536)/(
(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^6

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maple [B]  time = 0.06, size = 224, normalized size = 1.71 \begin {gather*} \frac {243 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{612500}+\frac {3159 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{21437500}-\frac {27 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{306250}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {29 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{56000 \left (x +\frac {3}{2}\right )^{5}}-\frac {\left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4000 \left (x +\frac {3}{2}\right )^{4}}-\frac {9 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{70000 \left (x +\frac {3}{2}\right )^{3}}-\frac {93 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1225000 \left (x +\frac {3}{2}\right )^{2}}-\frac {1053 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{21437500 \left (x +\frac {3}{2}\right )}+\frac {36 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{5359375}+\frac {27 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{306250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^7,x)

[Out]

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

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maxima [B]  time = 1.32, size = 252, normalized size = 1.92 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.86, size = 223, normalized size = 1.70 \begin {gather*} \frac {27\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{306250}-\frac {27\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{306250}-\frac {5977\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{89600\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {577\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{5120\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70000\,\left (x+\frac {3}{2}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6144\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{28000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {2829\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{224000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^7,x)

[Out]

(27*35^(1/2)*log(x + 3/2))/306250 - (27*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/306250
- (5977*3^(1/2)*(x^2 + 2/3)^(1/2))/(89600*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (577*3^(1/2)*(x^2
+ 2/3)^(1/2))/(5120*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (9*3^(1/2)*(x^2 + 2
/3)^(1/2))/(70000*(x + 3/2)) - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(6144*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2
+ (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (9*3^(1/2)*(x^2 + 2/3)^(1/2))/(28000*(3*x + x^2 + 9/4)) + (2829*3^(1
/2)*(x^2 + 2/3)^(1/2))/(224000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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