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

Optimal. Leaf size=148 \[ -\frac {1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac {111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac {281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac {1017 (4-9 x) \sqrt {3 x^2+2}}{7503125 (2 x+3)^2}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \]

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Rubi [A]  time = 0.09, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac {111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac {281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac {1017 (4-9 x) \sqrt {3 x^2+2}}{7503125 (2 x+3)^2}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1017*(4 - 9*x)*Sqrt[2 + 3*x^2])/(7503125*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(210*(3 + 2*x)^6) - (281*(2 +
3*x^2)^(3/2))/(12250*(3 + 2*x)^5) - (111*(2 + 3*x^2)^(3/2))/(17500*(3 + 2*x)^4) - (1207*(2 + 3*x^2)^(3/2))/(8
57500*(3 + 2*x)^3) - (6102*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*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) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {1}{210} \int \frac {(-246+117 x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}+\frac {\int \frac {(8730-5058 x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx}{36750}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {\int \frac {(-233352+97902 x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx}{5145000}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac {2034 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{214375}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac {6102 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{7503125}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac {6102 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{7503125}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{7503125 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 80, normalized size = 0.54 \begin {gather*} \frac {-36612 \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 (642132 x^5+5388660 x^4+18236055 x^3+30753930 x^2+18651300 x+22308548\right )}{(2 x+3)^6}}{1575656250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(22308548 + 18651300*x + 30753930*x^2 + 18236055*x^3 + 5388660*x^4 + 642132*x^5))/(3 + 2
*x)^6 - 36612*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/1575656250

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IntegrateAlgebraic [A]  time = 1.61, size = 96, normalized size = 0.65 \begin {gather*} \frac {12204 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{7503125 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-642132 x^5-5388660 x^4-18236055 x^3-30753930 x^2-18651300 x-22308548\right )}{45018750 (2 x+3)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-22308548 - 18651300*x - 30753930*x^2 - 18236055*x^3 - 5388660*x^4 - 642132*x^5))/(45018750*
(3 + 2*x)^6) + (12204*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(7503125*Sqrt[35]
)

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fricas [A]  time = 0.42, size = 149, normalized size = 1.01 \begin {gather*} \frac {18306 \, \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 (642132 \, x^{5} + 5388660 \, x^{4} + 18236055 \, x^{3} + 30753930 \, x^{2} + 18651300 \, x + 22308548\right )} \sqrt {3 \, x^{2} + 2}}{1575656250 \, {\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)^(1/2)/(3+2*x)^7,x, algorithm="fricas")

[Out]

1/1575656250*(18306*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*(642132*x^5 + 5388660*x^4 + 18236055
*x^3 + 30753930*x^2 + 18651300*x + 22308548)*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 = 0.27, size = 367, normalized size = 2.48 \begin {gather*} \frac {6102}{262609375} \, \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 (21696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 1073952 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 6978880 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 87678735 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 66333990 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 258582989 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 426764436 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 755892540 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 355133440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 207134880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 19853952 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 2283136\right )}}{240100000 \, {\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)^(1/2)/(3+2*x)^7,x, algorithm="giac")

[Out]

6102/262609375*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))) - 3/240100000*sqrt(3)*(21696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 +
1073952*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 6978880*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 87678735*(sqrt(3)
*x - sqrt(3*x^2 + 2))^8 - 66333990*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 258582989*(sqrt(3)*x - sqrt(3*x^2
+ 2))^6 - 426764436*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 755892540*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 355
133440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 207134880*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 19853952*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 2)) + 2283136)/((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 [A]  time = 0.07, size = 191, normalized size = 1.29 \begin {gather*} \frac {27459 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{262609375}-\frac {6102 \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 )}{262609375}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {281 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{392000 \left (x +\frac {3}{2}\right )^{5}}-\frac {111 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{280000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1207 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6860000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1017 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )^{2}}-\frac {9153 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{262609375 \left (x +\frac {3}{2}\right )}+\frac {6102 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{262609375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^7,x)

[Out]

-13/13440/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-281/392000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-111/28000
0/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-1207/6860000/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-1017/15006250/(
x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-9153/262609375/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+6102/262609375*(-3
6*x+12*(x+3/2)^2-19)^(1/2)-6102/262609375*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2
))+27459/262609375*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x

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maxima [A]  time = 1.23, size = 229, normalized size = 1.55 \begin {gather*} \frac {6102}{262609375} \, \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 {3051}{15006250} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {281 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{12250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {111 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{17500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1207 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{857500 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2034 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{7503125 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {9153 \, \sqrt {3 \, x^{2} + 2}}{15006250 \, {\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)^(1/2)/(3+2*x)^7,x, algorithm="maxima")

[Out]

6102/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 3051/15006250*sqrt(3*
x^2 + 2) - 13/210*(3*x^2 + 2)^(3/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 281/1
2250*(3*x^2 + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 111/17500*(3*x^2 + 2)^(3/2)/(16
*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1207/857500*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2034/7503
125*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 9153/15006250*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 0.13, size = 223, normalized size = 1.51 \begin {gather*} \frac {6102\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{262609375}-\frac {6102\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{262609375}+\frac {127\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1568000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {109\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{44800\,\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 {53511\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{240100000\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536\,\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 {2727\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{13720000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {479\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3920000\,\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)^(1/2)*(x - 5))/(2*x + 3)^7,x)

[Out]

(6102*35^(1/2)*log(x + 3/2))/262609375 - (6102*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))
/262609375 + (127*3^(1/2)*(x^2 + 2/3)^(1/2))/(1568000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (109*3^
(1/2)*(x^2 + 2/3)^(1/2))/(44800*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (53511*
3^(1/2)*(x^2 + 2/3)^(1/2))/(240100000*(x + 3/2)) - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(1536*((729*x)/16 + (1215*x^
2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) - (2727*3^(1/2)*(x^2 + 2/3)^(1/2))/(13720000*(3*x +
x^2 + 9/4)) - (479*3^(1/2)*(x^2 + 2/3)^(1/2))/(3920000*((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)**(1/2)/(3+2*x)**7,x)

[Out]

Timed out

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