3.13.20 \(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=158 \[ -\frac {773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac {233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac {699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac {6291 (4-9 x) \sqrt {3 x^2+2}}{84035000 (2 x+3)^2}-\frac {18873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42017500 \sqrt {35}} \]

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Rubi [A]  time = 0.08, antiderivative size = 158, 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 {773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac {233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac {699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac {6291 (4-9 x) \sqrt {3 x^2+2}}{84035000 (2 x+3)^2}-\frac {18873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42017500 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

(-6291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(84035000*(3 + 2*x)^2) - (699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(2401000*(3 + 2*x
)^4) - (233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(171500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) - (77
3*(2 + 3*x^2)^(7/2))/(68600*(3 + 2*x)^7) - (18873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42017500*Sqr
t[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 )^{5/2}}{(3+2 x)^9} \, dx &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {1}{280} \int \frac {(-328+39 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac {699 \int \frac {\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{2450}\\ &=-\frac {233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac {699 \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{17150}\\ &=-\frac {699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac {233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac {6291 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{1200500}\\ &=-\frac {6291 (4-9 x) \sqrt {2+3 x^2}}{84035000 (3+2 x)^2}-\frac {699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac {233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac {18873 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{42017500}\\ &=-\frac {6291 (4-9 x) \sqrt {2+3 x^2}}{84035000 (3+2 x)^2}-\frac {699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac {233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}-\frac {18873 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{42017500}\\ &=-\frac {6291 (4-9 x) \sqrt {2+3 x^2}}{84035000 (3+2 x)^2}-\frac {699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac {233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac {773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}-\frac {18873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42017500 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 144, normalized size = 0.91 \begin {gather*} \frac {1}{280} \left (-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}-\frac {13 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^8}+\frac {466 (9 x-4) \left (3 x^2+2\right )^{5/2}}{1225 (2 x+3)^6}+\frac {699 \left (\frac {35 \sqrt {3 x^2+2} \left (1269 x^3+408 x^2+927 x-604\right )}{(2 x+3)^4}-54 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )}{10504375}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

((466*(-4 + 9*x)*(2 + 3*x^2)^(5/2))/(1225*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(3 + 2*x)^8 - (773*(2 + 3*x^2)
^(7/2))/(245*(3 + 2*x)^7) + (699*((35*Sqrt[2 + 3*x^2]*(-604 + 927*x + 408*x^2 + 1269*x^3))/(3 + 2*x)^4 - 54*Sq
rt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/10504375)/280

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IntegrateAlgebraic [A]  time = 2.64, size = 106, normalized size = 0.67 \begin {gather*} \frac {18873 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{21008750 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (49626 x^7+2206008 x^6+210306726 x^5+33613440 x^4+226355535 x^3-178164896 x^2-38788883 x-104577556\right )}{84035000 (2 x+3)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

(Sqrt[2 + 3*x^2]*(-104577556 - 38788883*x - 178164896*x^2 + 226355535*x^3 + 33613440*x^4 + 210306726*x^5 + 220
6008*x^6 + 49626*x^7))/(84035000*(3 + 2*x)^8) + (18873*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x
^2])/Sqrt[35]])/(21008750*Sqrt[35])

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fricas [A]  time = 0.46, size = 179, normalized size = 1.13 \begin {gather*} \frac {18873 \, \sqrt {35} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 (49626 \, x^{7} + 2206008 \, x^{6} + 210306726 \, x^{5} + 33613440 \, x^{4} + 226355535 \, x^{3} - 178164896 \, x^{2} - 38788883 \, x - 104577556\right )} \sqrt {3 \, x^{2} + 2}}{2941225000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="fricas")

[Out]

1/2941225000*(18873*sqrt(35)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2
+ 34992*x + 6561)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(496
26*x^7 + 2206008*x^6 + 210306726*x^5 + 33613440*x^4 + 226355535*x^3 - 178164896*x^2 - 38788883*x - 104577556)*
sqrt(3*x^2 + 2))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x +
6561)

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giac [B]  time = 0.33, size = 457, normalized size = 2.89 \begin {gather*} \frac {18873}{1470612500} \, \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 \, \sqrt {3} {\left (178944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} + 138131220 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} + 30787400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 573375810 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 3328877720 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 8681082564 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 13787031160 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 1566458475 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 28541438480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 30582301680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 23140527424 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 12885596640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 1726278400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 9101541120 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 39843840 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 1411584\right )}}{10756480000 \, {\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 )}^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="giac")

[Out]

18873/1470612500*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqr
t(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/10756480000*sqrt(3)*(178944*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^
15 + 138131220*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 30787400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 57337581
0*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 3328877720*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 8681082564*(sqrt(3)
*x - sqrt(3*x^2 + 2))^10 - 13787031160*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 1566458475*(sqrt(3)*x - sqrt(
3*x^2 + 2))^8 - 28541438480*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 30582301680*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^6 - 23140527424*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 12885596640*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1726
278400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 9101541120*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 39843840*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 1411584)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 2)) - 2)^8

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maple [B]  time = 0.09, size = 299, normalized size = 1.89 \begin {gather*} \frac {78643791 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{9007501562500}+\frac {2208141 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{102942875000}+\frac {169857 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{2941225000}-\frac {18873 \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 )}{1470612500}-\frac {233 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{5488000 \left (x +\frac {3}{2}\right )^{6}}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{71680 \left (x +\frac {3}{2}\right )^{8}}-\frac {20271 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1680700000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2097 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{96040000 \left (x +\frac {3}{2}\right )^{5}}-\frac {207603 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{29412250000 \left (x +\frac {3}{2}\right )^{3}}-\frac {2258469 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{514714375000 \left (x +\frac {3}{2}\right )^{2}}-\frac {26214597 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{9007501562500 \left (x +\frac {3}{2}\right )}+\frac {150984 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{2251875390625}+\frac {12582 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12867859375}+\frac {18873 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1470612500}-\frac {773 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{8780800 \left (x +\frac {3}{2}\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^9,x)

[Out]

-233/5488000/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-13/71680/(x+3/2)^8*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-20271/16
80700000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-2097/96040000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-207603/
29412250000/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-2258469/514714375000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/
2)[phone]/9007501562500*(-9*x+3*(x+3/2)^2-19/4)^(5/2)*x-26214597/9007501562500/(x+3/2)*(-9*x+3*(x+3/2)^2-19/
4)^(7/2)+2208141/102942875000*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x+169857/2941225000*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*
x-18873/1470612500*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+150984/2251875390625
*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+12582/12867859375*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+18873/1470612500*(-36*x+12*(x+3
/2)^2-19)^(1/2)-773/8780800/(x+3/2)^7*(-9*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [B]  time = 1.45, size = 376, normalized size = 2.38 \begin {gather*} \frac {6775407}{514714375000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{280 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {773 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{68600 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {233 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{85750 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {2097 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3001250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {20271 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{105043750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {207603 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3676531250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2258469 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{128678593750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {2208141}{102942875000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {12582}{12867859375} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {26214597 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{514714375000 \, {\left (2 \, x + 3\right )}} + \frac {169857}{2941225000} \, \sqrt {3 \, x^{2} + 2} x + \frac {18873}{1470612500} \, \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 {18873}{735306250} \, \sqrt {3 \, x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="maxima")

[Out]

6775407/514714375000*(3*x^2 + 2)^(5/2) - 13/280*(3*x^2 + 2)^(7/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5
+ 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 773/68600*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 604
8*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 233/85750*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 +
2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 2097/3001250*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 +
1080*x^2 + 810*x + 243) - 20271/105043750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 207603/
3676531250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2258469/128678593750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12
*x + 9) + 2208141/102942875000*(3*x^2 + 2)^(3/2)*x + 12582/12867859375*(3*x^2 + 2)^(3/2) - 26214597/5147143750
00*(3*x^2 + 2)^(5/2)/(2*x + 3) + 169857/2941225000*sqrt(3*x^2 + 2)*x + 18873/1470612500*sqrt(35)*arcsinh(3/2*s
qrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 18873/735306250*sqrt(3*x^2 + 2)

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mupad [B]  time = 1.89, size = 326, normalized size = 2.06 \begin {gather*} \frac {18873\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1470612500}-\frac {18873\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1470612500}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{131072\,\left (x^8+12\,x^7+63\,x^6+189\,x^5+\frac {2835\,x^4}{8}+\frac {1701\,x^3}{4}+\frac {5103\,x^2}{16}+\frac {2187\,x}{16}+\frac {6561}{256}\right )}-\frac {4816641\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70246400\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {861381\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4014080\,\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 {24813\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{10756480000\,\left (x+\frac {3}{2}\right )}-\frac {81899\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{229376\,\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 {48141\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{614656000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {20705\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{65536\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {1573857\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{175616000\,\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)^(5/2)*(x - 5))/(2*x + 3)^9,x)

[Out]

(18873*35^(1/2)*log(x + 3/2))/1470612500 - (18873*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/
9))/1470612500 - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(131072*((2187*x)/16 + (5103*x^2)/16 + (1701*x^3)/4 + (2835
*x^4)/8 + 189*x^5 + 63*x^6 + 12*x^7 + x^8 + 6561/256)) - (4816641*3^(1/2)*(x^2 + 2/3)^(1/2))/(70246400*((27*x)
/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (861381*3^(1/2)*(x^2 + 2/3)^(1/2))/(4014080*((405*x)/16 + (135*x^2)/
4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) + (24813*3^(1/2)*(x^2 + 2/3)^(1/2))/(10756480000*(x + 3/2)) - (81
899*3^(1/2)*(x^2 + 2/3)^(1/2))/(229376*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 +
729/64)) + (48141*3^(1/2)*(x^2 + 2/3)^(1/2))/(614656000*(3*x + x^2 + 9/4)) + (20705*3^(1/2)*(x^2 + 2/3)^(1/2)
)/(65536*((5103*x)/64 + (5103*x^2)/32 + (2835*x^3)/16 + (945*x^4)/8 + (189*x^5)/4 + (21*x^6)/2 + x^7 + 2187/12
8)) + (1573857*3^(1/2)*(x^2 + 2/3)^(1/2))/(175616000*((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)**(5/2)/(3+2*x)**9,x)

[Out]

Timed out

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