3.13.21 \(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=180 \[ -\frac {4741 \left (3 x^2+2\right )^{7/2}}{1800750 (2 x+3)^7}-\frac {27 \left (3 x^2+2\right )^{7/2}}{2450 (2 x+3)^8}-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac {949 (4-9 x) \left (3 x^2+2\right )^{5/2}}{3001250 (2 x+3)^6}-\frac {2847 (4-9 x) \left (3 x^2+2\right )^{3/2}}{42017500 (2 x+3)^4}-\frac {25623 (4-9 x) \sqrt {3 x^2+2}}{1470612500 (2 x+3)^2}-\frac {76869 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{735306250 \sqrt {35}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {4741 \left (3 x^2+2\right )^{7/2}}{1800750 (2 x+3)^7}-\frac {27 \left (3 x^2+2\right )^{7/2}}{2450 (2 x+3)^8}-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac {949 (4-9 x) \left (3 x^2+2\right )^{5/2}}{3001250 (2 x+3)^6}-\frac {2847 (4-9 x) \left (3 x^2+2\right )^{3/2}}{42017500 (2 x+3)^4}-\frac {25623 (4-9 x) \sqrt {3 x^2+2}}{1470612500 (2 x+3)^2}-\frac {76869 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{735306250 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

(-25623*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1470612500*(3 + 2*x)^2) - (2847*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(42017500*(3
+ 2*x)^4) - (949*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(3001250*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9)
- (27*(2 + 3*x^2)^(7/2))/(2450*(3 + 2*x)^8) - (4741*(2 + 3*x^2)^(7/2))/(1800750*(3 + 2*x)^7) - (76869*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(735306250*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 )^{5/2}}{(3+2 x)^{10}} \, dx &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {1}{315} \int \frac {(-369+78 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}+\frac {\int \frac {(24072-2916 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{88200}\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {2847 \int \frac {\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{42875}\\ &=-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {2847 \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{300125}\\ &=-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {25623 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{21008750}\\ &=-\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {76869 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{735306250}\\ &=-\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{735306250}\\ &=-\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{735306250 \sqrt {35}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 185, normalized size = 1.03 \begin {gather*} \frac {1}{315} \left (-\frac {243 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}-\frac {13 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^9}-\frac {3 \left (406540750 \left (3 x^2+2\right )^{7/2}+2847 (2 x+3) \left (-945 (9 x-4) \sqrt {3 x^2+2} (2 x+3)^4-3675 (9 x-4) \left (3 x^2+2\right )^{3/2} (2 x+3)^2-17150 (9 x-4) \left (3 x^2+2\right )^{5/2}+162 \sqrt {35} (2 x+3)^6 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )\right )}{1470612500 (2 x+3)^7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

((-13*(2 + 3*x^2)^(7/2))/(3 + 2*x)^9 - (243*(2 + 3*x^2)^(7/2))/(70*(3 + 2*x)^8) - (3*(406540750*(2 + 3*x^2)^(7
/2) + 2847*(3 + 2*x)*(-945*(3 + 2*x)^4*(-4 + 9*x)*Sqrt[2 + 3*x^2] - 3675*(3 + 2*x)^2*(-4 + 9*x)*(2 + 3*x^2)^(3
/2) - 17150*(-4 + 9*x)*(2 + 3*x^2)^(5/2) + 162*Sqrt[35]*(3 + 2*x)^6*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])))/(1470612500*(3 + 2*x)^7))/315

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.13, size = 111, normalized size = 0.62 \begin {gather*} \frac {76869 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{367653125 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-10968696 x^8-30006612 x^7+620594352 x^6+25197346566 x^5-9750269970 x^4+11567526201 x^3-42455611758 x^2-11990965797 x-15948113036\right )}{13235512500 (2 x+3)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

(Sqrt[2 + 3*x^2]*(-15948113036 - 11990965797*x - 42455611758*x^2 + 11567526201*x^3 - 9750269970*x^4 + 25197346
566*x^5 + 620594352*x^6 - 30006612*x^7 - 10968696*x^8))/(13235512500*(3 + 2*x)^9) + (76869*ArcTanh[3*Sqrt[3/35
] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(367653125*Sqrt[35])

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 194, normalized size = 1.08 \begin {gather*} \frac {691821 \, \sqrt {35} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\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 (10968696 \, x^{8} + 30006612 \, x^{7} - 620594352 \, x^{6} - 25197346566 \, x^{5} + 9750269970 \, x^{4} - 11567526201 \, x^{3} + 42455611758 \, x^{2} + 11990965797 \, x + 15948113036\right )} \sqrt {3 \, x^{2} + 2}}{463242937500 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\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)^10,x, algorithm="fricas")

[Out]

1/463242937500*(691821*sqrt(35)*(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 48988
8*x^3 + 314928*x^2 + 118098*x + 19683)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 +
12*x + 9)) - 35*(10968696*x^8 + 30006612*x^7 - 620594352*x^6 - 25197346566*x^5 + 9750269970*x^4 - 11567526201
*x^3 + 42455611758*x^2 + 11990965797*x + 15948113036)*sqrt(3*x^2 + 2))/(512*x^9 + 6912*x^8 + 41472*x^7 + 14515
2*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683)

________________________________________________________________________________________

giac [B]  time = 0.31, size = 502, normalized size = 2.79 \begin {gather*} \frac {76869}{25735718750} \, \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 (364416 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{17} + 27877824 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{16} + 1042205258 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} - 956098170 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} + 1003625490 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} - 85987901496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 60468401868 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 331045664193 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 22913148915 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 544736640510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 284856270864 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 908850124224 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 90616216992 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 115517223360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 52895204480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 565618176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 140708352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 17333248\right )}}{94119200000 \, {\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 )}^{9}} \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)^10,x, algorithm="giac")

[Out]

76869/25735718750*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sq
rt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/94119200000*sqrt(3)*(364416*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))
^17 + 27877824*(sqrt(3)*x - sqrt(3*x^2 + 2))^16 + 1042205258*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 - 956098
170*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 1003625490*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 - 85987901496*(sqrt
(3)*x - sqrt(3*x^2 + 2))^12 - 60468401868*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 331045664193*(sqrt(3)*x -
sqrt(3*x^2 + 2))^10 - 22913148915*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 544736640510*(sqrt(3)*x - sqrt(3*
x^2 + 2))^8 + 284856270864*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 908850124224*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^6 + 90616216992*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 115517223360*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 528
95204480*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 565618176*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 140708352*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 17333248)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 2)) - 2)^9

________________________________________________________________________________________

maple [B]  time = 0.10, size = 320, normalized size = 1.78 \begin {gather*} \frac {320313123 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{157631277343750}+\frac {8993673 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{1801500312500}+\frac {691821 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{51471437500}-\frac {76869 \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 )}{25735718750}-\frac {949 \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 )^{6}}-\frac {27 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{627200 \left (x +\frac {3}{2}\right )^{8}}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{161280 \left (x +\frac {3}{2}\right )^{9}}-\frac {82563 \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 )^{4}}-\frac {8541 \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 )^{5}}-\frac {845559 \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 )^{3}}-\frac {9198657 \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 )^{2}}-\frac {106771041 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{157631277343750 \left (x +\frac {3}{2}\right )}+\frac {1229904 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{78815638671875}+\frac {102492 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{450375078125}+\frac {76869 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{25735718750}-\frac {4741 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{230496000 \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)^10,x)

[Out]

-949/96040000/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-27/627200/(x+3/2)^8*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-13/161
280/(x+3/2)^9*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-82563/29412250000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-8541/168
0700000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-845559/514714375000/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-91
98657/9007501562500/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/2)[phone]/157631277343750*(-9*x+3*(x+3/2)^2-19/4)^
(5/2)*x-106771041/157631277343750/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+8993673/1801500312500*(-9*x+3*(x+3/2)^
2-19/4)^(3/2)*x+691821/51471437500*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-76869/25735718750*35^(1/2)*arctanh(2/35*(-9
*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+1229904/78815638671875*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+102492/4503
75078125*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+76869/25735718750*(-36*x+12*(x+3/2)^2-19)^(1/2)-4741/230496000/(x+3/2)^
7*(-9*x+3*(x+3/2)^2-19/4)^(7/2)

________________________________________________________________________________________

maxima [B]  time = 1.71, size = 434, normalized size = 2.41 \begin {gather*} \frac {27595971}{9007501562500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{315 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2450 \, {\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 {4741 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1800750 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {949 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1500625 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {8541 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{52521875 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {82563 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1838265625 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {845559 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{64339296875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9198657 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2251875390625 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {8993673}{1801500312500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {102492}{450375078125} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {106771041 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{9007501562500 \, {\left (2 \, x + 3\right )}} + \frac {691821}{51471437500} \, \sqrt {3 \, x^{2} + 2} x + \frac {76869}{25735718750} \, \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 {76869}{12867859375} \, \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)^10,x, algorithm="maxima")

[Out]

27595971/9007501562500*(3*x^2 + 2)^(5/2) - 13/315*(3*x^2 + 2)^(7/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x
^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 27/2450*(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) - 4741/1800750*(3*
x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 949/1500
625*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 8541/52521875*(3*x^
2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 82563/1838265625*(3*x^2 + 2)^(7/2)/(16*x^
4 + 96*x^3 + 216*x^2 + 216*x + 81) - 845559/64339296875*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 91986
57/2251875390625*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 8993673/1801500312500*(3*x^2 + 2)^(3/2)*x + 102492/450
375078125*(3*x^2 + 2)^(3/2) - 106771041/9007501562500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 691821/51471437500*sqrt(3*
x^2 + 2)*x + 76869/25735718750*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 76869
/12867859375*sqrt(3*x^2 + 2)

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 385, normalized size = 2.14 \begin {gather*} \frac {76869\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{25735718750}-\frac {76869\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{25735718750}+\frac {4515\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32768\,\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 {1838301\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{614656000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{294912\,\left (x^9+\frac {27\,x^8}{2}+81\,x^7+\frac {567\,x^6}{2}+\frac {5103\,x^5}{8}+\frac {15309\,x^4}{16}+\frac {15309\,x^3}{16}+\frac {19683\,x^2}{32}+\frac {59049\,x}{256}+\frac {19683}{512}\right )}-\frac {923241\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{35123200\,\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 {152343\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{94119200000\,\left (x+\frac {3}{2}\right )}+\frac {35213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{401408\,\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 {80649\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{5378240000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {52201\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{344064\,\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 {55473\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536640000\,\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)^10,x)

[Out]

(76869*35^(1/2)*log(x + 3/2))/25735718750 - (76869*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4
/9))/25735718750 + (4515*3^(1/2)*(x^2 + 2/3)^(1/2))/(32768*((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)) + (1838301*3^(1/2)*(x^2 + 2/3)^(1/2))/(614656000*((27*x
)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(294912*((59049*x)/256 + (19683*x
^2)/32 + (15309*x^3)/16 + (15309*x^4)/16 + (5103*x^5)/8 + (567*x^6)/2 + 81*x^7 + (27*x^8)/2 + x^9 + 19683/512)
) - (923241*3^(1/2)*(x^2 + 2/3)^(1/2))/(35123200*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 2
43/32)) - (152343*3^(1/2)*(x^2 + 2/3)^(1/2))/(94119200000*(x + 3/2)) + (35213*3^(1/2)*(x^2 + 2/3)^(1/2))/(4014
08*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (80649*3^(1/2)*(x^2 + 2/
3)^(1/2))/(5378240000*(3*x + x^2 + 9/4)) - (52201*3^(1/2)*(x^2 + 2/3)^(1/2))/(344064*((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 + 2187/128)) + (55473*3^(1/2)*(x^2 + 2/3)^(
1/2))/(1536640000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

________________________________________________________________________________________

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)**10,x)

[Out]

Timed out

________________________________________________________________________________________