3.2431 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=199 \[ -\frac {3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac {717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac {19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac {51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac {153927 (8 x+7) \sqrt {3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac {153927 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{64000000 \sqrt {5}} \]

[Out]

51309/800000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-13/40*(3*x^2+5*x+2)^(5/2)/(3+2*x)^8-19/50*(3*x^2+5*x+2)^(5/
2)/(3+2*x)^7-717/2000*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6-3879/12500*(3*x^2+5*x+2)^(5/2)/(3+2*x)^5+153927/320000000*
arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-153927/32000000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^
2

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Rubi [A]  time = 0.13, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \[ -\frac {3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac {717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac {19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac {51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac {153927 (8 x+7) \sqrt {3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac {153927 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{64000000 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-153927*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(32000000*(3 + 2*x)^2) + (51309*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(
800000*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(40*(3 + 2*x)^8) - (19*(2 + 5*x + 3*x^2)^(5/2))/(50*(3 + 2*
x)^7) - (717*(2 + 5*x + 3*x^2)^(5/2))/(2000*(3 + 2*x)^6) - (3879*(2 + 5*x + 3*x^2)^(5/2))/(12500*(3 + 2*x)^5)
+ (153927*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(64000000*Sqrt[5])

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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {1}{40} \int \frac {\left (-\frac {181}{2}+117 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}+\frac {\int \frac {\left (\frac {5481}{2}-3192 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx}{1400}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {\int \frac {\left (-\frac {190323}{2}+45171 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{42000}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac {51309 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{20000}\\ &=\frac {51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}-\frac {153927 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1600000}\\ &=-\frac {153927 (7+8 x) \sqrt {2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac {51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac {153927 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{64000000}\\ &=-\frac {153927 (7+8 x) \sqrt {2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac {51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}-\frac {153927 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{32000000}\\ &=-\frac {153927 (7+8 x) \sqrt {2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac {51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac {153927 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{64000000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 182, normalized size = 0.91 \[ \frac {1}{40} \left (-\frac {7758 \left (3 x^2+5 x+2\right )^{5/2}}{625 (2 x+3)^5}-\frac {717 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^6}-\frac {76 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8}+\frac {51309 \left (\frac {10 \sqrt {3 x^2+5 x+2} \left (864 x^3+2068 x^2+1572 x+371\right )}{(2 x+3)^4}-3 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{8000000}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-13*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^8 - (76*(2 + 5*x + 3*x^2)^(5/2))/(5*(3 + 2*x)^7) - (717*(2 + 5*x + 3*
x^2)^(5/2))/(50*(3 + 2*x)^6) - (7758*(2 + 5*x + 3*x^2)^(5/2))/(625*(3 + 2*x)^5) + (51309*((10*Sqrt[2 + 5*x + 3
*x^2]*(371 + 1572*x + 2068*x^2 + 864*x^3))/(3 + 2*x)^4 - 3*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x
+ 3*x^2])]))/8000000)/40

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fricas [A]  time = 0.87, size = 185, normalized size = 0.93 \[ \frac {153927 \, \sqrt {5} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (5681664 \, x^{7} + 60161472 \, x^{6} + 272314944 \, x^{5} + 682163760 \, x^{4} + 1007243840 \, x^{3} + 924451956 \, x^{2} + 512781828 \, x + 131091161\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{640000000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/640000000*(153927*sqrt(5)*(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((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) -
20*(5681664*x^7 + 60161472*x^6 + 272314944*x^5 + 682163760*x^4 + 1007243840*x^3 + 924451956*x^2 + 512781828*x
+ 131091161)*sqrt(3*x^2 + 5*x + 2))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 816
48*x^2 + 34992*x + 6561)

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giac [B]  time = 0.35, size = 512, normalized size = 2.57 \[ \frac {153927}{320000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {19702656 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 443309760 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 13775440320 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 88813739520 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 1135723030560 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 3326100961968 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 20795205897360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 31719485197440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 108381222834920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 93303707056820 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 182905948708404 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 90199904722080 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 98616726439110 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 25302796273485 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 12323187970155 \, \sqrt {3} x + 954490882968 \, \sqrt {3} - 12323187970155 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{32000000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

153927/320000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3
)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/32000000*(19702656*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^15 + 443309760*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 13775440320*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^13 + 88813739520*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 1135723030560*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^11 + 3326100961968*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 20795205897360*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^9 + 31719485197440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 108381222834920*(sqrt
(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 93303707056820*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 182905948708
404*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 90199904722080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 986
16726439110*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 25302796273485*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^2 + 12323187970155*sqrt(3)*x + 954490882968*sqrt(3) - 12323187970155*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^8

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maple [A]  time = 0.14, size = 274, normalized size = 1.38 \[ -\frac {153927 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{320000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}-\frac {51309 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{3200000 \left (x +\frac {3}{2}\right )^{4}}-\frac {3879 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{400000 \left (x +\frac {3}{2}\right )^{5}}-\frac {51309 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{2000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1590579 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {1487961 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{50000000}-\frac {1487961 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{25000000 \left (x +\frac {3}{2}\right )}-\frac {153927 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{40000000}-\frac {19 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{6400 \left (x +\frac {3}{2}\right )^{7}}+\frac {153927 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{320000000}+\frac {51309 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{200000000}-\frac {717 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{128000 \left (x +\frac {3}{2}\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/10240/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-51309/3200000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-3879/4
00000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-51309/2000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1590579/40
000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+1487961/50000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-1487961/
25000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-153927/40000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-153927/32
0000000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-19/6400/(x+3/2)^7*(-4*x+3*(x+3/2
)^2-19/4)^(5/2)+153927/320000000*(-16*x+12*(x+3/2)^2-19)^(1/2)+51309/200000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-7
17/128000/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(5/2)

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maxima [B]  time = 1.31, size = 394, normalized size = 1.98 \[ \frac {4771737}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{40 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {19 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {717 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{2000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {3879 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{12500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {51309 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{200000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {51309 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{250000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1590579 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {461781}{20000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {153927}{320000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {2924613}{160000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1487961 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4771737/40000000*(3*x^2 + 5*x + 2)^(3/2) - 13/40*(3*x^2 + 5*x + 2)^(5/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 483
84*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 19/50*(3*x^2 + 5*x + 2)^(5/2)/(128*x^7 + 1344*
x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 717/2000*(3*x^2 + 5*x + 2)^(5/2)/(64*x^
6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 3879/12500*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 24
0*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 51309/200000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81) - 51309/250000*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1590579/10000000*(3*x^2 + 5*
x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 461781/20000000*sqrt(3*x^2 + 5*x + 2)*x - 153927/320000000*sqrt(5)*log(sqrt(
5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 2924613/160000000*sqrt(3*x^2 + 5*x + 2) - 1487
961/10000000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 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), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 69
12*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683)
, x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 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), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(5
12*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 11809
8*x + 19683), x)

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