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

Optimal. Leaf size=179 \[ -\frac {107 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}+\frac {1517 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{24000 (2 x+3)^6}-\frac {1517 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{384000 (2 x+3)^4}+\frac {1517 (8 x+7) \sqrt {3 x^2+5 x+2}}{5120000 (2 x+3)^2}-\frac {1517 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10240000 \sqrt {5}} \]

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Rubi [A]  time = 0.10, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {107 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}+\frac {1517 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{24000 (2 x+3)^6}-\frac {1517 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{384000 (2 x+3)^4}+\frac {1517 (8 x+7) \sqrt {3 x^2+5 x+2}}{5120000 (2 x+3)^2}-\frac {1517 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10240000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1517*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(5120000*(3 + 2*x)^2) - (1517*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(38400
0*(3 + 2*x)^4) + (1517*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(24000*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(
40*(3 + 2*x)^8) - (107*(2 + 5*x + 3*x^2)^(7/2))/(350*(3 + 2*x)^7) - (1517*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/(10240000*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 )^{5/2}}{(3+2 x)^9} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {1}{40} \int \frac {\left (-\frac {311}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac {1517}{400} \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac {1517 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{9600}\\ &=-\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac {1517 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{256000}\\ &=\frac {1517 (7+8 x) \sqrt {2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac {1517 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{10240000}\\ &=\frac {1517 (7+8 x) \sqrt {2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac {1517 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{5120000}\\ &=\frac {1517 (7+8 x) \sqrt {2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac {1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac {107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac {1517 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10240000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 182, normalized size = 1.02 \begin {gather*} \frac {1}{40} \left (-\frac {428 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^8}+\frac {1517 \left (\frac {32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac {2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac {3 (8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac {3 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )}{19200}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-13*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8 - (428*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x)^7) + (1517*((3*(7 + 8
*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - (2*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 + (32*(7 + 8*x
)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6 + (3*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40*Sqrt[5]
)))/19200)/40

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IntegrateAlgebraic [A]  time = 0.77, size = 96, normalized size = 0.54 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (35495424 x^7+395685952 x^6+2141523904 x^5+5486222160 x^4+7363989440 x^3+5395613996 x^2+2061624348 x+325079151\right )}{107520000 (2 x+3)^8}-\frac {1517 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{5120000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(325079151 + 2061624348*x + 5395613996*x^2 + 7363989440*x^3 + 5486222160*x^4 + 21415239
04*x^5 + 395685952*x^6 + 35495424*x^7))/(107520000*(3 + 2*x)^8) - (1517*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]
*(1 + x))])/(5120000*Sqrt[5])

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fricas [A]  time = 0.43, size = 186, normalized size = 1.04 \begin {gather*} \frac {31857 \, \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 (35495424 \, x^{7} + 395685952 \, x^{6} + 2141523904 \, x^{5} + 5486222160 \, x^{4} + 7363989440 \, x^{3} + 5395613996 \, x^{2} + 2061624348 \, x + 325079151\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{2150400000 \, {\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+5*x+2)^(5/2)/(3+2*x)^9,x, algorithm="fricas")

[Out]

1/2150400000*(31857*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*(35495424*x^7 + 395685952*x^6 + 2141523904*x^5 + 5486222160*x^4 + 7363989440*x^3 + 5395613996*x^2 + 206162
4348*x + 325079151)*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 + 81648*x^2 + 34992*x + 6561)

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giac [B]  time = 0.35, size = 512, normalized size = 2.86 \begin {gather*} -\frac {1517}{51200000} \, \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 {4077696 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 2811291840 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 54242130880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 23829496320 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 4407279220960 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 22617729467088 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 195051199819760 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 377875254407040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1580087388997720 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 1627784736400620 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 3742975645158764 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2115026806109280 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2573382759804010 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 709918795444635 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 358308332266605 \, \sqrt {3} x + 27766562618088 \, \sqrt {3} - 358308332266605 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{107520000 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1517/51200000*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))) + 1/107520000*(4077696*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
))^15 - 2811291840*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 54242130880*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^13 + 23829496320*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 4407279220960*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^11 + 22617729467088*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 195051199819760*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^9 + 377875254407040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 1580087388997720*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 1627784736400620*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 3742975
645158764*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2115026806109280*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^4 + 2573382759804010*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 709918795444635*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^2 + 358308332266605*sqrt(3)*x + 27766562618088*sqrt(3) - 358308332266605*sqrt(3*x^2 + 5*x + 2))/(2
*(sqrt(3)*x - sqrt(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 [B]  time = 0.08, size = 311, normalized size = 1.74 \begin {gather*} \frac {1517 \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 )}{51200000}-\frac {1517 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{384000 \left (x +\frac {3}{2}\right )^{6}}-\frac {107 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{44800 \left (x +\frac {3}{2}\right )^{7}}-\frac {92537 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{6000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1517 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{240000 \left (x +\frac {3}{2}\right )^{5}}-\frac {2820103 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{120000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {881377 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{50000000}-\frac {43993 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{24000000}-\frac {881377 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{25000000 \left (x +\frac {3}{2}\right )}+\frac {1517 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{6400000}-\frac {1517 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{51200000}-\frac {31857 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{3200000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1517 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{96000000}-\frac {1517 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1517/384000/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-107/44800/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-92537/6
000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-1517/240000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-2820103/120
000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+881377/50000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-43993/240
00000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-881377/25000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+1517/6400000
*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+1517/51200000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^
2-19)^(1/2))-1517/51200000*(-16*x+12*(x+3/2)^2-19)^(1/2)-31857/3200000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(7/2)
-1517/96000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-1517/200000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-13/10240/(x+3/2)^8*(
-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [B]  time = 1.63, size = 423, normalized size = 2.36 \begin {gather*} \frac {2820103}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{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 {107 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{350 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {1517 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{6000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {1517 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{7500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {31857 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{200000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {92537 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2820103 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {43993}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {881377}{96000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {881377 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} + \frac {4551}{3200000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1517}{51200000} \, \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 {28823}{25600000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2820103/40000000*(3*x^2 + 5*x + 2)^(5/2) - 13/40*(3*x^2 + 5*x + 2)^(7/2)/(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) - 107/350*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 134
4*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 1517/6000*(3*x^2 + 5*x + 2)^(7/2)/(64
*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1517/7500*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 +
240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 31857/200000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2
+ 216*x + 81) - 92537/750000*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2820103/30000000*(3*x^2 +
5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 43993/4000000*(3*x^2 + 5*x + 2)^(3/2)*x - 881377/96000000*(3*x^2 + 5*x + 2
)^(3/2) - 881377/10000000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 4551/3200000*sqrt(3*x^2 + 5*x + 2)*x + 1517/5120
0000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 28823/25600000*sqrt(3*x^
2 + 5*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \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 {96 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 {165 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 \left (- \frac {113 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}\right )\, dx - \int \left (- \frac {15 x^{4} \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 {9 x^{5} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-20*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(-96*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(-165*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(-113*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 + 1
18098*x + 19683), x) - Integral(-15*x**4*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(9*x**5*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 + 314
928*x**2 + 118098*x + 19683), x)

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