3.23.20 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=184 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac {329 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac {329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000 \sqrt {5}} \]

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Rubi [A]  time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \begin {gather*} -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac {329 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac {329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(-329*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000*(3 + 2*x)^2) + (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(15360
00*(3 + 2*x)^4) - (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(96000*(3 + 2*x)^6) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2
)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (329*ArcTanh[(7 + 8*x)/(2*Sqrt[5]
*Sqrt[2 + 5*x + 3*x^2])])/(40960000*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]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {47}{10} \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600}\\ &=-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{38400}\\ &=\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40960000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20480000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 185, normalized size = 1.01 \begin {gather*} -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 \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 )}{3072000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) - (32
9*((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])))/3072000

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IntegrateAlgebraic [A]  time = 0.86, size = 101, normalized size = 0.55 \begin {gather*} \frac {329 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{20480000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (28394496 x^8+2848109952 x^7+15895201728 x^6+38558367264 x^5+51825176720 x^4+41530110824 x^3+19810691268 x^2+5201574542 x+578701331\right )}{184320000 (2 x+3)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(578701331 + 5201574542*x + 19810691268*x^2 + 41530110824*x^3 + 51825176720*x^4 + 38558
367264*x^5 + 15895201728*x^6 + 2848109952*x^7 + 28394496*x^8))/(184320000*(3 + 2*x)^9) + (329*ArcTanh[Sqrt[2 +
5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(20480000*Sqrt[5])

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fricas [A]  time = 0.43, size = 200, normalized size = 1.09 \begin {gather*} \frac {2961 \, \sqrt {5} {\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 {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 (28394496 \, x^{8} + 2848109952 \, x^{7} + 15895201728 \, x^{6} + 38558367264 \, x^{5} + 51825176720 \, x^{4} + 41530110824 \, x^{3} + 19810691268 \, x^{2} + 5201574542 \, x + 578701331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3686400000 \, {\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+5*x+2)^(7/2)/(3+2*x)^10,x, algorithm="fricas")

[Out]

1/3686400000*(2961*sqrt(5)*(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)*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*(28394496*x^8 + 2848109952*x^7 + 15895201728*x^6 + 38558367264*x^5 + 51825176720*x^4 + 415
30110824*x^3 + 19810691268*x^2 + 5201574542*x + 578701331)*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)

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giac [B]  time = 0.38, size = 563, normalized size = 3.06 \begin {gather*} \frac {329}{204800000} \, \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 {14930678016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 204061569408 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} + 3866707486848 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 14840812733760 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 114102022608000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 198779998219488 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 649357338634272 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 207317438979984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2217334591351040 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 5247913396815000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 20151247122371016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 17924557725783828 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 35125577732048328 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 16953161853593070 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 17752204726475250 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 4253745315948057 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1882391465118753 \, \sqrt {3} x - 129047626217736 \, \sqrt {3} + 1882391465118753 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{184320000 \, {\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 )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^10,x, algorithm="giac")

[Out]

329/204800000*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))) - 1/184320000*(14930678016*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^17 + 204061569408*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 + 3866707486848*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^15 + 14840812733760*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 114102022608000*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^13 + 198779998219488*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 649357338634272*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 207317438979984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 2217
334591351040*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 5247913396815000*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^8 - 20151247122371016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 17924557725783828*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^6 - 35125577732048328*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 16953161853593070*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 17752204726475250*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 4253745315948057
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1882391465118753*sqrt(3)*x - 129047626217736*sqrt(3) + 188239
1465118753*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)^9

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maple [B]  time = 0.11, size = 369, normalized size = 2.01 \begin {gather*} -\frac {329 \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 )}{204800000}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{51200 \left (x +\frac {3}{2}\right )^{8}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{32000 \left (x +\frac {3}{2}\right )^{7}}-\frac {90287 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2621237 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{200000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {259393 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {491479 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{50000000}-\frac {191149 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200000000}-\frac {491479 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{25000000 \left (x +\frac {3}{2}\right )}+\frac {9541 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{96000000}-\frac {329 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{25600000}+\frac {329 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{204800000}+\frac {329 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{384000000}+\frac {329 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800000000}+\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{200000000}-\frac {1457 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{400000 \left (x +\frac {3}{2}\right )^{5}}-\frac {893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{384000 \left (x +\frac {3}{2}\right )^{6}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{23040 \left (x +\frac {3}{2}\right )^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(2*x+3)^10,x)

[Out]

-47/51200/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-47/32000/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-90287/16000
000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-2621237/200000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-259393/3
0000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+491479/50000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-191149/2
00000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-491479/25000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+9541/9600
0000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-329/25600000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-329/204800000*5^
(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+329/204800000*(-16*x+12*(x+3/2)^2-19)^(1/2
)+329/384000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+329/800000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+47/200000000*(-4*x+3
*(x+3/2)^2-19/4)^(7/2)-1457/400000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-893/384000/(x+3/2)^6*(-4*x+3*(x+3/2
)^2-19/4)^(9/2)-13/23040/(x+3/2)^9*(-4*x+3*(x+3/2)^2-19/4)^(9/2)

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maxima [B]  time = 1.40, size = 513, normalized size = 2.79 \begin {gather*} \frac {7863711}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{45 \, {\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 {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{200 \, {\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 {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{6000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {90287 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {259393 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2621237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {573447}{100000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {3822651}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {491479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} + \frac {9541}{16000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {191149}{384000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {987}{12800000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {329}{204800000} \, \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 {6251}{102400000} \, \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)^(7/2)/(3+2*x)^10,x, algorithm="maxima")

[Out]

7863711/200000000*(3*x^2 + 5*x + 2)^(7/2) - 13/45*(3*x^2 + 5*x + 2)^(9/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 14
5152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 47/200*(3*x^2 + 5*x + 2)^(9
/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) - 47/25
0*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)
- 893/6000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1457/
12500*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 90287/1000000*(3*x^2 + 5
*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 259393/3750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^
2 + 54*x + 27) - 2621237/50000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 573447/100000000*(3*x^2 + 5*x +
2)^(5/2)*x - 3822651/800000000*(3*x^2 + 5*x + 2)^(5/2) - 491479/10000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) +
9541/16000000*(3*x^2 + 5*x + 2)^(3/2)*x + 191149/384000000*(3*x^2 + 5*x + 2)^(3/2) - 987/12800000*sqrt(3*x^2 +
5*x + 2)*x - 329/204800000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 6
251/102400000*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 )}^{7/2}}{{\left (2\,x+3\right )}^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^10,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^10, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**10,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 195
9552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-292*x*sqrt(3*x**2 +
5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 +
2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 +
15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x
**2 + 393660*x + 59049), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**
8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049)
, x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 10886
40*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-396*x*
*5*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 +
2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/
(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x*
*3 + 1180980*x**2 + 393660*x + 59049), x)

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