3.23.19 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=197 \[ \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac {3 (434104 x+559841) \sqrt {3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{8192000 \sqrt {5}} \]

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Rubi [A]  time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {810, 843, 621, 206, 724} \begin {gather*} \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac {3 (434104 x+559841) \sqrt {3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{8192000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]

[Out]

(3*(559841 + 434104*x)*Sqrt[2 + 5*x + 3*x^2])/(4096000*(3 + 2*x)^2) + ((20959 + 17096*x)*(2 + 5*x + 3*x^2)^(3/
2))/(102400*(3 + 2*x)^4) + ((881 + 664*x)*(2 + 5*x + 3*x^2)^(5/2))/(6400*(3 + 2*x)^6) + ((757 + 808*x)*(2 + 5*
x + 3*x^2)^(7/2))/(1120*(3 + 2*x)^8) - (27*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/512 +
(1673211*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(8192000*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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 810

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx &=\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {1}{320} \int \frac {(291+240 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac {\int \frac {(-36690-43200 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{76800}\\ &=\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {\int \frac {(4488660+5184000 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{12288000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac {\int \frac {-265774680-311040000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{983040000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {81}{512} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1673211 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{8192000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {81}{256} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {1673211 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{4096000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{8192000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 130, normalized size = 0.66 \begin {gather*} \frac {-11712477 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-15120000 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+\frac {10 \sqrt {3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{(2 x+3)^8}}{286720000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(9818427389 + 48950756372*x + 105874603844*x^2 + 129405924160*x^3 + 97176896240*x^4
+ 45214440256*x^5 + 12182619328*x^6 + 1478785536*x^7))/(3 + 2*x)^8 - 11712477*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*S
qrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 15120000*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/286720000

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IntegrateAlgebraic [A]  time = 0.99, size = 131, normalized size = 0.66 \begin {gather*} -\frac {27}{256} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{4096000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{28672000 (2 x+3)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(9818427389 + 48950756372*x + 105874603844*x^2 + 129405924160*x^3 + 97176896240*x^4 + 4
5214440256*x^5 + 12182619328*x^6 + 1478785536*x^7))/(28672000*(3 + 2*x)^8) - (27*Sqrt[3]*ArcTanh[Sqrt[2 + 5*x
+ 3*x^2]/(Sqrt[3]*(1 + x))])/256 + (1673211*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(4096000*Sqrt[5]
)

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fricas [A]  time = 0.45, size = 263, normalized size = 1.34 \begin {gather*} \frac {15120000 \, \sqrt {3} {\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 (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 11712477 \, \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 (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{573440000 \, {\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)^(7/2)/(3+2*x)^9,x, algorithm="fricas")

[Out]

1/573440000*(15120000*sqrt(3)*(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(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 11712477*sqrt(5)*(2
56*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*(1478785536*x^7 + 121826193
28*x^6 + 45214440256*x^5 + 97176896240*x^4 + 129405924160*x^3 + 105874603844*x^2 + 48950756372*x + 9818427389)
*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 + 349
92*x + 6561)

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giac [B]  time = 0.51, size = 546, normalized size = 2.77 \begin {gather*} \frac {1673211}{40960000} \, \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 {27}{512} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {25982914944 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 475461282240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 12329944383680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 66497191380480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 747738478510240 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 2056338758898032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 12823219634258640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 20470141041874560 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 75774797457107080 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 72179382871515780 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 157788604924552196 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 86325470670757920 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 102935771527447390 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 28057073003987265 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 14067886443441495 \, \sqrt {3} x + 1086949713645432 \, \sqrt {3} - 14067886443441495 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{28672000 \, {\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)^(7/2)/(3+2*x)^9,x, algorithm="giac")

[Out]

1673211/40960000*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))) + 27/512*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5)) + 1/28672000*(25982914944*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 + 475461282240*sqrt(3
)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 12329944383680*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 66497191380
480*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 747738478510240*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 +
2056338758898032*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 12823219634258640*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^9 + 20470141041874560*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 75774797457107080*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^7 + 72179382871515780*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 1577886049245521
96*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 86325470670757920*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1
02935771527447390*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 28057073003987265*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^2 + 14067886443441495*sqrt(3)*x + 1086949713645432*sqrt(3) - 14067886443441495*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 = 379, normalized size = 1.92 \begin {gather*} -\frac {1673211 \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 )}{40960000}-\frac {27 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{512}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}-\frac {81 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{44800 \left (x +\frac {3}{2}\right )^{7}}-\frac {158331 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{22400000 \left (x +\frac {3}{2}\right )^{4}}-\frac {664383 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{40000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {150503 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{14000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {767427 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{70000000}-\frac {135591 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40000000}-\frac {767427 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{35000000 \left (x +\frac {3}{2}\right )}-\frac {25627 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6400000}-\frac {53211 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{5120000}+\frac {1673211 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{40960000}+\frac {557737 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{25600000}+\frac {1673211 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{160000000}+\frac {1673211 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{280000000}-\frac {363 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{80000 \left (x +\frac {3}{2}\right )^{5}}-\frac {523 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{179200 \left (x +\frac {3}{2}\right )^{6}} \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)^9,x)

[Out]

-13/10240/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-81/44800/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-158331/2240
0000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-664383/40000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-150503/14
000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+767427/70000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-135591/40
000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-767427/35000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-25627/64000
00*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-53211/5120000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-1673211/40960000*
5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-27/512*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+
(-4*x+3*(x+3/2)^2-19/4)^(1/2))+1673211/40960000*(-16*x+12*(x+3/2)^2-19)^(1/2)+557737/25600000*(-4*x+3*(x+3/2)^
2-19/4)^(3/2)+1673211/160000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+1673211/280000000*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-
363/80000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-523/179200/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(9/2)

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maxima [B]  time = 1.64, size = 479, normalized size = 2.43 \begin {gather*} \frac {1993149}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{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 {81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{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 {523 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2800 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {158331 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1400000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {150503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {664383 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {406773}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {1038609}{160000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {767427 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{14000000 \, {\left (2 \, x + 3\right )}} - \frac {76881}{3200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {45197}{25600000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {159633}{2560000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {27}{512} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1673211}{40960000} \, \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 {608991}{20480000} \, \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)^9,x, algorithm="maxima")

[Out]

1993149/40000000*(3*x^2 + 5*x + 2)^(7/2) - 13/40*(3*x^2 + 5*x + 2)^(9/2)/(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) - 81/350*(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) - 523/2800*(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) - 363/2500*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240
*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 158331/1400000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81) - 150503/1750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 664383/10000000*(3*x^2 +
5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 406773/20000000*(3*x^2 + 5*x + 2)^(5/2)*x - 1038609/160000000*(3*x^2 + 5*x
+ 2)^(5/2) - 767427/14000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 76881/3200000*(3*x^2 + 5*x + 2)^(3/2)*x + 45
197/25600000*(3*x^2 + 5*x + 2)^(3/2) - 159633/2560000*sqrt(3*x^2 + 5*x + 2)*x - 27/512*sqrt(3)*log(sqrt(3)*sqr
t(3*x^2 + 5*x + 2) + 3*x + 5/2) - 1673211/40960000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/
2/abs(2*x + 3) - 2) + 608991/20480000*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 )}^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)^(7/2))/(2*x + 3)^9,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(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 {40 \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 {292 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 {870 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 {1339 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 {1090 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 \left (- \frac {396 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}\right )\, dx - \int \frac {27 x^{7} \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)**(7/2)/(3+2*x)**9,x)

[Out]

-Integral(-40*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(-292*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6
912*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(-870*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(-1339*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), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 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), x) - Integral(-396*x**5*sq
rt(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(27*x**7*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 4147
2*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x)

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