3.22.89 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=174 \[ -\frac {4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac {4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac {4663 (8 x+7) \sqrt {3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac {4663 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600000 \sqrt {5}} \]

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Rubi [A]  time = 0.11, antiderivative size = 174, 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 {4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac {4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac {4663 (8 x+7) \sqrt {3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac {4663 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

(-4663*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800000*(3 + 2*x)^2) + (4663*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(60000
*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^7) - (433*(2 + 5*x + 3*x^2)^(5/2))/(1050*(3 + 2*x)^
6) - (4892*(2 + 5*x + 3*x^2)^(5/2))/(13125*(3 + 2*x)^5) + (4663*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/(1600000*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)^8} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {1}{35} \int \frac {\left (-\frac {199}{2}+78 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}+\frac {\int \frac {\left (\frac {5887}{2}-1299 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{1050}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{1500}\\ &=\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac {4663 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{40000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1600000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac {4663 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{800000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1600000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 157, normalized size = 0.90 \begin {gather*} \frac {1}{35} \left (-\frac {4892 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}+\frac {32641 \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 )}{4800000}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

((-13*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^7 - (433*(2 + 5*x + 3*x^2)^(5/2))/(30*(3 + 2*x)^6) - (4892*(2 + 5*x +
3*x^2)^(5/2))/(375*(3 + 2*x)^5) + (32641*((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])]))/4800000)/35

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IntegrateAlgebraic [A]  time = 0.72, size = 91, normalized size = 0.52 \begin {gather*} \frac {4663 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{800000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (191232 x^6+2893088 x^5+16376240 x^4+55403520 x^3+64140640 x^2+15759118 x-6554463\right )}{16800000 (2 x+3)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-6554463 + 15759118*x + 64140640*x^2 + 55403520*x^3 + 16376240*x^4 + 2893088*x^5 + 191
232*x^6))/(16800000*(3 + 2*x)^7) + (4663*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(800000*Sqrt[5])

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fricas [A]  time = 0.42, size = 170, normalized size = 0.98 \begin {gather*} \frac {97923 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\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 (191232 \, x^{6} + 2893088 \, x^{5} + 16376240 \, x^{4} + 55403520 \, x^{3} + 64140640 \, x^{2} + 15759118 \, x - 6554463\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{336000000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \end {gather*}

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)^8,x, algorithm="fricas")

[Out]

1/336000000*(97923*sqrt(5)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187
)*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*(191232*x^6
+ 2893088*x^5 + 16376240*x^4 + 55403520*x^3 + 64140640*x^2 + 15759118*x - 6554463)*sqrt(3*x^2 + 5*x + 2))/(128
*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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giac [B]  time = 0.33, size = 461, normalized size = 2.65 \begin {gather*} \frac {4663}{8000000} \, \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 {6267072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 122207904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 3852187808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 18344551344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 131374293680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 134399090784 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 264419126976 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 1446858601104 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 6675760646156 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 5954681858370 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 10149146991914 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3640765552263 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 2268672558411 \, \sqrt {3} x - 208833935688 \, \sqrt {3} + 2268672558411 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16800000 \, {\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 )}^{7}} \end {gather*}

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)^8,x, algorithm="giac")

[Out]

4663/8000000*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))) - 1/16800000*(6267072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
13 + 122207904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 3852187808*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^11 + 18344551344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 131374293680*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^9 + 134399090784*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 264419126976*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^7 - 1446858601104*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 6675760646156*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^5 - 5954681858370*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 10149146991914*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^3 - 3640765552263*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 2268672558411*sqrt(3)
*x - 208833935688*sqrt(3) + 2268672558411*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)^7

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maple [A]  time = 0.07, size = 253, normalized size = 1.45 \begin {gather*} -\frac {4663 \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 )}{8000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {433 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{67200 \left (x +\frac {3}{2}\right )^{6}}-\frac {1223 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{105000 \left (x +\frac {3}{2}\right )^{5}}-\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{240000 \left (x +\frac {3}{2}\right )^{4}}-\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{150000 \left (x +\frac {3}{2}\right )^{3}}-\frac {144553 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{3000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {135227 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1875000 \left (x +\frac {3}{2}\right )}+\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15000000}-\frac {4663 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1000000}+\frac {4663 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{8000000}+\frac {135227 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3750000} \end {gather*}

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)^8,x)

[Out]

-13/4480/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-433/67200/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1223/105000
/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-4663/240000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-4663/150000/(x+3/
2)^3*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-144553/3000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-135227/1875000/(x+3/
2)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+4663/15000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-4663/1000000*(6*x+5)*(-4*x+3*(x+3
/2)^2-19/4)^(1/2)+4663/8000000*(-16*x+12*(x+3/2)^2-19)^(1/2)-4663/8000000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/
2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+135227/3750000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)

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maxima [B]  time = 1.30, size = 338, normalized size = 1.94 \begin {gather*} \frac {144553}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {433 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{1050 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {4892 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{13125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{18750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {144553 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{750000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {13989}{500000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {4663}{8000000} \, \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 {88597}{4000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {135227 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{750000 \, {\left (2 \, x + 3\right )}} \end {gather*}

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)^8,x, algorithm="maxima")

[Out]

144553/1000000*(3*x^2 + 5*x + 2)^(3/2) - 13/35*(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) - 433/1050*(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) - 4892/13125*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*
x^2 + 810*x + 243) - 4663/15000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 4663/18750*
(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 144553/750000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9
) - 13989/500000*sqrt(3*x^2 + 5*x + 2)*x - 4663/8000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3)
+ 5/2/abs(2*x + 3) - 2) - 88597/4000000*sqrt(3*x^2 + 5*x + 2) - 135227/750000*(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 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^8} \,d x \end {gather*}

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)^8,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^8, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \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}\right )\, dx - \int \left (- \frac {23 x \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}\right )\, dx - \int \left (- \frac {10 x^{2} \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}\right )\, dx - \int \frac {3 x^{3} \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}\, dx \end {gather*}

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)**8,x)

[Out]

-Integral(-10*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), x) - Integral(-23*x*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), x) - Integral(-10*x**2*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), x) - Integral(3*x**3*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), x)

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