3.22.75 \(\int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=144 \[ -\frac {87 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^3}-\frac {339 \left (3 x^2+5 x+2\right )^{3/2}}{500 (2 x+3)^4}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}+\frac {3159 (8 x+7) \sqrt {3 x^2+5 x+2}}{20000 (2 x+3)^2}-\frac {3159 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40000 \sqrt {5}} \]

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Rubi [A]  time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {87 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^3}-\frac {339 \left (3 x^2+5 x+2\right )^{3/2}}{500 (2 x+3)^4}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}+\frac {3159 (8 x+7) \sqrt {3 x^2+5 x+2}}{20000 (2 x+3)^2}-\frac {3159 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^6,x]

[Out]

(3159*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20000*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(25*(3 + 2*x)^5) - (
339*(2 + 5*x + 3*x^2)^(3/2))/(500*(3 + 2*x)^4) - (87*(2 + 5*x + 3*x^2)^(3/2))/(125*(3 + 2*x)^3) - (3159*ArcTan
h[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40000*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) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {1}{25} \int \frac {\left (-\frac {105}{2}+78 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^5} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}+\frac {1}{500} \int \frac {\left (\frac {2169}{2}-1017 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^4} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac {3159 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1000}\\ &=\frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40000}\\ &=\frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac {3159 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20000}\\ &=\frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 146, normalized size = 1.01 \begin {gather*} \frac {1}{25} \left (-\frac {87 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac {339 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}+\frac {3159 \left (\frac {10 \sqrt {3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{8000}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^6,x]

[Out]

((-13*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^5 - (339*(2 + 5*x + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) - (87*(2 + 5*x + 3
*x^2)^(3/2))/(5*(3 + 2*x)^3) + (3159*((10*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + Sqrt[5]*ArcTanh[(-7 -
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/8000)/25

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IntegrateAlgebraic [A]  time = 0.58, size = 81, normalized size = 0.56 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (35136 x^4+225816 x^3+549516 x^2+606326 x+244331\right )}{20000 (2 x+3)^5}-\frac {3159 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{20000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^6,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(244331 + 606326*x + 549516*x^2 + 225816*x^3 + 35136*x^4))/(20000*(3 + 2*x)^5) - (3159*
ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(20000*Sqrt[5])

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fricas [A]  time = 0.42, size = 141, normalized size = 0.98 \begin {gather*} \frac {3159 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (35136 \, x^{4} + 225816 \, x^{3} + 549516 \, x^{2} + 606326 \, x + 244331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{400000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/400000*(3159*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*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*(35136*x^4 + 225816*x^3 + 549516*x^2 + 606326*
x + 244331)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.41, size = 363, normalized size = 2.52 \begin {gather*} -\frac {3159}{200000} \, \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 {\sqrt {3} {\left (50544 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 2047032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11747352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 121295556 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 269183136 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1164571962 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1077361162 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 1845838971 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 592102521 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 244862928\right )}}{60000 \, {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

-3159/200000*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))) + 1/60000*sqrt(3)*(50544*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^9 + 2047032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11747352*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^7 + 121295556*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 269183136*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^5 + 1164571962*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1077361162*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^3 + 1845838971*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 592102521*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
)) + 244862928)/(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)
^5

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maple [A]  time = 0.06, size = 174, normalized size = 1.21 \begin {gather*} \frac {3159 \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 )}{200000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {339 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {87 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {3159 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {3159 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12500 \left (x +\frac {3}{2}\right )}-\frac {3159 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{200000}+\frac {3159 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{25000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(1/2)/(2*x+3)^6,x)

[Out]

-13/800/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-339/8000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-87/1000/(x+3/
2)^3*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-3159/20000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-3159/12500/(x+3/2)*(-4*x
+3*(x+3/2)^2-19/4)^(3/2)-3159/200000*(-16*x+12*(x+3/2)^2-19)^(1/2)+3159/200000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*
5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+3159/25000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)

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maxima [A]  time = 1.42, size = 212, normalized size = 1.47 \begin {gather*} \frac {3159}{200000} \, \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 {9477}{20000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {339 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {87 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {3159 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3159 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{5000 \, {\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)^(1/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

3159/200000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 9477/20000*sqrt(3
*x^2 + 5*x + 2) - 13/25*(3*x^2 + 5*x + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 339/50
0*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 87/125*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 3
6*x^2 + 54*x + 27) - 3159/5000*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 3159/5000*sqrt(3*x^2 + 5*x + 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 )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^6,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**6,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x
) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729),
x)

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