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

Optimal. Leaf size=139 \[ -\frac {681 \sqrt {3 x^2+5 x+2}}{250 (2 x+3)}-\frac {41 \sqrt {3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac {86 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac {5771 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2000 \sqrt {5}} \]

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Rubi [A]  time = 0.10, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \begin {gather*} -\frac {681 \sqrt {3 x^2+5 x+2}}{250 (2 x+3)}-\frac {41 \sqrt {3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac {86 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac {5771 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) - (86*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (41*Sqrt[2 + 5*x
+ 3*x^2])/(24*(3 + 2*x)^2) - (681*Sqrt[2 + 5*x + 3*x^2])/(250*(3 + 2*x)) + (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*
Sqrt[2 + 5*x + 3*x^2])])/(2000*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 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}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {1}{20} \int \frac {\frac {7}{2}+117 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac {1}{300} \int \frac {-\frac {1067}{2}-2064 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {\int \frac {\frac {5265}{2}+15375 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{3000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}+\frac {5771 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}-\frac {5771 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}+\frac {5771 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 79, normalized size = 0.57 \begin {gather*} \frac {-17313 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \sqrt {3 x^2+5 x+2} \left (65376 x^3+314692 x^2+509668 x+279039\right )}{(2 x+3)^4}}{30000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(279039 + 509668*x + 314692*x^2 + 65376*x^3))/(3 + 2*x)^4 - 17313*Sqrt[5]*ArcTanh[
(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/30000

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IntegrateAlgebraic [A]  time = 0.64, size = 76, normalized size = 0.55 \begin {gather*} \frac {5771 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{1000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-65376 x^3-314692 x^2-509668 x-279039\right )}{3000 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-279039 - 509668*x - 314692*x^2 - 65376*x^3))/(3000*(3 + 2*x)^4) + (5771*ArcTanh[Sqrt[
2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(1000*Sqrt[5])

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fricas [A]  time = 0.40, size = 125, normalized size = 0.90 \begin {gather*} \frac {17313 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (65376 \, x^{3} + 314692 \, x^{2} + 509668 \, x + 279039\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{60000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60000*(17313*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(65376*x^3 + 314692*x^2 + 509668*x + 279039)*sqrt(3*x^2 + 5*
x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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giac [A]  time = 0.37, size = 193, normalized size = 1.39 \begin {gather*} \frac {1}{10000} \, \sqrt {5} {\left (2724 \, \sqrt {5} \sqrt {3} + 5771 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{6000} \, {\left (\frac {5 \, {\left (\frac {2 \, {\left (\frac {344}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {1025}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {8172}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {5771 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{10000 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10000*sqrt(5)*(2724*sqrt(5)*sqrt(3) + 5771*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) - 1/6000*(5*(2*(344/s
gn(1/(2*x + 3)) + 195/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) + 1025/sgn(1/(2*x + 3)))/(2*x + 3) + 8172/sgn(1/
(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 5771/10000*sqrt(5)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5
/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))/sgn(1/(2*x + 3))

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maple [A]  time = 0.01, size = 116, normalized size = 0.83 \begin {gather*} -\frac {5771 \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 )}{10000}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{320 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{300 \left (x +\frac {3}{2}\right )^{3}}-\frac {41 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{96 \left (x +\frac {3}{2}\right )^{2}}-\frac {681 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{500 \left (x +\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/320/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-43/300/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-41/96/(x+3/2)^2
*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-681/500/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-5771/10000*5^(1/2)*arctanh(2/5*(-
4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.27, size = 157, normalized size = 1.13 \begin {gather*} -\frac {5771}{10000} \, \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 {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {86 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {41 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{24 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {681 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{250 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5771/10000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/20*sqrt(3*x^2
+ 5*x + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 86/75*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27)
- 41/24*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 681/250*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 {x-5}{{\left (2\,x+3\right )}^5\,\sqrt {3\,x^2+5\,x+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 720*x**3*sqrt(3*x**2 + 5*x + 2
) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x) - Integr
al(-5/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 720*x**3*sqrt(3*x**2 + 5*x + 2) + 10
80*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x)

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