3.2415 \(\int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=119 \[ -\frac {4 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}+\frac {153 (8 x+7) \sqrt {3 x^2+5 x+2}}{800 (2 x+3)^2}-\frac {153 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600 \sqrt {5}} \]

[Out]

-13/20*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-4/5*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3-153/8000*arctanh(1/10*(7+8*x)*5^(1/2)/(
3*x^2+5*x+2)^(1/2))*5^(1/2)+153/800*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2

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Rubi [A]  time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {4 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}+\frac {153 (8 x+7) \sqrt {3 x^2+5 x+2}}{800 (2 x+3)^2}-\frac {153 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(153*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) - (4*(
2 + 5*x + 3*x^2)^(3/2))/(5*(3 + 2*x)^3) - (153*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1600*Sqr
t[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)^5} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac {1}{20} \int \frac {\left (-\frac {123}{2}+39 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}}{20 (3+2 x)^4}-\frac {4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}+\frac {153}{40} \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=\frac {153 (7+8 x) \sqrt {2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac {4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}-\frac {153 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1600}\\ &=\frac {153 (7+8 x) \sqrt {2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac {4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}+\frac {153}{800} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {153 (7+8 x) \sqrt {2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac {4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}-\frac {153 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1600 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 119, normalized size = 1.00 \[ \frac {1}{20} \left (-\frac {16 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac {153 (8 x+7) \sqrt {3 x^2+5 x+2}}{40 (2 x+3)^2}+\frac {153 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{80 \sqrt {5}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((153*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 - (16*(2 +
5*x + 3*x^2)^(3/2))/(3 + 2*x)^3 + (153*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(80*Sqrt[5]))/20

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fricas [A]  time = 0.69, size = 126, normalized size = 1.06 \[ \frac {153 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1056 \, x^{3} + 5252 \, x^{2} + 9108 \, x + 4759\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{16000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/16000*(153*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*(1056*x^3 + 5252*x^2 + 9108*x + 4759)*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 = 0.27, size = 183, normalized size = 1.54 \[ -\frac {3}{8000} \, \sqrt {5} {\left (44 \, \sqrt {5} \sqrt {3} + 51 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {153}{8000} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{1600} \, {\left (\frac {5 \, {\left (\frac {2 \, {\left (\frac {65 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 24 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 25 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 132 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \]

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)^5,x, algorithm="giac")

[Out]

-3/8000*sqrt(5)*(44*sqrt(5)*sqrt(3) + 51*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) + 153/8000*sqrt(5)*log(ab
s(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 1/1600*(5*(2*(
65*sgn(1/(2*x + 3))/(2*x + 3) - 24*sgn(1/(2*x + 3)))/(2*x + 3) - 25*sgn(1/(2*x + 3)))/(2*x + 3) - 132*sgn(1/(2
*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)

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maple [A]  time = 0.06, size = 153, normalized size = 1.29 \[ \frac {153 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{8000}-\frac {\left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{10 \left (x +\frac {3}{2}\right )^{3}}-\frac {153 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{800 \left (x +\frac {3}{2}\right )^{2}}-\frac {153 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{500 \left (x +\frac {3}{2}\right )}-\frac {153 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{8000}+\frac {153 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}} \]

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)^5,x)

[Out]

-1/10/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-153/800/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-153/500/(x+3/2)*
(-4*x+3*(x+3/2)^2-19/4)^(3/2)-153/8000*(-16*x+12*(x+3/2)^2-19)^(1/2)+153/8000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5
^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+153/1000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-13/320/(x+3/2)^4*(-4*x+3*
(x+3/2)^2-19/4)^(3/2)

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maxima [A]  time = 1.25, size = 171, normalized size = 1.44 \[ \frac {153}{8000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {459}{800} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {4 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {153 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{200 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {153 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{200 \, {\left (2 \, x + 3\right )}} \]

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)^5,x, algorithm="maxima")

[Out]

153/8000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 459/800*sqrt(3*x^2 +
5*x + 2) - 13/20*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 4/5*(3*x^2 + 5*x + 2)^(3/
2)/(8*x^3 + 36*x^2 + 54*x + 27) - 153/200*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 153/200*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 \[ -\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^5} \,d x \]

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)^5,x)

[Out]

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

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

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)**5,x)

[Out]

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

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