3.2503 \(\int \frac{5-x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{72 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{49 \sqrt{3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^3}+\frac{331 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{100 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^3) - (49*Sqrt[2 + 5*x + 3*x^2])/(30*(3 + 2*x)^2) - (72*Sqrt[2 + 5*x
+ 3*x^2])/(25*(3 + 2*x)) + (331*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(100*Sqrt[5])

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Rubi [A]  time = 0.07495, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ -\frac{72 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{49 \sqrt{3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^3}+\frac{331 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{100 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^3) - (49*Sqrt[2 + 5*x + 3*x^2])/(30*(3 + 2*x)^2) - (72*Sqrt[2 + 5*x
+ 3*x^2])/(25*(3 + 2*x)) + (331*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(100*Sqrt[5])

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])

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 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 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])

Rubi steps

\begin{align*} \int \frac{5-x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{13 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^3}-\frac{1}{15} \int \frac{-\frac{11}{2}+78 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^3}-\frac{49 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}+\frac{1}{150} \int \frac{-\frac{45}{2}-735 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^3}-\frac{49 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{72 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{331}{100} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^3}-\frac{49 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{72 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}-\frac{331}{50} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^3}-\frac{49 \sqrt{2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac{72 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{331 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{100 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0443091, size = 74, normalized size = 0.65 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (1728 x^2+5674 x+4753\right )}{(2 x+3)^3}-993 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1500} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(4753 + 5674*x + 1728*x^2))/(3 + 2*x)^3 - 993*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5
]*Sqrt[2 + 5*x + 3*x^2])])/1500

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Maple [A]  time = 0.009, size = 95, normalized size = 0.8 \begin{align*} -{\frac{49}{120}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{36}{25}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{331\,\sqrt{5}}{500}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{13}{120}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2),x)

[Out]

-49/120/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(1/2)-36/25/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-331/500*5^(1/2)*arct
anh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 2.01038, size = 163, normalized size = 1.43 \begin{align*} -\frac{331}{500} \, \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}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{49 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{30 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{72 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{25 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-331/500*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/15*sqrt(3*x^2 + 5
*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 49/30*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 72/25*sqrt(3*x^2 + 5*x
+ 2)/(2*x + 3)

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Fricas [A]  time = 1.8161, size = 304, normalized size = 2.67 \begin{align*} \frac{993 \, \sqrt{5}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 (1728 \, x^{2} + 5674 \, x + 4753\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/3000*(993*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 21
2*x + 89)/(4*x^2 + 12*x + 9)) - 20*(1728*x^2 + 5674*x + 4753)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x +
27)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{16 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 96 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 216 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 216 x \sqrt{3 x^{2} + 5 x + 2} + 81 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{16 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 96 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 216 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 216 x \sqrt{3 x^{2} + 5 x + 2} + 81 \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)**4/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(16*x**4*sqrt(3*x**2 + 5*x + 2) + 96*x**3*sqrt(3*x**2 + 5*x + 2) + 216*x**2*sqrt(3*x**2 + 5*x + 2)
+ 216*x*sqrt(3*x**2 + 5*x + 2) + 81*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(16*x**4*sqrt(3*x**2 + 5*x + 2)
+ 96*x**3*sqrt(3*x**2 + 5*x + 2) + 216*x**2*sqrt(3*x**2 + 5*x + 2) + 216*x*sqrt(3*x**2 + 5*x + 2) + 81*sqrt(3
*x**2 + 5*x + 2)), x)

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Giac [B]  time = 1.20053, size = 347, normalized size = 3.04 \begin{align*} \frac{331}{500} \, \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{3972 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 29790 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 255470 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 338835 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 632175 \, \sqrt{3} x + 149502 \, \sqrt{3} - 632175 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{150 \,{\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

331/500*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*s
qrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/150*(3972*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 29790*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 255470*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 338835*sqrt(3)*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 632175*sqrt(3)*x + 149502*sqrt(3) - 632175*sqrt(3*x^2 + 5*x + 2))/(2*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3