3.2561 \(\int \frac{(5-x) \sqrt{3+2 x}}{(2+5 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x]] + (316*ArcTanh[Sqrt[3/5]*Sqrt[3 +
2*x]])/Sqrt[15]

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Rubi [A]  time = 0.0399227, antiderivative size = 66, 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 = {820, 826, 1166, 207} \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x]] + (316*ArcTanh[Sqrt[3/5]*Sqrt[3 +
2*x]])/Sqrt[15]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{\sqrt{3+2 x} (29+35 x)}{2+5 x+3 x^2}-\int \frac{76+35 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{\sqrt{3+2 x} (29+35 x)}{2+5 x+3 x^2}-2 \operatorname{Subst}\left (\int \frac{47+35 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{\sqrt{3+2 x} (29+35 x)}{2+5 x+3 x^2}+246 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )-316 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{\sqrt{3+2 x} (29+35 x)}{2+5 x+3 x^2}-82 \tanh ^{-1}\left (\sqrt{3+2 x}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )}{\sqrt{15}}\\ \end{align*}

Mathematica [A]  time = 0.0726592, size = 66, normalized size = 1. \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x]] + (316*ArcTanh[Sqrt[3/5]*Sqrt[3 +
2*x]])/Sqrt[15]

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Maple [A]  time = 0.016, size = 86, normalized size = 1.3 \begin{align*} -{\frac{34}{3}\sqrt{3+2\,x} \left ( 2\,x+{\frac{4}{3}} \right ) ^{-1}}+{\frac{316\,\sqrt{15}}{15}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-41\,\ln \left ( 1+\sqrt{3+2\,x} \right ) -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+41\,\ln \left ( -1+\sqrt{3+2\,x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-34/3*(3+2*x)^(1/2)/(2*x+4/3)+316/15*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-6/(1+(3+2*x)^(1/2))-41*ln(1+
(3+2*x)^(1/2))-6/(-1+(3+2*x)^(1/2))+41*ln(-1+(3+2*x)^(1/2))

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Maxima [A]  time = 1.46746, size = 132, normalized size = 2. \begin{align*} -\frac{158}{15} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (35 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 47 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 41 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-158/15*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 2*(35*(2*x + 3)^(3/2) - 47*
sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 41*log(sqrt(2*x + 3) + 1) + 41*log(sqrt(2*x + 3) - 1)

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Fricas [B]  time = 1.5939, size = 313, normalized size = 4.74 \begin{align*} \frac{158 \, \sqrt{15}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{15} \sqrt{2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 615 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 615 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 15 \,{\left (35 \, x + 29\right )} \sqrt{2 \, x + 3}}{15 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(158*sqrt(15)*(3*x^2 + 5*x + 2)*log((sqrt(15)*sqrt(2*x + 3) + 3*x + 7)/(3*x + 2)) - 615*(3*x^2 + 5*x + 2)
*log(sqrt(2*x + 3) + 1) + 615*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 15*(35*x + 29)*sqrt(2*x + 3))/(3*x^2
+ 5*x + 2)

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Sympy [A]  time = 66.3154, size = 212, normalized size = 3.21 \begin{align*} 340 \left (\begin{cases} \frac{\sqrt{15} \left (- \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1\right )}\right )}{75} & \text{for}\: x \geq - \frac{3}{2} \wedge x < - \frac{2}{3} \end{cases}\right ) - 282 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right ) + 41 \log{\left (\sqrt{2 x + 3} - 1 \right )} - 41 \log{\left (\sqrt{2 x + 3} + 1 \right )} - \frac{6}{\sqrt{2 x + 3} + 1} - \frac{6}{\sqrt{2 x + 3} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

340*Piecewise((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/4 - 1/(4*(sq
rt(15)*sqrt(2*x + 3)/5 + 1)) - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 - 1)))/75, (x >= -3/2) & (x < -2/3))) - 282*Piec
ewise((-sqrt(15)*acoth(sqrt(15)*sqrt(2*x + 3)/5)/15, 2*x + 3 > 5/3), (-sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5
)/15, 2*x + 3 < 5/3)) + 41*log(sqrt(2*x + 3) - 1) - 41*log(sqrt(2*x + 3) + 1) - 6/(sqrt(2*x + 3) + 1) - 6/(sqr
t(2*x + 3) - 1)

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Giac [A]  time = 1.08987, size = 138, normalized size = 2.09 \begin{align*} -\frac{158}{15} \, \sqrt{15} \log \left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{2 \,{\left (35 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 47 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 41 \, \log \left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-158/15*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 2*(35*(2*x + 3)^(3
/2) - 47*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 41*log(sqrt(2*x + 3) + 1) + 41*log(abs(sqrt(2*x + 3) - 1
))