3.2566 \(\int \frac{(5-x) (3+2 x)^{9/2}}{(2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac{(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac{3983}{9} \sqrt{2 x+3}+1962 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{13675}{9} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(-3983*Sqrt[3 + 2*x])/9 - ((3 + 2*x)^(7/2)*(121 + 139*x))/(6*(2 + 5*x + 3*x^2)^2) + ((3 + 2*x)^(3/2)*(10832 +
12473*x))/(18*(2 + 5*x + 3*x^2)) + 1962*ArcTanh[Sqrt[3 + 2*x]] - (13675*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2
*x]])/9

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Rubi [A]  time = 0.0820516, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {818, 824, 826, 1166, 207} \[ -\frac{(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{(12473 x+10832) (2 x+3)^{3/2}}{18 \left (3 x^2+5 x+2\right )}-\frac{3983}{9} \sqrt{2 x+3}+1962 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{13675}{9} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

(-3983*Sqrt[3 + 2*x])/9 - ((3 + 2*x)^(7/2)*(121 + 139*x))/(6*(2 + 5*x + 3*x^2)^2) + ((3 + 2*x)^(3/2)*(10832 +
12473*x))/(18*(2 + 5*x + 3*x^2)) + 1962*ArcTanh[Sqrt[3 + 2*x]] - (13675*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2
*x]])/9

Rule 818

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

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]

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) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{1}{6} \int \frac{(3+2 x)^{5/2} (-416+131 x)}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac{1}{18} \int \frac{(5709-11949 x) \sqrt{3+2 x}}{2+5 x+3 x^2} \, dx\\ &=-\frac{3983}{9} \sqrt{3+2 x}-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac{1}{54} \int \frac{99177+46203 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{3983}{9} \sqrt{3+2 x}-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+\frac{1}{27} \operatorname{Subst}\left (\int \frac{59745+46203 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{3983}{9} \sqrt{3+2 x}-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}-5886 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+\frac{68375}{9} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{3983}{9} \sqrt{3+2 x}-\frac{(3+2 x)^{7/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{(3+2 x)^{3/2} (10832+12473 x)}{18 \left (2+5 x+3 x^2\right )}+1962 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{13675}{9} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.12305, size = 86, normalized size = 0.75 \[ \frac{1}{54} \left (-\frac{3 \sqrt{2 x+3} \left (192 x^4-45083 x^3-112467 x^2-90465 x-23327\right )}{\left (3 x^2+5 x+2\right )^2}-27350 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right )+1962 \tanh ^{-1}\left (\sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

1962*ArcTanh[Sqrt[3 + 2*x]] + ((-3*Sqrt[3 + 2*x]*(-23327 - 90465*x - 112467*x^2 - 45083*x^3 + 192*x^4))/(2 + 5
*x + 3*x^2)^2 - 27350*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/54

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Maple [A]  time = 0.02, size = 133, normalized size = 1.2 \begin{align*} -{\frac{32}{27}\sqrt{3+2\,x}}+{\frac{1250}{3\, \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{77}{10} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{265}{18}\sqrt{3+2\,x}} \right ) }-{\frac{13675\,\sqrt{15}}{27}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+104\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+981\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+104\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-981\,\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)^(9/2)/(3*x^2+5*x+2)^3,x)

[Out]

-32/27*(3+2*x)^(1/2)+1250/3*(77/10*(3+2*x)^(3/2)-265/18*(3+2*x)^(1/2))/(6*x+4)^2-13675/27*arctanh(1/5*15^(1/2)
*(3+2*x)^(1/2))*15^(1/2)-3/(1+(3+2*x)^(1/2))^2+104/(1+(3+2*x)^(1/2))+981*ln(1+(3+2*x)^(1/2))+3/(-1+(3+2*x)^(1/
2))^2+104/(-1+(3+2*x)^(1/2))-981*ln(-1+(3+2*x)^(1/2))

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Maxima [A]  time = 1.50288, size = 193, normalized size = 1.68 \begin{align*} \frac{13675}{54} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{32}{27} \, \sqrt{2 \, x + 3} + \frac{137169 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 554983 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 717035 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 297925 \, \sqrt{2 \, x + 3}}{27 \,{\left (9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 981 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 981 \, \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)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

13675/54*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 32/27*sqrt(2*x + 3) + 1/27
*(137169*(2*x + 3)^(7/2) - 554983*(2*x + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(9*(2*x + 3
)^4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160*x - 215) + 981*log(sqrt(2*x + 3) + 1) - 981*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 1.63137, size = 498, normalized size = 4.33 \begin{align*} \frac{13675 \, \sqrt{5} \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 52974 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 52974 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 3 \,{\left (192 \, x^{4} - 45083 \, x^{3} - 112467 \, x^{2} - 90465 \, x - 23327\right )} \sqrt{2 \, x + 3}}{54 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

1/54*(13675*sqrt(5)*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(-(sqrt(5)*sqrt(3)*sqrt(2*x + 3) - 3*x - 7
)/(3*x + 2)) + 52974*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3) + 1) - 52974*(9*x^4 + 30*x^3 + 37*
x^2 + 20*x + 4)*log(sqrt(2*x + 3) - 1) - 3*(192*x^4 - 45083*x^3 - 112467*x^2 - 90465*x - 23327)*sqrt(2*x + 3))
/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.10539, size = 174, normalized size = 1.51 \begin{align*} \frac{13675}{54} \, \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{32}{27} \, \sqrt{2 \, x + 3} + \frac{137169 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 554983 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 717035 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 297925 \, \sqrt{2 \, x + 3}}{27 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 981 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 981 \, \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)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

13675/54*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 32/27*sqrt(2*x +
3) + 1/27*(137169*(2*x + 3)^(7/2) - 554983*(2*x + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(3
*(2*x + 3)^2 - 16*x - 19)^2 + 981*log(sqrt(2*x + 3) + 1) - 981*log(abs(sqrt(2*x + 3) - 1))