3.2574 \(\int \frac{5-x}{(3+2 x)^{7/2} (2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=141 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac{9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24409}{3125 \sqrt{2 x+3}}+\frac{102697}{1875 (2 x+3)^{3/2}}+\frac{56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{806841 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{3125} \]

[Out]

56399/(625*(3 + 2*x)^(5/2)) + 102697/(1875*(3 + 2*x)^(3/2)) - 24409/(3125*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10
*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^2) + (8852 + 9957*x)/(50*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 266*ArcTanh[S
qrt[3 + 2*x]] - (806841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

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Rubi [A]  time = 0.116358, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac{9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24409}{3125 \sqrt{2 x+3}}+\frac{102697}{1875 (2 x+3)^{3/2}}+\frac{56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{806841 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{3125} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3),x]

[Out]

56399/(625*(3 + 2*x)^(5/2)) + 102697/(1875*(3 + 2*x)^(3/2)) - 24409/(3125*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10
*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^2) + (8852 + 9957*x)/(50*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 266*ArcTanh[S
qrt[3 + 2*x]] - (806841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

Rule 822

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

Rule 828

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

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)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}-\frac{1}{10} \int \frac{1772+1551 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \frac{76349+69699 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{56399}{625 (3+2 x)^{5/2}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{1}{250} \int \frac{202447+169197 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{56399}{625 (3+2 x)^{5/2}}+\frac{102697}{1875 (3+2 x)^{3/2}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{\int \frac{474341+308091 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx}{1250}\\ &=\frac{56399}{625 (3+2 x)^{5/2}}+\frac{102697}{1875 (3+2 x)^{3/2}}-\frac{24409}{3125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{\int \frac{758023-73227 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{6250}\\ &=\frac{56399}{625 (3+2 x)^{5/2}}+\frac{102697}{1875 (3+2 x)^{3/2}}-\frac{24409}{3125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{1735727-73227 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )}{3125}\\ &=\frac{56399}{625 (3+2 x)^{5/2}}+\frac{102697}{1875 (3+2 x)^{3/2}}-\frac{24409}{3125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac{2420523 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )}{3125}-798 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{56399}{625 (3+2 x)^{5/2}}+\frac{102697}{1875 (3+2 x)^{3/2}}-\frac{24409}{3125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac{8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+266 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{806841 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )}{3125}\\ \end{align*}

Mathematica [A]  time = 0.191879, size = 121, normalized size = 0.86 \[ \frac{-\frac{28125 (47 x+37)}{\left (3 x^2+5 x+2\right )^2}+\frac{1875 (9957 x+8852)}{3 x^2+5 x+2}+2 (2 x+3) \left (21 (2 x+3) \left (593750 \sqrt{2 x+3} \tanh ^{-1}\left (\sqrt{2 x+3}\right )-115263 \sqrt{30 x+45} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )-17435\right )+2567425\right )+8459850}{93750 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3),x]

[Out]

(8459850 - (28125*(37 + 47*x))/(2 + 5*x + 3*x^2)^2 + (1875*(8852 + 9957*x))/(2 + 5*x + 3*x^2) + 2*(3 + 2*x)*(2
567425 + 21*(3 + 2*x)*(-17435 + 593750*Sqrt[3 + 2*x]*ArcTanh[Sqrt[3 + 2*x]] - 115263*Sqrt[45 + 30*x]*ArcTanh[S
qrt[3/5]*Sqrt[3 + 2*x]])))/(93750*(3 + 2*x)^(5/2))

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Maple [A]  time = 0.023, size = 151, normalized size = 1.1 \begin{align*} -{\frac{416}{625} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}}-{\frac{9824}{1875} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{137184}{3125}{\frac{1}{\sqrt{3+2\,x}}}}+{\frac{13122}{3125\, \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{775}{18} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4045}{54}\sqrt{3+2\,x}} \right ) }-{\frac{806841\,\sqrt{15}}{15625}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+8\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+133\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+8\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-133\,\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)^(7/2)/(3*x^2+5*x+2)^3,x)

[Out]

-416/625/(3+2*x)^(5/2)-9824/1875/(3+2*x)^(3/2)-137184/3125/(3+2*x)^(1/2)+13122/3125*(775/18*(3+2*x)^(3/2)-4045
/54*(3+2*x)^(1/2))/(6*x+4)^2-806841/15625*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-3/(1+(3+2*x)^(1/2))^2+8
/(1+(3+2*x)^(1/2))+133*ln(1+(3+2*x)^(1/2))+3/(-1+(3+2*x)^(1/2))^2+8/(-1+(3+2*x)^(1/2))-133*ln(-1+(3+2*x)^(1/2)
)

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Maxima [A]  time = 1.73369, size = 217, normalized size = 1.54 \begin{align*} \frac{806841}{31250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{659043 \,{\left (2 \, x + 3\right )}^{6} - 8136261 \,{\left (2 \, x + 3\right )}^{5} + 23916753 \,{\left (2 \, x + 3\right )}^{4} - 24720095 \,{\left (2 \, x + 3\right )}^{3} + 6945760 \,{\left (2 \, x + 3\right )}^{2} + 1457600 \, x + 2342400}{9375 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 25 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}\right )}} + 133 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 133 \, \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)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

806841/31250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 1/9375*(659043*(2*x +
3)^6 - 8136261*(2*x + 3)^5 + 23916753*(2*x + 3)^4 - 24720095*(2*x + 3)^3 + 6945760*(2*x + 3)^2 + 1457600*x + 2
342400)/(9*(2*x + 3)^(13/2) - 48*(2*x + 3)^(11/2) + 94*(2*x + 3)^(9/2) - 80*(2*x + 3)^(7/2) + 25*(2*x + 3)^(5/
2)) + 133*log(sqrt(2*x + 3) + 1) - 133*log(sqrt(2*x + 3) - 1)

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Fricas [B]  time = 1.67971, size = 792, normalized size = 5.62 \begin{align*} \frac{2420523 \, \sqrt{5} \sqrt{3}{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 12468750 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 12468750 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 5 \,{\left (5272344 \, x^{6} + 14906052 \, x^{5} - 18312714 \, x^{4} - 114099329 \, x^{3} - 160041829 \, x^{2} - 94082723 \, x - 20250051\right )} \sqrt{2 \, x + 3}}{93750 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

1/93750*(2420523*sqrt(5)*sqrt(3)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)*
log(-(sqrt(5)*sqrt(3)*sqrt(2*x + 3) - 3*x - 7)/(3*x + 2)) + 12468750*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 +
3560*x^3 + 2223*x^2 + 756*x + 108)*log(sqrt(2*x + 3) + 1) - 12468750*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4
+ 3560*x^3 + 2223*x^2 + 756*x + 108)*log(sqrt(2*x + 3) - 1) - 5*(5272344*x^6 + 14906052*x^5 - 18312714*x^4 - 1
14099329*x^3 - 160041829*x^2 - 94082723*x - 20250051)*sqrt(2*x + 3))/(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 +
3560*x^3 + 2223*x^2 + 756*x + 108)

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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)**(7/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.08824, size = 193, normalized size = 1.37 \begin{align*} \frac{806841}{31250} \, \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{202995 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 745077 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 831169 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 259087 \, \sqrt{2 \, x + 3}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} - \frac{32 \,{\left (12861 \,{\left (2 \, x + 3\right )}^{2} + 3070 \, x + 4800\right )}}{9375 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 133 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 133 \, \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)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

806841/31250*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 1/625*(202995
*(2*x + 3)^(7/2) - 745077*(2*x + 3)^(5/2) + 831169*(2*x + 3)^(3/2) - 259087*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16
*x - 19)^2 - 32/9375*(12861*(2*x + 3)^2 + 3070*x + 4800)/(2*x + 3)^(5/2) + 133*log(sqrt(2*x + 3) + 1) - 133*lo
g(abs(sqrt(2*x + 3) - 1))