3.1389 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=133 \[ -\frac{(x+8) \left (3 x^2+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (12 x+37) \left (3 x^2+2\right )^{3/2}}{12 (2 x+3)^2}-\frac{15 (37 x+119) \sqrt{3 x^2+2}}{8 (2 x+3)}+\frac{3657}{16} \sqrt{\frac{5}{7}} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{1785}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(-15*(119 + 37*x)*Sqrt[2 + 3*x^2])/(8*(3 + 2*x)) + (5*(37 + 12*x)*(2 + 3*x^2)^(3/2))/(12*(3 + 2*x)^2) - ((8 +
x)*(2 + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) + (1785*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (3657*Sqrt[5/7]*ArcTanh[(4 -
9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/16

________________________________________________________________________________________

Rubi [A]  time = 0.0792698, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {813, 844, 215, 725, 206} \[ -\frac{(x+8) \left (3 x^2+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (12 x+37) \left (3 x^2+2\right )^{3/2}}{12 (2 x+3)^2}-\frac{15 (37 x+119) \sqrt{3 x^2+2}}{8 (2 x+3)}+\frac{3657}{16} \sqrt{\frac{5}{7}} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{1785}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*(119 + 37*x)*Sqrt[2 + 3*x^2])/(8*(3 + 2*x)) + (5*(37 + 12*x)*(2 + 3*x^2)^(3/2))/(12*(3 + 2*x)^2) - ((8 +
x)*(2 + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) + (1785*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (3657*Sqrt[5/7]*ArcTanh[(4 -
9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/16

Rule 813

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

Rule 844

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

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) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx &=-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac{5}{72} \int \frac{(24-288 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx\\ &=\frac{5 (37+12 x) \left (2+3 x^2\right )^{3/2}}{12 (3+2 x)^2}-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{5}{768} \int \frac{(4608-21312 x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac{15 (119+37 x) \sqrt{2+3 x^2}}{8 (3+2 x)}+\frac{5 (37+12 x) \left (2+3 x^2\right )^{3/2}}{12 (3+2 x)^2}-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac{5 \int \frac{170496-822528 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{6144}\\ &=-\frac{15 (119+37 x) \sqrt{2+3 x^2}}{8 (3+2 x)}+\frac{5 (37+12 x) \left (2+3 x^2\right )^{3/2}}{12 (3+2 x)^2}-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{5355}{16} \int \frac{1}{\sqrt{2+3 x^2}} \, dx-\frac{18285}{16} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{15 (119+37 x) \sqrt{2+3 x^2}}{8 (3+2 x)}+\frac{5 (37+12 x) \left (2+3 x^2\right )^{3/2}}{12 (3+2 x)^2}-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{1785}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{18285}{16} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{15 (119+37 x) \sqrt{2+3 x^2}}{8 (3+2 x)}+\frac{5 (37+12 x) \left (2+3 x^2\right )^{3/2}}{12 (3+2 x)^2}-\frac{(8+x) \left (2+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{1785}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{3657}{16} \sqrt{\frac{5}{7}} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.164208, size = 97, normalized size = 0.73 \[ \frac{1}{336} \left (-\frac{14 \sqrt{3 x^2+2} \left (36 x^5-432 x^4+3408 x^3+37974 x^2+77061 x+46103\right )}{(2 x+3)^3}+10971 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+37485 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-14*Sqrt[2 + 3*x^2]*(46103 + 77061*x + 37974*x^2 + 3408*x^3 - 432*x^4 + 36*x^5))/(3 + 2*x)^3 + 37485*Sqrt[3]
*ArcSinh[Sqrt[3/2]*x] + 10971*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/336

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 206, normalized size = 1.6 \begin{align*} -{\frac{13}{840} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{37}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{2819}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{7314}{42875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{591\,x}{490} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{1143\,x}{56}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{1785\,\sqrt{3}}{16}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{1219}{490} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{3657}{112}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{3657\,\sqrt{35}}{112}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{8457\,x}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)+37/4900/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)-2819/85750/(x+3/
2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)-7314/42875*(3*(x+3/2)^2-9*x-19/4)^(5/2)+591/490*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)
+1143/56*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+1785/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)-1219/490*(3*(x+3/2)^2-9*x-19/4)
^(3/2)-3657/112*(12*(x+3/2)^2-36*x-19)^(1/2)+3657/112*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*
x-19)^(1/2))+8457/85750*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.54499, size = 234, normalized size = 1.76 \begin{align*} -\frac{111}{4900} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{37 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{1225 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{591}{490} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x - \frac{1219}{490} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{2819 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{4900 \,{\left (2 \, x + 3\right )}} + \frac{1143}{56} \, \sqrt{3 \, x^{2} + 2} x + \frac{1785}{16} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{3657}{112} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{3657}{56} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-111/4900*(3*x^2 + 2)^(5/2) - 13/105*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 37/1225*(3*x^2 + 2)^(7/2
)/(4*x^2 + 12*x + 9) + 591/490*(3*x^2 + 2)^(3/2)*x - 1219/490*(3*x^2 + 2)^(3/2) - 2819/4900*(3*x^2 + 2)^(5/2)/
(2*x + 3) + 1143/56*sqrt(3*x^2 + 2)*x + 1785/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 3657/112*sqrt(35)*arcsinh(3/2
*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 3657/56*sqrt(3*x^2 + 2)

________________________________________________________________________________________

Fricas [A]  time = 2.05823, size = 470, normalized size = 3.53 \begin{align*} \frac{10971 \, \sqrt{7} \sqrt{5}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\frac{\sqrt{7} \sqrt{5} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 37485 \, \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - 28 \,{\left (36 \, x^{5} - 432 \, x^{4} + 3408 \, x^{3} + 37974 \, x^{2} + 77061 \, x + 46103\right )} \sqrt{3 \, x^{2} + 2}}{672 \,{\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*x^2+2)^(5/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/672*(10971*sqrt(7)*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*log((sqrt(7)*sqrt(5)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*
x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) + 37485*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-sqrt(3)*sqrt(3*x^2 + 2)
*x - 3*x^2 - 1) - 28*(36*x^5 - 432*x^4 + 3408*x^3 + 37974*x^2 + 77061*x + 46103)*sqrt(3*x^2 + 2))/(8*x^3 + 36*
x^2 + 54*x + 27)

________________________________________________________________________________________

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*x**2+2)**(5/2)/(3+2*x)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.2886, size = 366, normalized size = 2.75 \begin{align*} -\frac{1}{32} \,{\left (3 \,{\left (2 \, x - 33\right )} x + 973\right )} \sqrt{3 \, x^{2} + 2} - \frac{1785}{16} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3657}{112} \, \sqrt{35} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{122001 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 589140 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 1403190 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1939920 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 1757100 \, \sqrt{3} x - 166304 \, \sqrt{3} - 1757100 \, \sqrt{3 \, x^{2} + 2}}{128 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(3*(2*x - 33)*x + 973)*sqrt(3*x^2 + 2) - 1785/16*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 3657/112*sq
rt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) -
2*sqrt(3*x^2 + 2))) - 1/128*(122001*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 589140*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 2))^4 + 1403190*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1939920*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 1757100*
sqrt(3)*x - 166304*sqrt(3) - 1757100*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x
- sqrt(3*x^2 + 2)) - 2)^3