3.1388 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=126 \[ -\frac{(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (29 x+178) \left (3 x^2+2\right )^{3/2}}{32 (2 x+3)}+\frac{15}{64} (859-267 x) \sqrt{3 x^2+2}-\frac{12885}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(15*(859 - 267*x)*Sqrt[2 + 3*x^2])/64 + (5*(178 + 29*x)*(2 + 3*x^2)^(3/2))/(32*(3 + 2*x)) - ((29 + 2*x)*(2 + 3
*x^2)^(5/2))/(16*(3 + 2*x)^2) - (43995*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (12885*Sqrt[35]*ArcTanh[(4 - 9*x)/(
Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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Rubi [A]  time = 0.0790293, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {813, 815, 844, 215, 725, 206} \[ -\frac{(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (29 x+178) \left (3 x^2+2\right )^{3/2}}{32 (2 x+3)}+\frac{15}{64} (859-267 x) \sqrt{3 x^2+2}-\frac{12885}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*(859 - 267*x)*Sqrt[2 + 3*x^2])/64 + (5*(178 + 29*x)*(2 + 3*x^2)^(3/2))/(32*(3 + 2*x)) - ((29 + 2*x)*(2 + 3
*x^2)^(5/2))/(16*(3 + 2*x)^2) - (43995*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (12885*Sqrt[35]*ArcTanh[(4 - 9*x)/(
Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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)^3} \, dx &=-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{5}{64} \int \frac{(16-348 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx\\ &=\frac{5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac{5}{512} \int \frac{(2784-25632 x) \sqrt{2+3 x^2}}{3+2 x} \, dx\\ &=\frac{15}{64} (859-267 x) \sqrt{2+3 x^2}+\frac{5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac{5 \int \frac{1056384-5068224 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{12288}\\ &=\frac{15}{64} (859-267 x) \sqrt{2+3 x^2}+\frac{5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{131985}{128} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{450975}{128} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{15}{64} (859-267 x) \sqrt{2+3 x^2}+\frac{5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{450975}{128} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{15}{64} (859-267 x) \sqrt{2+3 x^2}+\frac{5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac{(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{12885}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.150015, size = 97, normalized size = 0.77 \[ \frac{1}{128} \left (-\frac{2 \sqrt{3 x^2+2} \left (72 x^5-696 x^4+2826 x^3-19268 x^2-127403 x-126181\right )}{(2 x+3)^2}-12885 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-43995 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[2 + 3*x^2]*(-126181 - 127403*x - 19268*x^2 + 2826*x^3 - 696*x^4 + 72*x^5))/(3 + 2*x)^2 - 43995*Sqrt[
3]*ArcSinh[Sqrt[3/2]*x] - 12885*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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Maple [A]  time = 0.012, size = 185, normalized size = 1.5 \begin{align*}{\frac{421}{4900} \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{2577}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{807\,x}{224} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{4005\,x}{64}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{43995\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{859}{112} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{12885}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{12885\,\sqrt{35}}{128}{\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{1263\,x}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{13}{280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-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)^3,x)

[Out]

421/4900/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+2577/4900*(3*(x+3/2)^2-9*x-19/4)^(5/2)-807/224*x*(3*(x+3/2)^2-9*
x-19/4)^(3/2)-4005/64*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-43995/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)+859/112*(3*(x+3/
2)^2-9*x-19/4)^(3/2)+12885/128*(12*(x+3/2)^2-36*x-19)^(1/2)-12885/128*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(
12*(x+3/2)^2-36*x-19)^(1/2))-1263/4900*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-13/280/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^
(7/2)

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Maxima [A]  time = 1.55607, size = 196, normalized size = 1.56 \begin{align*} \frac{39}{280} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{807}{224} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{859}{112} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{421 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{280 \,{\left (2 \, x + 3\right )}} - \frac{4005}{64} \, \sqrt{3 \, x^{2} + 2} x - \frac{43995}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{12885}{128} \, \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{12885}{64} \, \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)^3,x, algorithm="maxima")

[Out]

39/280*(3*x^2 + 2)^(5/2) - 13/70*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 807/224*(3*x^2 + 2)^(3/2)*x + 859/112*
(3*x^2 + 2)^(3/2) + 421/280*(3*x^2 + 2)^(5/2)/(2*x + 3) - 4005/64*sqrt(3*x^2 + 2)*x - 43995/128*sqrt(3)*arcsin
h(1/2*sqrt(6)*x) + 12885/128*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 12885/6
4*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.98381, size = 412, normalized size = 3.27 \begin{align*} \frac{43995 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12885 \, \sqrt{35}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \,{\left (72 \, x^{5} - 696 \, x^{4} + 2826 \, x^{3} - 19268 \, x^{2} - 127403 \, x - 126181\right )} \sqrt{3 \, x^{2} + 2}}{256 \,{\left (4 \, x^{2} + 12 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/256*(43995*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 12885*sqrt(35)*(4*x^2 + 1
2*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 4*(72*x^5 - 696*
x^4 + 2826*x^3 - 19268*x^2 - 127403*x - 126181)*sqrt(3*x^2 + 2))/(4*x^2 + 12*x + 9)

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

[Out]

Timed out

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Giac [B]  time = 1.27543, size = 311, normalized size = 2.47 \begin{align*} -\frac{1}{32} \,{\left (3 \,{\left ({\left (3 \, x - 38\right )} x + 225\right )} x - 4177\right )} \sqrt{3 \, x^{2} + 2} + \frac{43995}{128} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{12885}{128} \, \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{35 \,{\left (11472 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 25829 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 57912 \, \sqrt{3} x + 8984 \, \sqrt{3} + 57912 \, \sqrt{3 \, x^{2} + 2}\right )}}{256 \,{\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 )}^{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)^3,x, algorithm="giac")

[Out]

-1/32*(3*((3*x - 38)*x + 225)*x - 4177)*sqrt(3*x^2 + 2) + 43995/128*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
+ 12885/128*sqrt(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))) + 35/256*(11472*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 25829*sqrt(3)*(sqrt(3)*x
- sqrt(3*x^2 + 2))^2 - 57912*sqrt(3)*x + 8984*sqrt(3) + 57912*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)^2