3.1386 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{3+2 x} \, dx\)

Optimal. Leaf size=112 \[ \frac{1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac{7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac{7}{64} (2275-691 x) \sqrt{3 x^2+2}-\frac{15925}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}} \]

[Out]

(7*(2275 - 691*x)*Sqrt[2 + 3*x^2])/64 + (7*(130 - 53*x)*(2 + 3*x^2)^(3/2))/96 + ((39 - 5*x)*(2 + 3*x^2)^(5/2))
/60 - (162673*ArcSinh[Sqrt[3/2]*x])/(128*Sqrt[3]) - (15925*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/128

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Rubi [A]  time = 0.0761678, antiderivative size = 112, 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 = {815, 844, 215, 725, 206} \[ \frac{1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac{7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac{7}{64} (2275-691 x) \sqrt{3 x^2+2}-\frac{15925}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2275 - 691*x)*Sqrt[2 + 3*x^2])/64 + (7*(130 - 53*x)*(2 + 3*x^2)^(3/2))/96 + ((39 - 5*x)*(2 + 3*x^2)^(5/2))
/60 - (162673*ArcSinh[Sqrt[3/2]*x])/(128*Sqrt[3]) - (15925*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/128

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} \, dx &=\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac{1}{72} \int \frac{(756-2226 x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac{7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac{\int \frac{(152712-1044792 x) \sqrt{2+3 x^2}}{3+2 x} \, dx}{3456}\\ &=\frac{7}{64} (2275-691 x) \sqrt{2+3 x^2}+\frac{7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac{\int \frac{44942688-210824208 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{82944}\\ &=\frac{7}{64} (2275-691 x) \sqrt{2+3 x^2}+\frac{7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac{162673}{128} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{557375}{128} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{7}{64} (2275-691 x) \sqrt{2+3 x^2}+\frac{7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}}-\frac{557375}{128} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{7}{64} (2275-691 x) \sqrt{2+3 x^2}+\frac{7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}}-\frac{15925}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.058676, size = 90, normalized size = 0.8 \[ \frac{-2 \sqrt{3 x^2+2} \left (720 x^5-5616 x^4+12090 x^3-34788 x^2+80295 x-259571\right )-238875 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-813365 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{1920} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + 3*x^2]*(-259571 + 80295*x - 34788*x^2 + 12090*x^3 - 5616*x^4 + 720*x^5) - 813365*Sqrt[3]*ArcSinh[
Sqrt[3/2]*x] - 238875*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/1920

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Maple [A]  time = 0.008, size = 162, normalized size = 1.5 \begin{align*} -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{5\,x}{24} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x}{8}\sqrt{3\,{x}^{2}+2}}-{\frac{162673\,\sqrt{3}}{384}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{117\,x}{32} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{4797\,x}{64}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{455}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{15925}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{15925\,\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 ) } \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),x)

[Out]

-1/12*x*(3*x^2+2)^(5/2)-5/24*x*(3*x^2+2)^(3/2)-5/8*x*(3*x^2+2)^(1/2)-162673/384*arcsinh(1/2*x*6^(1/2))*3^(1/2)
+13/20*(3*(x+3/2)^2-9*x-19/4)^(5/2)-117/32*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-4797/64*x*(3*(x+3/2)^2-9*x-19/4)^(1/
2)+455/48*(3*(x+3/2)^2-9*x-19/4)^(3/2)+15925/128*(12*(x+3/2)^2-36*x-19)^(1/2)-15925/128*35^(1/2)*arctanh(2/35*
(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 1.48746, size = 157, normalized size = 1.4 \begin{align*} -\frac{1}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{13}{20} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{371}{96} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{455}{48} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{4837}{64} \, \sqrt{3 \, x^{2} + 2} x - \frac{162673}{384} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{15925}{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{15925}{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),x, algorithm="maxima")

[Out]

-1/12*(3*x^2 + 2)^(5/2)*x + 13/20*(3*x^2 + 2)^(5/2) - 371/96*(3*x^2 + 2)^(3/2)*x + 455/48*(3*x^2 + 2)^(3/2) -
4837/64*sqrt(3*x^2 + 2)*x - 162673/384*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 15925/128*sqrt(35)*arcsinh(3/2*sqrt(6)
*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 15925/64*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.92288, size = 346, normalized size = 3.09 \begin{align*} -\frac{1}{960} \,{\left (720 \, x^{5} - 5616 \, x^{4} + 12090 \, x^{3} - 34788 \, x^{2} + 80295 \, x - 259571\right )} \sqrt{3 \, x^{2} + 2} + \frac{162673}{768} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac{15925}{256} \, \sqrt{35} \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 ) \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),x, algorithm="fricas")

[Out]

-1/960*(720*x^5 - 5616*x^4 + 12090*x^3 - 34788*x^2 + 80295*x - 259571)*sqrt(3*x^2 + 2) + 162673/768*sqrt(3)*lo
g(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 15925/256*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^
2 - 36*x + 43)/(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),x)

[Out]

Timed out

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Giac [A]  time = 1.18072, size = 169, normalized size = 1.51 \begin{align*} -\frac{1}{960} \,{\left (3 \,{\left (2 \,{\left ({\left (24 \,{\left (5 \, x - 39\right )} x + 2015\right )} x - 5798\right )} x + 26765\right )} x - 259571\right )} \sqrt{3 \, x^{2} + 2} + \frac{162673}{384} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{15925}{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 ) \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),x, algorithm="giac")

[Out]

-1/960*(3*(2*((24*(5*x - 39)*x + 2015)*x - 5798)*x + 26765)*x - 259571)*sqrt(3*x^2 + 2) + 162673/384*sqrt(3)*l
og(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 15925/128*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)))