3.13.4 \(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=109 \[ -\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac {369 (4-9 x) \sqrt {3 x^2+2}}{171500 (2 x+3)^2}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{85750 \sqrt {35}} \]

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Rubi [A]  time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {807, 721, 725, 206} \begin {gather*} -\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac {369 (4-9 x) \sqrt {3 x^2+2}}{171500 (2 x+3)^2}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{85750 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(4900*(3 + 2*x)^4) -
(13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) - (1107*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(85750*Sqrt[35
])

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])

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac {41}{35} \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac {369 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{2450}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{171500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}+\frac {1107 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{85750}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{171500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}-\frac {1107 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{85750}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{171500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{85750 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 112, normalized size = 1.03 \begin {gather*} \frac {1}{350} \left (-\frac {26 \left (3 x^2+2\right )^{5/2}}{(2 x+3)^5}+\frac {41 (9 x-4) \left (3 x^2+2\right )^{3/2}}{14 (2 x+3)^4}-\frac {369 \left (6 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 (9 x-4) \sqrt {3 x^2+2}}{(2 x+3)^2}\right )}{17150}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

((41*(-4 + 9*x)*(2 + 3*x^2)^(3/2))/(14*(3 + 2*x)^4) - (26*(2 + 3*x^2)^(5/2))/(3 + 2*x)^5 - (369*((-35*(-4 + 9*
x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 + 6*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/17150)/350

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IntegrateAlgebraic [A]  time = 1.37, size = 91, normalized size = 0.83 \begin {gather*} \frac {1107 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{42875 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-10602 x^4+189543 x^3-26682 x^2+64493 x-125252\right )}{171500 (2 x+3)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(Sqrt[2 + 3*x^2]*(-125252 + 64493*x - 26682*x^2 + 189543*x^3 - 10602*x^4))/(171500*(3 + 2*x)^5) + (1107*ArcTan
h[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(42875*Sqrt[35])

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fricas [A]  time = 0.43, size = 134, normalized size = 1.23 \begin {gather*} \frac {1107 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 ) - 35 \, {\left (10602 \, x^{4} - 189543 \, x^{3} + 26682 \, x^{2} - 64493 \, x + 125252\right )} \sqrt {3 \, x^{2} + 2}}{6002500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/6002500*(1107*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*
(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(10602*x^4 - 189543*x^3 + 26682*x^2 - 64493*x + 12525
2)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.27, size = 318, normalized size = 2.92 \begin {gather*} \frac {1107}{3001250} \, \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 {9 \, {\left (89686 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 138886 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 1224478 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 245133 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 1224531 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 4374874 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 4855928 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1339152 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 586816 \, \sqrt {3} x - 37696 \, \sqrt {3} + 586816 \, \sqrt {3 \, x^{2} + 2}\right )}}{2744000 \, {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

1107/3001250*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))) - 9/2744000*(89686*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 138886*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 2))^8 + 1224478*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 245133*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
2))^6 - 1224531*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4374874*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 4855928*(
sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1339152*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 586816*sqrt(3)*x - 37696*sq
rt(3) + 586816*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
)^5

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maple [B]  time = 0.07, size = 203, normalized size = 1.86 \begin {gather*} \frac {9963 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{6002500}+\frac {129519 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{210087500}-\frac {1107 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{3001250}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {41 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{39200 \left (x +\frac {3}{2}\right )^{4}}-\frac {369 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{686000 \left (x +\frac {3}{2}\right )^{3}}-\frac {3813 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12005000 \left (x +\frac {3}{2}\right )^{2}}-\frac {43173 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{210087500 \left (x +\frac {3}{2}\right )}+\frac {1476 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{52521875}+\frac {1107 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{3001250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^6,x)

[Out]

-13/5600/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-41/39200/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-369/686000/(
x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-3813/12005000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-43173/210087500/(
x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+1476/52521875*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+9963/6002500*(-9*x+3*(x+3/2)^
2-19/4)^(1/2)*x+1107/3001250*(-36*x+12*(x+3/2)^2-19)^(1/2)-1107/3001250*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2
)/(-36*x+12*(x+3/2)^2-19)^(1/2))+129519/210087500*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x

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maxima [B]  time = 1.32, size = 209, normalized size = 1.92 \begin {gather*} \frac {11439}{12005000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{2450 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {369 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{85750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {3813 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3001250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {9963}{6002500} \, \sqrt {3 \, x^{2} + 2} x + \frac {1107}{3001250} \, \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 {1107}{1500625} \, \sqrt {3 \, x^{2} + 2} - \frac {43173 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{12005000 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

11439/12005000*(3*x^2 + 2)^(3/2) - 13/175*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 2
43) - 41/2450*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 369/85750*(3*x^2 + 2)^(5/2)/(8*x^3
+ 36*x^2 + 54*x + 27) - 3813/3001250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 9963/6002500*sqrt(3*x^2 + 2)*x + 1
107/3001250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1107/1500625*sqrt(3*x^2
+ 2) - 43173/12005000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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mupad [B]  time = 1.98, size = 179, normalized size = 1.64 \begin {gather*} \frac {1107\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{3001250}-\frac {1107\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{3001250}+\frac {731\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2560\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {91\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {5301\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2744000\,\left (x+\frac {3}{2}\right )}+\frac {7233\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{156800\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {8349\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{44800\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^6,x)

[Out]

(1107*35^(1/2)*log(x + 3/2))/3001250 - (1107*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/3
001250 + (731*3^(1/2)*(x^2 + 2/3)^(1/2))/(2560*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (91*3^(1/2)*(x
^2 + 2/3)^(1/2))/(512*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (5301*3^(1/2)*(x^
2 + 2/3)^(1/2))/(2744000*(x + 3/2)) + (7233*3^(1/2)*(x^2 + 2/3)^(1/2))/(156800*(3*x + x^2 + 9/4)) - (8349*3^(1
/2)*(x^2 + 2/3)^(1/2))/(44800*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**6,x)

[Out]

Timed out

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