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

Optimal. Leaf size=133 \[ -\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac {3 (1917 x+6125) \sqrt {3 x^2+2}}{448 (2 x+3)}-\frac {188379 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{896 \sqrt {35}}-\frac {2625}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {813, 811, 844, 215, 725, 206} \begin {gather*} -\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac {3 (1917 x+6125) \sqrt {3 x^2+2}}{448 (2 x+3)}-\frac {188379 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{896 \sqrt {35}}-\frac {2625}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(6125 + 1917*x)*Sqrt[2 + 3*x^2])/(448*(3 + 2*x)) - ((5003 + 5517*x)*(2 + 3*x^2)^(3/2))/(672*(3 + 2*x)^3) -
((19 + 4*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (2625*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (188379*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(896*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 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 811

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

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]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx &=-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {5}{64} \int \frac {(32-228 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=-\frac {(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}+\frac {\int \frac {(-35904+184032 x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx}{7168}\\ &=\frac {3 (6125+1917 x) \sqrt {2+3 x^2}}{448 (3+2 x)}-\frac {(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {\int \frac {-1472256+7056000 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{57344}\\ &=\frac {3 (6125+1917 x) \sqrt {2+3 x^2}}{448 (3+2 x)}-\frac {(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {7875}{128} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {188379}{896} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {3 (6125+1917 x) \sqrt {2+3 x^2}}{448 (3+2 x)}-\frac {(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {2625}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {188379}{896} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {3 (6125+1917 x) \sqrt {2+3 x^2}}{448 (3+2 x)}-\frac {(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac {(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {2625}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {188379 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{896 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 97, normalized size = 0.73 \begin {gather*} \frac {-565137 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {70 \sqrt {3 x^2+2} \left (3024 x^5-57456 x^4-898734 x^3-2762820 x^2-3335009 x-1421955\right )}{(2 x+3)^4}-1929375 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{94080} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-70*Sqrt[2 + 3*x^2]*(-1421955 - 3335009*x - 2762820*x^2 - 898734*x^3 - 57456*x^4 + 3024*x^5))/(3 + 2*x)^4 -
1929375*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 565137*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/94080

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IntegrateAlgebraic [A]  time = 1.31, size = 126, normalized size = 0.95 \begin {gather*} \frac {2625}{128} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )+\frac {188379 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{448 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-3024 x^5+57456 x^4+898734 x^3+2762820 x^2+3335009 x+1421955\right )}{1344 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(1421955 + 3335009*x + 2762820*x^2 + 898734*x^3 + 57456*x^4 - 3024*x^5))/(1344*(3 + 2*x)^4) +
(188379*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(448*Sqrt[35]) + (2625*Sqrt[3]
*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/128

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fricas [A]  time = 0.46, size = 176, normalized size = 1.32 \begin {gather*} \frac {1929375 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 565137 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) - 140 \, {\left (3024 \, x^{5} - 57456 \, x^{4} - 898734 \, x^{3} - 2762820 \, x^{2} - 3335009 \, x - 1421955\right )} \sqrt {3 \, x^{2} + 2}}{188160 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/188160*(1929375*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
+ 565137*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 -
36*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3024*x^5 - 57456*x^4 - 898734*x^3 - 2762820*x^2 - 3335009*x - 1421955)*
sqrt(3*x^2 + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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giac [B]  time = 0.82, size = 440, normalized size = 3.31 \begin {gather*} -\frac {188379}{31360} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {2625}{128} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{10752} \, {\left (\frac {7 \, {\left (\frac {35 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 2129 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 57681 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 242979 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {9 \, {\left (256 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 93 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 582 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 225 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{64 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-188379/31360*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1
/(2*x + 3)) + 2625/128*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35
)/(2*x + 3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 1/1
0752*(7*(35*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 2129*sgn(1/(2*x + 3)))/(2*x + 3) + 57681*sgn(1/(2*x + 3)))/(2*x
+ 3) - 242979*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 9/64*(256*(sqrt(-18/(2*x + 3) + 35
/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sgn(1/(2*x + 3)) - 93*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2
+ 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 582*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x
+ 3))*sgn(1/(2*x + 3)) + 225*sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/
(2*x + 3))^2 - 3)^2

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maple [B]  time = 0.07, size = 227, normalized size = 1.71 \begin {gather*} -\frac {58629 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{274400}-\frac {58491 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{15680}-\frac {89151 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{6002500}-\frac {2625 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{128}-\frac {188379 \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 )}{31360}+\frac {23 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{117600 \left (x +\frac {3}{2}\right )^{3}}-\frac {1041 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{343000 \left (x +\frac {3}{2}\right )^{2}}+\frac {29717 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{6002500 \left (x +\frac {3}{2}\right )}+\frac {188379 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{6002500}+\frac {62793 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{137200}+\frac {188379 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{31360}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23/117600/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-1041/343000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+29717/60
02500/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+188379/6002500*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-58629/274400*(-9*x+3*
(x+3/2)^2-19/4)^(3/2)*x-58491/15680*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-2625/128*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+62
793/137200*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+188379/31360*(-36*x+12*(x+3/2)^2-19)^(1/2)-188379/31360*35^(1/2)*arct
anh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))-89151/6002500*(-9*x+3*(x+3/2)^2-19/4)^(5/2)*x-13/224
0/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [A]  time = 1.54, size = 206, normalized size = 1.55 \begin {gather*} \frac {3123}{343000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac {23 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{14700 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1041 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{85750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {58629}{274400} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {62793}{137200} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {29717 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{343000 \, {\left (2 \, x + 3\right )}} - \frac {58491}{15680} \, \sqrt {3 \, x^{2} + 2} x - \frac {2625}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {188379}{31360} \, \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 {188379}{15680} \, \sqrt {3 \, x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3123/343000*(3*x^2 + 2)^(5/2) - 13/140*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) + 23/14700*(
3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1041/85750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 58629/274400
*(3*x^2 + 2)^(3/2)*x + 62793/137200*(3*x^2 + 2)^(3/2) + 29717/343000*(3*x^2 + 2)^(5/2)/(2*x + 3) - 58491/15680
*sqrt(3*x^2 + 2)*x - 2625/128*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 188379/31360*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs
(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 188379/15680*sqrt(3*x^2 + 2)

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mupad [B]  time = 0.13, size = 180, normalized size = 1.35 \begin {gather*} \frac {188379\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{31360}+\frac {225\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64}-\frac {2625\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {188379\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{31360}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {80993\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3584\,\left (x+\frac {3}{2}\right )}-\frac {19227\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1024\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {9\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{64}+\frac {74515\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6144\,\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)^(5/2)*(x - 5))/(2*x + 3)^5,x)

[Out]

(188379*35^(1/2)*log(x + 3/2))/31360 + (225*3^(1/2)*(x^2 + 2/3)^(1/2))/64 - (2625*3^(1/2)*asinh((2^(1/2)*3^(1/
2)*x)/2))/128 - (188379*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/31360 - (15925*3^(1/2)
*(x^2 + 2/3)^(1/2))/(4096*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (80993*3^(1/2)*(x^2 + 2/3)^(1/2))/(
3584*(x + 3/2)) - (19227*3^(1/2)*(x^2 + 2/3)^(1/2))/(1024*(3*x + x^2 + 9/4)) - (9*3^(1/2)*x*(x^2 + 2/3)^(1/2))
/64 + (74515*3^(1/2)*(x^2 + 2/3)^(1/2))/(6144*((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)**(5/2)/(3+2*x)**5,x)

[Out]

Timed out

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