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

Optimal. Leaf size=117 \[ -\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}-\frac {7}{16} (775-243 x) \sqrt {3 x^2+2}+\frac {5425}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )+\frac {18543}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 117, 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, 815, 844, 215, 725, 206} \begin {gather*} -\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}-\frac {7}{16} (775-243 x) \sqrt {3 x^2+2}+\frac {5425}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )+\frac {18543}{32} \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)^2,x]

[Out]

(-7*(775 - 243*x)*Sqrt[2 + 3*x^2])/16 - ((310 - 153*x)*(2 + 3*x^2)^(3/2))/24 - ((34 + x)*(2 + 3*x^2)^(5/2))/(1
0*(3 + 2*x)) + (18543*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (5425*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3
*x^2])])/32

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 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]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx &=-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{8} \int \frac {(8-408 x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{384} \int \frac {(15456-163296 x) \sqrt {2+3 x^2}}{3+2 x} \, dx\\ &=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {\int \frac {6620544-32042304 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{9216}\\ &=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {55629}{32} \int \frac {1}{\sqrt {2+3 x^2}} \, dx-\frac {189875}{32} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {189875}{32} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {5425}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 97, normalized size = 0.83 \begin {gather*} \frac {1}{480} \left (81375 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {2 \sqrt {3 x^2+2} \left (216 x^5-1836 x^4+5118 x^3-19458 x^2+89521 x+265989\right )}{2 x+3}+278145 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[2 + 3*x^2]*(265989 + 89521*x - 19458*x^2 + 5118*x^3 - 1836*x^4 + 216*x^5))/(3 + 2*x) + 278145*Sqrt[3
]*ArcSinh[Sqrt[3/2]*x] + 81375*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/480

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IntegrateAlgebraic [A]  time = 0.66, size = 126, normalized size = 1.08 \begin {gather*} -\frac {18543}{32} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )-\frac {5425}{16} \sqrt {35} \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )+\frac {\sqrt {3 x^2+2} \left (-216 x^5+1836 x^4-5118 x^3+19458 x^2-89521 x-265989\right )}{240 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-265989 - 89521*x + 19458*x^2 - 5118*x^3 + 1836*x^4 - 216*x^5))/(240*(3 + 2*x)) - (5425*Sqrt
[35]*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/16 - (18543*Sqrt[3]*Log[-(Sqrt[3]*
x) + Sqrt[2 + 3*x^2]])/32

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fricas [A]  time = 0.44, size = 131, normalized size = 1.12 \begin {gather*} \frac {278145 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 81375 \, \sqrt {35} {\left (2 \, x + 3\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 (216 \, x^{5} - 1836 \, x^{4} + 5118 \, x^{3} - 19458 \, x^{2} + 89521 \, x + 265989\right )} \sqrt {3 \, x^{2} + 2}}{960 \, {\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)^(5/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/960*(278145*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 81375*sqrt(35)*(2*x + 3)*log((sq
rt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 4*(216*x^5 - 1836*x^4 + 5118*x^3
- 19458*x^2 + 89521*x + 265989)*sqrt(3*x^2 + 2))/(2*x + 3)

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giac [B]  time = 1.02, size = 665, normalized size = 5.68 \begin {gather*} \frac {5425}{32} \, \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 {18543}{32} \, \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 {15925}{128} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {9 \, {\left (238455 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 149045 \, \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 )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 697600 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 719040 \, \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 )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 4150566 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2707250 \, \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 )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 6756120 \, {\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 ) + 4557000 \, \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 ) + 3563595 \, {\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 ) - 2833425 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{320 \, {\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 )}^{5}} \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)^2,x, algorithm="giac")

[Out]

5425/32*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)) - 18543/32*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)) - 15925/128
*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 9/320*(238455*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^
2 + 3) + sqrt(35)/(2*x + 3))^9*sgn(1/(2*x + 3)) - 149045*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) +
sqrt(35)/(2*x + 3))^8*sgn(1/(2*x + 3)) - 697600*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)
)^7*sgn(1/(2*x + 3)) + 719040*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^6*sgn(1
/(2*x + 3)) + 4150566*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 270
7250*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 6756120*(sq
rt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 4557000*sqrt(35)*(sqrt(-18/(
2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 3563595*(sqrt(-18/(2*x + 3) + 35/(2*
x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sgn(1/(2*x + 3)) - 2833425*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)^5

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maple [A]  time = 0.05, size = 164, normalized size = 1.40 \begin {gather*} \frac {51 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{8}+\frac {1701 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{16}+\frac {39 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{70}+\frac {18543 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{32}+\frac {5425 \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 )}{32}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{70 \left (x +\frac {3}{2}\right )}-\frac {31 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{35}-\frac {155 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {5425 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32} \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)^2,x)

[Out]

-13/70/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-31/35*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+51/8*(-9*x+3*(x+3/2)^2-19/4)^
(3/2)*x+1701/16*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+18543/32*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-155/12*(-9*x+3*(x+3/2)
^2-19/4)^(3/2)-5425/32*(-36*x+12*(x+3/2)^2-19)^(1/2)+5425/32*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12
*(x+3/2)^2-19)^(1/2))+39/70*x*(-9*x+3*(x+3/2)^2-19/4)^(5/2)

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maxima [A]  time = 1.24, size = 122, normalized size = 1.04 \begin {gather*} -\frac {1}{20} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {51}{8} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x - \frac {155}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {1701}{16} \, \sqrt {3 \, x^{2} + 2} x + \frac {18543}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {5425}{32} \, \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 {5425}{16} \, \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)^2,x, algorithm="maxima")

[Out]

-1/20*(3*x^2 + 2)^(5/2) + 51/8*(3*x^2 + 2)^(3/2)*x - 155/12*(3*x^2 + 2)^(3/2) - 13/4*(3*x^2 + 2)^(5/2)/(2*x +
3) + 1701/16*sqrt(3*x^2 + 2)*x + 18543/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 5425/32*sqrt(35)*arcsinh(3/2*sqrt(6
)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 5425/16*sqrt(3*x^2 + 2)

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mupad [B]  time = 0.13, size = 138, normalized size = 1.18 \begin {gather*} \frac {18543\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {275027\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{960}-\frac {5425\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{32}+\frac {5425\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{32}-\frac {1393\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{80}+\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{2}-\frac {9\,\sqrt {3}\,x^4\,\sqrt {x^2+\frac {2}{3}}}{20}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x+\frac {3}{2}\right )}+\frac {2133\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \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)^2,x)

[Out]

(18543*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 - (275027*3^(1/2)*(x^2 + 2/3)^(1/2))/960 - (5425*35^(1/2)*log(
x + 3/2))/32 + (5425*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/32 - (1393*3^(1/2)*x^2*(x
^2 + 2/3)^(1/2))/80 + (9*3^(1/2)*x^3*(x^2 + 2/3)^(1/2))/2 - (9*3^(1/2)*x^4*(x^2 + 2/3)^(1/2))/20 - (15925*3^(1
/2)*(x^2 + 2/3)^(1/2))/(128*(x + 3/2)) + (2133*3^(1/2)*x*(x^2 + 2/3)^(1/2))/32

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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)**2,x)

[Out]

Timed out

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