3.13.22 \(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^{11}} \, dx\)

Optimal. Leaf size=202 \[ -\frac {739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac {4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac {1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac {73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac {219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac {1977291 (4-9 x) \sqrt {3 x^2+2}}{514714375000 (2 x+3)^2}-\frac {5931873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{257357187500 \sqrt {35}} \]

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Rubi [A]  time = 0.13, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {835, 807, 721, 725, 206} \begin {gather*} -\frac {739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac {4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac {1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac {73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac {219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac {1977291 (4-9 x) \sqrt {3 x^2+2}}{514714375000 (2 x+3)^2}-\frac {5931873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{257357187500 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

(-1977291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(514714375000*(3 + 2*x)^2) - (219699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(147061
25000*(3 + 2*x)^4) - (73233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(1050437500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(35
0*(3 + 2*x)^10) - (1171*(2 + 3*x^2)^(7/2))/(110250*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/(1715000*(3 + 2*x)^
8) - (739619*(2 + 3*x^2)^(7/2))/(1260525000*(3 + 2*x)^7) - (5931873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/(257357187500*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]

Rule 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1}{350} \int \frac {(-410+117 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}+\frac {\int \frac {(28998-7026 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx}{110250}\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {\int \frac {(-1863024+237222 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{30870000}\\ &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac {219699 \int \frac {\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{15006250}\\ &=-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac {219699 \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{105043750}\\ &=-\frac {219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac {1977291 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{7353062500}\\ &=-\frac {1977291 (4-9 x) \sqrt {2+3 x^2}}{514714375000 (3+2 x)^2}-\frac {219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac {5931873 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{257357187500}\\ &=-\frac {1977291 (4-9 x) \sqrt {2+3 x^2}}{514714375000 (3+2 x)^2}-\frac {219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}-\frac {5931873 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{257357187500}\\ &=-\frac {1977291 (4-9 x) \sqrt {2+3 x^2}}{514714375000 (3+2 x)^2}-\frac {219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}-\frac {5931873 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{257357187500 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 207, normalized size = 1.02 \begin {gather*} \frac {1}{350} \left (-\frac {4393 \left (3 x^2+2\right )^{7/2}}{4900 (2 x+3)^8}-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac {13 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^{10}}-\frac {31711164625 \left (3 x^2+2\right )^{7/2}+219699 (2 x+3) \left (-945 (9 x-4) \sqrt {3 x^2+2} (2 x+3)^4-3675 (9 x-4) \left (3 x^2+2\right )^{3/2} (2 x+3)^2-17150 (9 x-4) \left (3 x^2+2\right )^{5/2}+162 \sqrt {35} (2 x+3)^6 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )}{154414312500 (2 x+3)^7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

((-13*(2 + 3*x^2)^(7/2))/(3 + 2*x)^10 - (1171*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/
(4900*(3 + 2*x)^8) - (31711164625*(2 + 3*x^2)^(7/2) + 219699*(3 + 2*x)*(-945*(3 + 2*x)^4*(-4 + 9*x)*Sqrt[2 + 3
*x^2] - 3675*(3 + 2*x)^2*(-4 + 9*x)*(2 + 3*x^2)^(3/2) - 17150*(-4 + 9*x)*(2 + 3*x^2)^(5/2) + 162*Sqrt[35]*(3 +
2*x)^6*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/(154414312500*(3 + 2*x)^7))/350

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IntegrateAlgebraic [A]  time = 3.64, size = 116, normalized size = 0.57 \begin {gather*} \frac {5931873 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{128678593750 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-7968937464 x^9-101311348104 x^8-544524933294 x^7-1541962687104 x^6+3078520541586 x^5-11369945485836 x^4-4704132871221 x^3-18888919063956 x^2-5421307926571 x-5288003538036\right )}{4632429375000 (2 x+3)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

(Sqrt[2 + 3*x^2]*(-5288003538036 - 5421307926571*x - 18888919063956*x^2 - 4704132871221*x^3 - 11369945485836*x
^4 + 3078520541586*x^5 - 1541962687104*x^6 - 544524933294*x^7 - 101311348104*x^8 - 7968937464*x^9))/(463242937
5000*(3 + 2*x)^10) + (5931873*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(12867859
3750*Sqrt[35])

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fricas [A]  time = 0.45, size = 209, normalized size = 1.03 \begin {gather*} \frac {53386857 \, \sqrt {35} {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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 (7968937464 \, x^{9} + 101311348104 \, x^{8} + 544524933294 \, x^{7} + 1541962687104 \, x^{6} - 3078520541586 \, x^{5} + 11369945485836 \, x^{4} + 4704132871221 \, x^{3} + 18888919063956 \, x^{2} + 5421307926571 \, x + 5288003538036\right )} \sqrt {3 \, x^{2} + 2}}{162135028125000 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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)^11,x, algorithm="fricas")

[Out]

1/162135028125000*(53386857*sqrt(35)*(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*
x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 9
3*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(7968937464*x^9 + 101311348104*x^8 + 544524933294*x^7 + 1541962687
104*x^6 - 3078520541586*x^5 + 11369945485836*x^4 + 4704132871221*x^3 + 18888919063956*x^2 + 5421307926571*x +
5288003538036)*sqrt(3*x^2 + 2))/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 +
2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)

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giac [B]  time = 0.53, size = 547, normalized size = 2.71 \begin {gather*} \frac {5931873}{9007501562500} \, \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 \, \sqrt {3} {\left (56242944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{19} + 4808771712 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{18} + 60161202432 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{17} + 2449600006086 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{16} + 650003734476 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} + 11324343251586 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} - 43249498138224 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} - 114750161469717 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 263561308381422 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 64560900263031 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 173527579922724 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 409007369125548 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 812515292998272 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 775661489485344 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 309262645005696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 53888888658816 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 21200045958144 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 6293205518848 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 348990277632 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 25185777664\right )}}{65883440000000 \, {\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 )}^{10}} \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)^11,x, algorithm="giac")

[Out]

5931873/9007501562500*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))) - 9/65883440000000*sqrt(3)*(56242944*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 2))^19 + 4808771712*(sqrt(3)*x - sqrt(3*x^2 + 2))^18 + 60161202432*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))
^17 + 2449600006086*(sqrt(3)*x - sqrt(3*x^2 + 2))^16 + 650003734476*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 +
11324343251586*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 - 43249498138224*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 - 1
14750161469717*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 263561308381422*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 6
4560900263031*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 173527579922724*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 409
007369125548*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 812515292998272*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 77566
1489485344*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 309262645005696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 5388888
8658816*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 21200045958144*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 62932055188
48*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 348990277632*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 25185777664)/((sqrt(
3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^10

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maple [B]  time = 0.15, size = 341, normalized size = 1.69 \begin {gather*} \frac {24718114791 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{55170947070312500}+\frac {694029141 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{630525109375000}+\frac {53386857 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{18015003125000}-\frac {5931873 \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 )}{9007501562500}-\frac {73233 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{33614000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {4393 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{439040000 \left (x +\frac {3}{2}\right )^{8}}-\frac {1171 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{56448000 \left (x +\frac {3}{2}\right )^{9}}-\frac {6371271 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10294287500000 \left (x +\frac {3}{2}\right )^{4}}-\frac {659097 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{588245000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {65250603 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{180150031250000 \left (x +\frac {3}{2}\right )^{3}}-\frac {709847469 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{3152625546875000 \left (x +\frac {3}{2}\right )^{2}}-\frac {8239371597 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{55170947070312500 \left (x +\frac {3}{2}\right )}+\frac {47454984 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{13792736767578125}+\frac {3954582 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{78815638671875}+\frac {5931873 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{9007501562500}-\frac {739619 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{161347200000 \left (x +\frac {3}{2}\right )^{7}}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{358400 \left (x +\frac {3}{2}\right )^{10}} \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)^11,x)

[Out]

-73233/33614000000/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-4393/439040000/(x+3/2)^8*(-9*x+3*(x+3/2)^2-19/4)^(7
/2)-1171/56448000/(x+3/2)^9*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-6371271/10294287500000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-1
9/4)^(7/2)-659097/588245000000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-65250603/180150031250000/(x+3/2)^3*(-9*
x+3*(x+3/2)^2-19/4)^(7/2)-709847469/3152625546875000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/2)[phone]/55170
947070312500*(-9*x+3*(x+3/2)^2-19/4)^(5/2)*x-8239371597/55170947070312500/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2
)[phone]/630525109375000*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x+53386857/18015003125000*(-9*x+3*(x+3/2)^2-19/4)^(1
/2)*x-5931873/9007501562500*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))[phone]/13
792736767578125*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+3954582/78815638671875*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+5931873/900
7501562500*(-36*x+12*(x+3/2)^2-19)^(1/2)-739619/161347200000/(x+3/2)^7*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-13/358400
/(x+3/2)^10*(-9*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [B]  time = 1.47, size = 497, normalized size = 2.46 \begin {gather*} \frac {2129542407}{3152625546875000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{350 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} - \frac {1171 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{110250 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {4393 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1715000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {739619 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1260525000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {73233 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{525218750 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {659097 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{18382656250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {6371271 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{643392968750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {65250603 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{22518753906250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {709847469 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{788156386718750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {694029141}{630525109375000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3954582}{78815638671875} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {8239371597 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3152625546875000 \, {\left (2 \, x + 3\right )}} + \frac {53386857}{18015003125000} \, \sqrt {3 \, x^{2} + 2} x + \frac {5931873}{9007501562500} \, \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 {5931873}{4503750781250} \, \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)^11,x, algorithm="maxima")

[Out]

2129542407/3152625546875000*(3*x^2 + 2)^(5/2) - 13/350*(3*x^2 + 2)^(7/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 +
414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 1171/1
10250*(3*x^2 + 2)^(7/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 +
314928*x^2 + 118098*x + 19683) - 4393/1715000*(3*x^2 + 2)^(7/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 +
90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 739619/1260525000*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6
+ 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 73233/525218750*(3*x^2 + 2)^(7/2)/(64*x^6
+ 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 659097/18382656250*(3*x^2 + 2)^(7/2)/(32*x^5 + 24
0*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 6371271/643392968750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2
+ 216*x + 81) - 65250603/22518753906250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 709847469/78815638671
8750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 694029141/630525109375000*(3*x^2 + 2)^(3/2)*x + 3954582/7881563867
1875*(3*x^2 + 2)^(3/2) - 8239371597/3152625546875000*(3*x^2 + 2)^(5/2)/(2*x + 3) + 53386857/18015003125000*sqr
t(3*x^2 + 2)*x + 5931873/9007501562500*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3))
+ 5931873/4503750781250*sqrt(3*x^2 + 2)

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mupad [B]  time = 1.90, size = 449, normalized size = 2.22 \begin {gather*} \frac {5931873\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{9007501562500}-\frac {5931873\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{9007501562500}-\frac {43213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{655360\,\left (x^8+12\,x^7+63\,x^6+189\,x^5+\frac {2835\,x^4}{8}+\frac {1701\,x^3}{4}+\frac {5103\,x^2}{16}+\frac {2187\,x}{16}+\frac {6561}{256}\right )}+\frac {4728159\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{430259200000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {36029\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{589824\,\left (x^9+\frac {27\,x^8}{2}+81\,x^7+\frac {567\,x^6}{2}+\frac {5103\,x^5}{8}+\frac {15309\,x^4}{16}+\frac {15309\,x^3}{16}+\frac {19683\,x^2}{32}+\frac {59049\,x}{256}+\frac {19683}{512}\right )}+\frac {27428781\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{24586240000\,\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 {3185\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{131072\,\left (x^{10}+15\,x^9+\frac {405\,x^8}{4}+405\,x^7+\frac {8505\,x^6}{8}+\frac {15309\,x^5}{8}+\frac {76545\,x^4}{32}+\frac {32805\,x^3}{16}+\frac {295245\,x^2}{256}+\frac {98415\,x}{256}+\frac {59049}{1024}\right )}-\frac {110679687\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{65883440000000\,\left (x+\frac {3}{2}\right )}-\frac {2988711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{280985600\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {4975641\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3764768000000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {1785563\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48168960\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {5833857\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1075648000000\,\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)^11,x)

[Out]

(5931873*35^(1/2)*log(x + 3/2))/9007501562500 - (5931873*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))
/9 - 4/9))/9007501562500 - (43213*3^(1/2)*(x^2 + 2/3)^(1/2))/(655360*((2187*x)/16 + (5103*x^2)/16 + (1701*x^3)
/4 + (2835*x^4)/8 + 189*x^5 + 63*x^6 + 12*x^7 + x^8 + 6561/256)) + (4728159*3^(1/2)*(x^2 + 2/3)^(1/2))/(430259
200000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (36029*3^(1/2)*(x^2 + 2/3)^(1/2))/(589824*((59049*x)/2
56 + (19683*x^2)/32 + (15309*x^3)/16 + (15309*x^4)/16 + (5103*x^5)/8 + (567*x^6)/2 + 81*x^7 + (27*x^8)/2 + x^9
+ 19683/512)) + (27428781*3^(1/2)*(x^2 + 2/3)^(1/2))/(24586240000*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (1
5*x^4)/2 + x^5 + 243/32)) - (3185*3^(1/2)*(x^2 + 2/3)^(1/2))/(131072*((98415*x)/256 + (295245*x^2)/256 + (3280
5*x^3)/16 + (76545*x^4)/32 + (15309*x^5)/8 + (8505*x^6)/8 + 405*x^7 + (405*x^8)/4 + 15*x^9 + x^10 + 59049/1024
)) - (110679687*3^(1/2)*(x^2 + 2/3)^(1/2))/(65883440000000*(x + 3/2)) - (2988711*3^(1/2)*(x^2 + 2/3)^(1/2))/(2
80985600*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (4975641*3^(1/2)*(
x^2 + 2/3)^(1/2))/(3764768000000*(3*x + x^2 + 9/4)) + (1785563*3^(1/2)*(x^2 + 2/3)^(1/2))/(48168960*((5103*x)/
64 + (5103*x^2)/32 + (2835*x^3)/16 + (945*x^4)/8 + (189*x^5)/4 + (21*x^6)/2 + x^7 + 2187/128)) + (5833857*3^(1
/2)*(x^2 + 2/3)^(1/2))/(1075648000000*((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)**11,x)

[Out]

Timed out

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