3.1380 \(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=153 \[ -\frac {822 \left (3 x^2+2\right )^{5/2}}{214375 (2 x+3)^5}-\frac {404 \left (3 x^2+2\right )^{5/2}}{25725 (2 x+3)^6}-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}-\frac {2689 (4-9 x) \left (3 x^2+2\right )^{3/2}}{6002500 (2 x+3)^4}-\frac {24201 (4-9 x) \sqrt {3 x^2+2}}{210087500 (2 x+3)^2}-\frac {72603 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{105043750 \sqrt {35}} \]

[Out]

-2689/6002500*(4-9*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-13/245*(3*x^2+2)^(5/2)/(3+2*x)^7-404/25725*(3*x^2+2)^(5/2)/(3+
2*x)^6-822/214375*(3*x^2+2)^(5/2)/(3+2*x)^5-72603/3676531250*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35
^(1/2)-24201/210087500*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2

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Rubi [A]  time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, 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 = {835, 807, 721, 725, 206} \[ -\frac {822 \left (3 x^2+2\right )^{5/2}}{214375 (2 x+3)^5}-\frac {404 \left (3 x^2+2\right )^{5/2}}{25725 (2 x+3)^6}-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}-\frac {2689 (4-9 x) \left (3 x^2+2\right )^{3/2}}{6002500 (2 x+3)^4}-\frac {24201 (4-9 x) \sqrt {3 x^2+2}}{210087500 (2 x+3)^2}-\frac {72603 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{105043750 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

(-24201*(4 - 9*x)*Sqrt[2 + 3*x^2])/(210087500*(3 + 2*x)^2) - (2689*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(6002500*(3 +
2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(245*(3 + 2*x)^7) - (404*(2 + 3*x^2)^(5/2))/(25725*(3 + 2*x)^6) - (822*(2 + 3
*x^2)^(5/2))/(214375*(3 + 2*x)^5) - (72603*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(105043750*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 )^{3/2}}{(3+2 x)^8} \, dx &=-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {1}{245} \int \frac {(-287+78 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}+\frac {\int \frac {(13626-2424 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{51450}\\ &=-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac {2689 \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{42875}\\ &=-\frac {2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac {24201 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{3001250}\\ &=-\frac {24201 (4-9 x) \sqrt {2+3 x^2}}{210087500 (3+2 x)^2}-\frac {2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac {72603 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{105043750}\\ &=-\frac {24201 (4-9 x) \sqrt {2+3 x^2}}{210087500 (3+2 x)^2}-\frac {2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}-\frac {72603 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{105043750}\\ &=-\frac {24201 (4-9 x) \sqrt {2+3 x^2}}{210087500 (3+2 x)^2}-\frac {2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}-\frac {72603 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{105043750 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 161, normalized size = 1.05 \[ \frac {1}{245} \left (-\frac {822 \left (3 x^2+2\right )^{5/2}}{875 (2 x+3)^5}-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}-\frac {13 \left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}-\frac {2689 \left (-315 (9 x-4) \sqrt {3 x^2+2} (2 x+3)^2-1225 (9 x-4) \left (3 x^2+2\right )^{3/2}+54 \sqrt {35} (2 x+3)^4 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )}{30012500 (2 x+3)^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

((-13*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7 - (404*(2 + 3*x^2)^(5/2))/(105*(3 + 2*x)^6) - (822*(2 + 3*x^2)^(5/2))/(87
5*(3 + 2*x)^5) - (2689*(-315*(3 + 2*x)^2*(-4 + 9*x)*Sqrt[2 + 3*x^2] - 1225*(-4 + 9*x)*(2 + 3*x^2)^(3/2) + 54*S
qrt[35]*(3 + 2*x)^4*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/(30012500*(3 + 2*x)^4))/245

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fricas [A]  time = 0.81, size = 164, normalized size = 1.07 \[ \frac {217809 \, \sqrt {35} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (5104296 \, x^{6} + 44301924 \, x^{5} + 148868010 \, x^{4} - 98810025 \, x^{3} + 740031210 \, x^{2} + 256388969 \, x + 471103116\right )} \sqrt {3 \, x^{2} + 2}}{22059187500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="fricas")

[Out]

1/22059187500*(217809*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x +
2187)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(5104296*x^6 + 4
4301924*x^5 + 148868010*x^4 - 98810025*x^3 + 740031210*x^2 + 256388969*x + 471103116)*sqrt(3*x^2 + 2))/(128*x^
7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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giac [B]  time = 0.28, size = 408, normalized size = 2.67 \[ \frac {72603}{3676531250} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, {\left (258144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 5033808 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} + 225898166 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 26360013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 555459995 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 2679767547 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 4252091247 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 6029804778 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 11677158028 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 7324195080 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 2245361152 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 675266496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 174039168 \, \sqrt {3} x - 6049536 \, \sqrt {3} - 174039168 \, \sqrt {3 \, x^{2} + 2}\right )}}{3361400000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="giac")

[Out]

72603/3676531250*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqr
t(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/3361400000*(258144*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 5033808*sqrt
(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 225898166*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 26360013*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2))^10 + 555459995*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 2679767547*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2))^8 - 4252091247*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 6029804778*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 +
11677158028*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 7324195080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 2245361152
*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 675266496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 174039168*sqrt(3)*x - 6
049536*sqrt(3) - 174039168*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)^7

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maple [A]  time = 0.08, size = 245, normalized size = 1.60 \[ \frac {653427 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{7353062500}+\frac {8494551 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{257357187500}-\frac {72603 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{3676531250}-\frac {101 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{411600 \left (x +\frac {3}{2}\right )^{6}}-\frac {411 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{3430000 \left (x +\frac {3}{2}\right )^{5}}-\frac {2689 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{48020000 \left (x +\frac {3}{2}\right )^{4}}-\frac {24201 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{840350000 \left (x +\frac {3}{2}\right )^{3}}-\frac {250077 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{14706125000 \left (x +\frac {3}{2}\right )^{2}}-\frac {2831517 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{257357187500 \left (x +\frac {3}{2}\right )}+\frac {96804 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{64339296875}+\frac {72603 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{3676531250}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{31360 \left (x +\frac {3}{2}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^8,x)

[Out]

-101/411600/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-411/3430000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-2689/4
8020000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-24201/840350000/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-250077
/14706125000/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-2831517/257357187500/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(5/2
)+96804/64339296875*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+653427/7353062500*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+72603/3676
531250*(-36*x+12*(x+3/2)^2-19)^(1/2)-72603/3676531250*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2
)^2-19)^(1/2))+8494551/257357187500*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x-13/31360/(x+3/2)^7*(-9*x+3*(x+3/2)^2-19/4)
^(5/2)

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maxima [B]  time = 1.43, size = 300, normalized size = 1.96 \[ \frac {750231}{14706125000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{245 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {404 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{25725 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {822 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{214375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {2689 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3001250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {24201 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{105043750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {250077 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3676531250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {653427}{7353062500} \, \sqrt {3 \, x^{2} + 2} x + \frac {72603}{3676531250} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {72603}{1838265625} \, \sqrt {3 \, x^{2} + 2} - \frac {2831517 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{14706125000 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="maxima")

[Out]

750231/14706125000*(3*x^2 + 2)^(3/2) - 13/245*(3*x^2 + 2)^(5/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 2
2680*x^3 + 20412*x^2 + 10206*x + 2187) - 404/25725*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729) - 822/214375*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
- 2689/3001250*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 24201/105043750*(3*x^2 + 2)^(5/2)
/(8*x^3 + 36*x^2 + 54*x + 27) - 250077/3676531250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 653427/7353062500*sqr
t(3*x^2 + 2)*x + 72603/3676531250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 72
603/1838265625*sqrt(3*x^2 + 2) - 2831517/14706125000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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mupad [B]  time = 0.14, size = 272, normalized size = 1.78 \[ \frac {72603\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{3676531250}-\frac {72603\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{3676531250}+\frac {92453\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{21952000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {507\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {212679\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3361400000\,\left (x+\frac {3}{2}\right )}+\frac {125\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2688\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {3897\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{192080000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {65\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2048\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {7569\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{54880000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^8,x)

[Out]

(72603*35^(1/2)*log(x + 3/2))/3676531250 - (72603*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/
9))/3676531250 + (92453*3^(1/2)*(x^2 + 2/3)^(1/2))/(21952000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) -
(507*3^(1/2)*(x^2 + 2/3)^(1/2))/(19600*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) -
(212679*3^(1/2)*(x^2 + 2/3)^(1/2))/(3361400000*(x + 3/2)) + (125*3^(1/2)*(x^2 + 2/3)^(1/2))/(2688*((729*x)/16
+ (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (3897*3^(1/2)*(x^2 + 2/3)^(1/2))/(19208
0000*(3*x + x^2 + 9/4)) - (65*3^(1/2)*(x^2 + 2/3)^(1/2))/(2048*((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)) + (7569*3^(1/2)*(x^2 + 2/3)^(1/2))/(54880000*((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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**8,x)

[Out]

Timed out

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