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

Optimal. Leaf size=136 \[ -\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{5/2}}{7350 (2 x+3)^6}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{34300 (2 x+3)^4}-\frac {369 (4-9 x) \sqrt {3 x^2+2}}{1200500 (2 x+3)^2}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{600250 \sqrt {35}} \]

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Rubi [A]  time = 0.07, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, 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 )^{7/2}}{245 (2 x+3)^7}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{5/2}}{7350 (2 x+3)^6}-\frac {41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{34300 (2 x+3)^4}-\frac {369 (4-9 x) \sqrt {3 x^2+2}}{1200500 (2 x+3)^2}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{600250 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1200500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(34300*(3 + 2*x)^4)
- (41*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(7350*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) - (1107*ArcTa
nh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(600250*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 )^{5/2}}{(3+2 x)^8} \, dx &=-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac {41}{35} \int \frac {\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac {41}{245} \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}}{34300 (3+2 x)^4}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac {369 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{17150}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{1200500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac {1107 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{600250}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{1200500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}-\frac {1107 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{600250}\\ &=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{1200500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}-\frac {1107 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{600250 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 122, normalized size = 0.90 \begin {gather*} \frac {1}{490} \left (-\frac {26 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^7}+\frac {41 (9 x-4) \left (3 x^2+2\right )^{5/2}}{15 (2 x+3)^6}+\frac {41 \left (\frac {35 \sqrt {3 x^2+2} \left (1269 x^3+408 x^2+927 x-604\right )}{(2 x+3)^4}-54 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )}{85750}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((41*(-4 + 9*x)*(2 + 3*x^2)^(5/2))/(15*(3 + 2*x)^6) - (26*(2 + 3*x^2)^(7/2))/(3 + 2*x)^7 + (41*((35*Sqrt[2 + 3
*x^2]*(-604 + 927*x + 408*x^2 + 1269*x^3))/(3 + 2*x)^4 - 54*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])]))/85750)/490

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IntegrateAlgebraic [A]  time = 2.09, size = 101, normalized size = 0.74 \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 )}{300125 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-656424 x^6+9455994 x^5+2997810 x^4+15015225 x^3-3488490 x^2+593639 x-4499004\right )}{3601500 (2 x+3)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-4499004 + 593639*x - 3488490*x^2 + 15015225*x^3 + 2997810*x^4 + 9455994*x^5 - 656424*x^6))/
(3601500*(3 + 2*x)^7) + (1107*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(300125*S
qrt[35])

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fricas [A]  time = 0.44, size = 164, normalized size = 1.21 \begin {gather*} \frac {3321 \, \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 (656424 \, x^{6} - 9455994 \, x^{5} - 2997810 \, x^{4} - 15015225 \, x^{3} + 3488490 \, x^{2} - 593639 \, x + 4499004\right )} \sqrt {3 \, x^{2} + 2}}{126052500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\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)^8,x, algorithm="fricas")

[Out]

1/126052500*(3321*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*(656424*x^6 - 945599
4*x^5 - 2997810*x^4 - 15015225*x^3 + 3488490*x^2 - 593639*x + 4499004)*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 = 0.32, size = 408, normalized size = 3.00 \begin {gather*} \frac {1107}{21008750} \, \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 (908247 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 3755004 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} + 52905908 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 114259794 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 422075810 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 16674486 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 1093657086 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 205745364 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 1886581864 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 1023977040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 660654976 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 94952448 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 9114816 \, \sqrt {3} x - 1555968 \, \sqrt {3} + 9114816 \, \sqrt {3 \, x^{2} + 2}\right )}}{38416000 \, {\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}} \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)^8,x, algorithm="giac")

[Out]

1107/21008750*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(3
5) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/38416000*(908247*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 3755004*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 52905908*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 114259794*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2))^10 + 422075810*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 16674486*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^8 - 1093657086*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 205745364*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 1886581
864*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 1023977040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 660654976*(sqrt(3)*
x - sqrt(3*x^2 + 2))^3 - 94952448*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 9114816*sqrt(3)*x - 1555968*sqrt(3
) + 9114816*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 [B]  time = 0.07, size = 278, normalized size = 2.04 \begin {gather*} \frac {4612869 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{128678593750}+\frac {129519 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{1470612500}+\frac {9963 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{42017500}-\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 )}{21008750}-\frac {41 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{235200 \left (x +\frac {3}{2}\right )^{6}}-\frac {1189 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{24010000 \left (x +\frac {3}{2}\right )^{4}}-\frac {123 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1372000 \left (x +\frac {3}{2}\right )^{5}}-\frac {12177 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{420175000 \left (x +\frac {3}{2}\right )^{3}}-\frac {132471 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{7353062500 \left (x +\frac {3}{2}\right )^{2}}-\frac {1537623 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{128678593750 \left (x +\frac {3}{2}\right )}+\frac {17712 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{64339296875}+\frac {1476 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{367653125}+\frac {1107 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{21008750}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{31360 \left (x +\frac {3}{2}\right )^{7}} \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)^8,x)

[Out]

-41/235200/(x+3/2)^6*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-1189/24010000/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-123/1
372000/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-12177/420175000/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-132471/
7353062500/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+4612869/128678593750*(-9*x+3*(x+3/2)^2-19/4)^(5/2)*x-153762
3/128678593750/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+129519/1470612500*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x+9963/42
017500*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-1107/21008750*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)
^2-19)^(1/2))+17712/64339296875*(-9*x+3*(x+3/2)^2-19/4)^(5/2)+1476/367653125*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+110
7/21008750*(-36*x+12*(x+3/2)^2-19)^(1/2)-13/31360/(x+3/2)^7*(-9*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [B]  time = 1.29, size = 323, normalized size = 2.38 \begin {gather*} \frac {397413}{7353062500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{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 {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3675 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {123 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{42875 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {1189 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1500625 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {12177 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{52521875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {132471 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1838265625 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {129519}{1470612500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {1476}{367653125} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {1537623 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{7353062500 \, {\left (2 \, x + 3\right )}} + \frac {9963}{42017500} \, \sqrt {3 \, x^{2} + 2} x + \frac {1107}{21008750} \, \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}{10504375} \, \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)^8,x, algorithm="maxima")

[Out]

397413/7353062500*(3*x^2 + 2)^(5/2) - 13/245*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22
680*x^3 + 20412*x^2 + 10206*x + 2187) - 41/3675*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 48
60*x^2 + 2916*x + 729) - 123/42875*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1
189/1500625*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 12177/52521875*(3*x^2 + 2)^(7/2)/(8*x
^3 + 36*x^2 + 54*x + 27) - 132471/1838265625*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 129519/1470612500*(3*x^2 +
2)^(3/2)*x + 1476/367653125*(3*x^2 + 2)^(3/2) - 1537623/7353062500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 9963/4201750
0*sqrt(3*x^2 + 2)*x + 1107/21008750*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) +
1107/10504375*sqrt(3*x^2 + 2)

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mupad [B]  time = 2.03, size = 272, normalized size = 2.00 \begin {gather*} \frac {1107\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{21008750}-\frac {1107\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{21008750}+\frac {34571\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{62720\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {6213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7168\,\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 {27351\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19208000\,\left (x+\frac {3}{2}\right )}+\frac {9095\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{12288\,\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 {73161\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2195200\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {2275\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8192\,\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 {122553\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{627200\,\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)^8,x)

[Out]

(1107*35^(1/2)*log(x + 3/2))/21008750 - (1107*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/
21008750 + (34571*3^(1/2)*(x^2 + 2/3)^(1/2))/(62720*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (6213*3^(
1/2)*(x^2 + 2/3)^(1/2))/(7168*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (27351*3^
(1/2)*(x^2 + 2/3)^(1/2))/(19208000*(x + 3/2)) + (9095*3^(1/2)*(x^2 + 2/3)^(1/2))/(12288*((729*x)/16 + (1215*x^
2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (73161*3^(1/2)*(x^2 + 2/3)^(1/2))/(2195200*(3*x +
x^2 + 9/4)) - (2275*3^(1/2)*(x^2 + 2/3)^(1/2))/(8192*((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)) - (122553*3^(1/2)*(x^2 + 2/3)^(1/2))/(627200*((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)**8,x)

[Out]

Timed out

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