3.13.54 \(\int \frac {5-x}{(3+2 x)^4 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=153 \[ -\frac {114-3331 x}{7350 (2 x+3)^3 \sqrt {3 x^2+2}}-\frac {5987 \sqrt {3 x^2+2}}{1500625 (2 x+3)}+\frac {541 \sqrt {3 x^2+2}}{42875 (2 x+3)^2}+\frac {1471 \sqrt {3 x^2+2}}{18375 (2 x+3)^3}+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-\frac {55344 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1500625 \sqrt {35}} \]

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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 = {823, 835, 807, 725, 206} \begin {gather*} -\frac {114-3331 x}{7350 (2 x+3)^3 \sqrt {3 x^2+2}}-\frac {5987 \sqrt {3 x^2+2}}{1500625 (2 x+3)}+\frac {541 \sqrt {3 x^2+2}}{42875 (2 x+3)^2}+\frac {1471 \sqrt {3 x^2+2}}{18375 (2 x+3)^3}+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-\frac {55344 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1500625 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*(2 + 3*x^2)^(5/2)),x]

[Out]

(26 + 41*x)/(210*(3 + 2*x)^3*(2 + 3*x^2)^(3/2)) - (114 - 3331*x)/(7350*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (1471*Sq
rt[2 + 3*x^2])/(18375*(3 + 2*x)^3) + (541*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)^2) - (5987*Sqrt[2 + 3*x^2])/(15006
25*(3 + 2*x)) - (55344*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*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 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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {1}{630} \int \frac {-1674-1230 x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {\int \frac {-16416+359748 x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx}{132300}\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}-\frac {\int \frac {-3873744-6672456 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx}{13891500}\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}+\frac {541 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}+\frac {\int \frac {123107040+36809640 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{972405000}\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}+\frac {541 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}-\frac {5987 \sqrt {2+3 x^2}}{1500625 (3+2 x)}+\frac {55344 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1500625}\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}+\frac {541 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}-\frac {5987 \sqrt {2+3 x^2}}{1500625 (3+2 x)}-\frac {55344 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1500625}\\ &=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}+\frac {541 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}-\frac {5987 \sqrt {2+3 x^2}}{1500625 (3+2 x)}-\frac {55344 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1500625 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 85, normalized size = 0.56 \begin {gather*} \frac {-332064 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \left (1293192 x^6+1834596 x^5-4920642 x^4-9795297 x^3-7866162 x^2-9103449 x-3788738\right )}{(2 x+3)^3 \left (3 x^2+2\right )^{3/2}}}{315131250} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*(2 + 3*x^2)^(5/2)),x]

[Out]

((-35*(-3788738 - 9103449*x - 7866162*x^2 - 9795297*x^3 - 4920642*x^4 + 1834596*x^5 + 1293192*x^6))/((3 + 2*x)
^3*(2 + 3*x^2)^(3/2)) - 332064*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/315131250

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IntegrateAlgebraic [A]  time = 1.10, size = 101, normalized size = 0.66 \begin {gather*} \frac {110688 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{1500625 \sqrt {35}}+\frac {-1293192 x^6-1834596 x^5+4920642 x^4+9795297 x^3+7866162 x^2+9103449 x+3788738}{9003750 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)/((3 + 2*x)^4*(2 + 3*x^2)^(5/2)),x]

[Out]

(3788738 + 9103449*x + 7866162*x^2 + 9795297*x^3 + 4920642*x^4 - 1834596*x^5 - 1293192*x^6)/(9003750*(3 + 2*x)
^3*(2 + 3*x^2)^(3/2)) + (110688*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(150062
5*Sqrt[35])

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fricas [A]  time = 0.45, size = 164, normalized size = 1.07 \begin {gather*} \frac {166032 \, \sqrt {35} {\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\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 (1293192 \, x^{6} + 1834596 \, x^{5} - 4920642 \, x^{4} - 9795297 \, x^{3} - 7866162 \, x^{2} - 9103449 \, x - 3788738\right )} \sqrt {3 \, x^{2} + 2}}{315131250 \, {\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/315131250*(166032*sqrt(35)*(72*x^7 + 324*x^6 + 582*x^5 + 675*x^4 + 680*x^3 + 468*x^2 + 216*x + 108)*log(-(sq
rt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(1293192*x^6 + 1834596*x^5 - 4
920642*x^4 - 9795297*x^3 - 7866162*x^2 - 9103449*x - 3788738)*sqrt(3*x^2 + 2))/(72*x^7 + 324*x^6 + 582*x^5 + 6
75*x^4 + 680*x^3 + 468*x^2 + 216*x + 108)

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giac [B]  time = 0.47, size = 257, normalized size = 1.68 \begin {gather*} \frac {55344}{52521875} \, \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 ({\left (49879 \, x + 344464\right )} x - 6729\right )} x + 2510374}{105043750 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {8 \, \sqrt {3} {\left (37652 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 695865 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 729630 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 3472470 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 1016800 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 259424\right )}}{52521875 \, {\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

55344/52521875*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))) + 1/105043750*(9*((49879*x + 344464)*x - 6729)*x + 2510374)/(3*x^2 + 2)^
(3/2) - 8/52521875*sqrt(3)*(37652*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 695865*(sqrt(3)*x - sqrt(3*x^2 + 2
))^4 + 729630*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 3472470*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 1016800*sqrt
(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 259424)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2)) - 2)^3

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maple [A]  time = 0.07, size = 161, normalized size = 1.05 \begin {gather*} -\frac {4071 x}{85750 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {17961 x}{3001250 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {55344 \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 )}{52521875}-\frac {79}{2450 \left (x +\frac {3}{2}\right )^{2} \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {516}{6125 \left (x +\frac {3}{2}\right ) \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {2306}{42875 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {27672}{1500625 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{840 \left (x +\frac {3}{2}\right )^{3} \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^4/(3*x^2+2)^(5/2),x)

[Out]

-79/2450/(x+3/2)^2/(-9*x+3*(x+3/2)^2-19/4)^(3/2)-516/6125/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(3/2)+2306/42875/(-9
*x+3*(x+3/2)^2-19/4)^(3/2)-4071/85750/(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x-17961/3001250/(-9*x+3*(x+3/2)^2-19/4)^(1
/2)*x+27672/1500625/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-55344/52521875*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*
x+12*(x+3/2)^2-19)^(1/2))-13/840/(x+3/2)^3/(-9*x+3*(x+3/2)^2-19/4)^(3/2)

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maxima [A]  time = 1.42, size = 207, normalized size = 1.35 \begin {gather*} \frac {55344}{52521875} \, \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 {17961 \, x}{3001250 \, \sqrt {3 \, x^{2} + 2}} + \frac {27672}{1500625 \, \sqrt {3 \, x^{2} + 2}} - \frac {4071 \, x}{85750 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{105 \, {\left (8 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + 36 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {158}{1225 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1032}{6125 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {2306}{42875 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

55344/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 17961/3001250*x/sqrt(
3*x^2 + 2) + 27672/1500625/sqrt(3*x^2 + 2) - 4071/85750*x/(3*x^2 + 2)^(3/2) - 13/105/(8*(3*x^2 + 2)^(3/2)*x^3
+ 36*(3*x^2 + 2)^(3/2)*x^2 + 54*(3*x^2 + 2)^(3/2)*x + 27*(3*x^2 + 2)^(3/2)) - 158/1225/(4*(3*x^2 + 2)^(3/2)*x^
2 + 12*(3*x^2 + 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 1032/6125/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) +
2306/42875/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 1.86, size = 330, normalized size = 2.16 \begin {gather*} \frac {55344\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{52521875}-\frac {55344\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{52521875}-\frac {6337\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{36015000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {49879\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {49879\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {6337\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{36015000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {129712\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{52521875\,\left (x+\frac {3}{2}\right )}-\frac {1256\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128625\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,3427{}\mathrm {i}}{72030000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,2288579{}\mathrm {i}}{2521050000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,2288579{}\mathrm {i}}{2521050000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,3427{}\mathrm {i}}{72030000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^4*(3*x^2 + 2)^(5/2)),x)

[Out]

(55344*35^(1/2)*log(x + 3/2))/52521875 - (55344*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)
)/52521875 - (6337*3^(1/2)*(x^2 + 2/3)^(1/2))/(36015000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (49879*3^(1/2)*(x^2
+ 2/3)^(1/2))/(210087500*(x - (6^(1/2)*1i)/3)) + (49879*3^(1/2)*(x^2 + 2/3)^(1/2))/(210087500*(x + (6^(1/2)*1i
)/3)) + (6337*3^(1/2)*(x^2 + 2/3)^(1/2))/(36015000*((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - (129712*3^(1/2)*(x^2 + 2/
3)^(1/2))/(52521875*(x + 3/2)) - (1256*3^(1/2)*(x^2 + 2/3)^(1/2))/(1500625*(3*x + x^2 + 9/4)) - (26*3^(1/2)*(x
^2 + 2/3)^(1/2))/(128625*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*3427i)/(720
30000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*2288579i)/(2521050000*(x - (6^(1/2)
*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*2288579i)/(2521050000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(
x^2 + 2/3)^(1/2)*3427i)/(72030000*((6^(1/2)*x*2i)/3 - x^2 + 2/3))

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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+2*x)**4/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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