3.24.33 \(\int \frac {5-x}{(3+2 x)^{7/2} (2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=141 \[ -\frac {3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac {9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24409}{3125 \sqrt {2 x+3}}+\frac {102697}{1875 (2 x+3)^{3/2}}+\frac {56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \]

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Rubi [A]  time = 0.12, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \begin {gather*} -\frac {3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac {9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24409}{3125 \sqrt {2 x+3}}+\frac {102697}{1875 (2 x+3)^{3/2}}+\frac {56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3),x]

[Out]

56399/(625*(3 + 2*x)^(5/2)) + 102697/(1875*(3 + 2*x)^(3/2)) - 24409/(3125*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10
*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^2) + (8852 + 9957*x)/(50*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 266*ArcTanh[S
qrt[3 + 2*x]] - (806841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 822

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1772+1551 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {76349+69699 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {56399}{625 (3+2 x)^{5/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {202447+169197 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {474341+308091 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx}{1250}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {758023-73227 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{6250}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1735727-73227 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )}{3125}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {2420523 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )}{3125}-798 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+266 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 121, normalized size = 0.86 \begin {gather*} \frac {-\frac {28125 (47 x+37)}{\left (3 x^2+5 x+2\right )^2}+\frac {1875 (9957 x+8852)}{3 x^2+5 x+2}+2 (2 x+3) \left (21 (2 x+3) \left (593750 \sqrt {2 x+3} \tanh ^{-1}\left (\sqrt {2 x+3}\right )-115263 \sqrt {30 x+45} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-17435\right )+2567425\right )+8459850}{93750 (2 x+3)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3),x]

[Out]

(8459850 - (28125*(37 + 47*x))/(2 + 5*x + 3*x^2)^2 + (1875*(8852 + 9957*x))/(2 + 5*x + 3*x^2) + 2*(3 + 2*x)*(2
567425 + 21*(3 + 2*x)*(-17435 + 593750*Sqrt[3 + 2*x]*ArcTanh[Sqrt[3 + 2*x]] - 115263*Sqrt[45 + 30*x]*ArcTanh[S
qrt[3/5]*Sqrt[3 + 2*x]])))/(93750*(3 + 2*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.30, size = 129, normalized size = 0.91 \begin {gather*} \frac {-659043 (2 x+3)^6+8136261 (2 x+3)^5-23916753 (2 x+3)^4+24720095 (2 x+3)^3-6945760 (2 x+3)^2-728800 (2 x+3)-156000}{9375 (2 x+3)^{5/2} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3),x]

[Out]

(-156000 - 728800*(3 + 2*x) - 6945760*(3 + 2*x)^2 + 24720095*(3 + 2*x)^3 - 23916753*(3 + 2*x)^4 + 8136261*(3 +
2*x)^5 - 659043*(3 + 2*x)^6)/(9375*(3 + 2*x)^(5/2)*(5 - 8*(3 + 2*x) + 3*(3 + 2*x)^2)^2) + 266*ArcTanh[Sqrt[3
+ 2*x]] - (806841*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3125

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fricas [B]  time = 0.40, size = 245, normalized size = 1.74 \begin {gather*} \frac {2420523 \, \sqrt {5} \sqrt {3} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (5272344 \, x^{6} + 14906052 \, x^{5} - 18312714 \, x^{4} - 114099329 \, x^{3} - 160041829 \, x^{2} - 94082723 \, x - 20250051\right )} \sqrt {2 \, x + 3}}{93750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

1/93750*(2420523*sqrt(5)*sqrt(3)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)*
log(-(sqrt(5)*sqrt(3)*sqrt(2*x + 3) - 3*x - 7)/(3*x + 2)) + 12468750*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 +
3560*x^3 + 2223*x^2 + 756*x + 108)*log(sqrt(2*x + 3) + 1) - 12468750*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4
+ 3560*x^3 + 2223*x^2 + 756*x + 108)*log(sqrt(2*x + 3) - 1) - 5*(5272344*x^6 + 14906052*x^5 - 18312714*x^4 - 1
14099329*x^3 - 160041829*x^2 - 94082723*x - 20250051)*sqrt(2*x + 3))/(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 +
3560*x^3 + 2223*x^2 + 756*x + 108)

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giac [A]  time = 0.19, size = 143, normalized size = 1.01 \begin {gather*} \frac {806841}{31250} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {202995 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 745077 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 831169 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 259087 \, \sqrt {2 \, x + 3}}{625 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} - \frac {32 \, {\left (12861 \, {\left (2 \, x + 3\right )}^{2} + 3070 \, x + 4800\right )}}{9375 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

806841/31250*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 1/625*(202995
*(2*x + 3)^(7/2) - 745077*(2*x + 3)^(5/2) + 831169*(2*x + 3)^(3/2) - 259087*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16
*x - 19)^2 - 32/9375*(12861*(2*x + 3)^2 + 3070*x + 4800)/(2*x + 3)^(5/2) + 133*log(sqrt(2*x + 3) + 1) - 133*lo
g(abs(sqrt(2*x + 3) - 1))

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maple [A]  time = 0.03, size = 151, normalized size = 1.07 \begin {gather*} -\frac {806841 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15625}-133 \ln \left (-1+\sqrt {2 x +3}\right )+133 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {22599 \left (2 x +3\right )^{\frac {3}{2}}}{125}-\frac {196587 \sqrt {2 x +3}}{625}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {8}{\sqrt {2 x +3}+1}-\frac {416}{625 \left (2 x +3\right )^{\frac {5}{2}}}-\frac {9824}{1875 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {137184}{3125 \sqrt {2 x +3}}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {8}{-1+\sqrt {2 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^(7/2)/(3*x^2+5*x+2)^3,x)

[Out]

13122/3125*(775/18*(2*x+3)^(3/2)-4045/54*(2*x+3)^(1/2))/(6*x+4)^2-806841/15625*arctanh(1/5*15^(1/2)*(2*x+3)^(1
/2))*15^(1/2)-3/((2*x+3)^(1/2)+1)^2+8/((2*x+3)^(1/2)+1)+133*ln((2*x+3)^(1/2)+1)-416/625/(2*x+3)^(5/2)-9824/187
5/(2*x+3)^(3/2)-137184/3125/(2*x+3)^(1/2)+3/(-1+(2*x+3)^(1/2))^2+8/(-1+(2*x+3)^(1/2))-133*ln(-1+(2*x+3)^(1/2))

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maxima [A]  time = 0.98, size = 161, normalized size = 1.14 \begin {gather*} \frac {806841}{31250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {659043 \, {\left (2 \, x + 3\right )}^{6} - 8136261 \, {\left (2 \, x + 3\right )}^{5} + 23916753 \, {\left (2 \, x + 3\right )}^{4} - 24720095 \, {\left (2 \, x + 3\right )}^{3} + 6945760 \, {\left (2 \, x + 3\right )}^{2} + 1457600 \, x + 2342400}{9375 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 25 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}\right )}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

806841/31250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 1/9375*(659043*(2*x +
3)^6 - 8136261*(2*x + 3)^5 + 23916753*(2*x + 3)^4 - 24720095*(2*x + 3)^3 + 6945760*(2*x + 3)^2 + 1457600*x + 2
342400)/(9*(2*x + 3)^(13/2) - 48*(2*x + 3)^(11/2) + 94*(2*x + 3)^(9/2) - 80*(2*x + 3)^(7/2) + 25*(2*x + 3)^(5/
2)) + 133*log(sqrt(2*x + 3) + 1) - 133*log(sqrt(2*x + 3) - 1)

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mupad [B]  time = 0.08, size = 127, normalized size = 0.90 \begin {gather*} 266\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {806841\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15625}-\frac {\frac {58304\,x}{3375}+\frac {1389152\,{\left (2\,x+3\right )}^2}{16875}-\frac {4944019\,{\left (2\,x+3\right )}^3}{16875}+\frac {2657417\,{\left (2\,x+3\right )}^4}{9375}-\frac {301343\,{\left (2\,x+3\right )}^5}{3125}+\frac {24409\,{\left (2\,x+3\right )}^6}{3125}+\frac {31232}{1125}}{\frac {25\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {80\,{\left (2\,x+3\right )}^{7/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{9/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{11/2}}{3}+{\left (2\,x+3\right )}^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^(7/2)*(5*x + 3*x^2 + 2)^3),x)

[Out]

266*atanh((2*x + 3)^(1/2)) - (806841*15^(1/2)*atanh((15^(1/2)*(2*x + 3)^(1/2))/5))/15625 - ((58304*x)/3375 + (
1389152*(2*x + 3)^2)/16875 - (4944019*(2*x + 3)^3)/16875 + (2657417*(2*x + 3)^4)/9375 - (301343*(2*x + 3)^5)/3
125 + (24409*(2*x + 3)^6)/3125 + 31232/1125)/((25*(2*x + 3)^(5/2))/9 - (80*(2*x + 3)^(7/2))/9 + (94*(2*x + 3)^
(9/2))/9 - (16*(2*x + 3)^(11/2))/3 + (2*x + 3)^(13/2))

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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)**(7/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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