3.20.95 \(\int \frac {(1+2 x)^{3/2}}{(2+3 x+5 x^2)^3} \, dx\)

Optimal. Leaf size=300 \[ -\frac {\sqrt {2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

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Rubi [A]  time = 0.42, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {738, 822, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {\sqrt {2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

-((5 - 4*x)*Sqrt[1 + 2*x])/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(67 + 120*x))/(1922*(2 + 3*x + 5*x^2)) -
(3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/961 + (3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(
-2 + Sqrt[35])]])/961 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 +
2*x] + 5*(1 + 2*x)])/1922 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/1922

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 738

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

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 1169

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

Rubi steps

\begin {align*} \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {17+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {1239+840 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1638+840 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1638-168 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1638-168 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{67270}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{67270}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1922}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1922}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{33635}-\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{33635}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}\\ \end {align*}

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Mathematica [C]  time = 0.50, size = 198, normalized size = 0.66 \begin {gather*} \frac {\frac {(2960 x+4391) (2 x+1)^{5/2}}{5 x^2+3 x+2}+\frac {217 (20 x+37) (2 x+1)^{5/2}}{\left (5 x^2+3 x+2\right )^2}-1184 (2 x+1)^{3/2}-3276 \sqrt {2 x+1}+\frac {42 \left (\sqrt {2-i \sqrt {31}} \left (1209-218 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {2+i \sqrt {31}} \left (1209+218 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{31 \sqrt {5}}}{94178} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

(-3276*Sqrt[1 + 2*x] - 1184*(1 + 2*x)^(3/2) + (217*(1 + 2*x)^(5/2)*(37 + 20*x))/(2 + 3*x + 5*x^2)^2 + ((1 + 2*
x)^(5/2)*(4391 + 2960*x))/(2 + 3*x + 5*x^2) + (42*(Sqrt[2 - I*Sqrt[31]]*(1209 - (218*I)*Sqrt[31])*ArcTanh[Sqrt
[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(1209 + (218*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[
2 + I*Sqrt[31]]]))/(31*Sqrt[5]))/94178

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IntegrateAlgebraic [C]  time = 2.02, size = 167, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {2 x+1} \left (300 (2 x+1)^3-205 (2 x+1)^2+640 (2 x+1)-819\right )}{961 \left (5 (2 x+1)^2-4 (2 x+1)+7\right )^2}+\frac {3}{961} \sqrt {\frac {1}{217} \left (15082+961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )+\frac {3}{961} \sqrt {\frac {1}{217} \left (15082-961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

(2*Sqrt[1 + 2*x]*(-819 + 640*(1 + 2*x) - 205*(1 + 2*x)^2 + 300*(1 + 2*x)^3))/(961*(7 - 4*(1 + 2*x) + 5*(1 + 2*
x)^2)^2) + (3*Sqrt[(15082 + (961*I)*Sqrt[31])/217]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/961 + (3
*Sqrt[(15082 - (961*I)*Sqrt[31])/217]*ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/961

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fricas [B]  time = 0.46, size = 625, normalized size = 2.08 \begin {gather*} -\frac {357492 \cdot 256095875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{971794421886819908125} \cdot 256095875^{\frac {3}{4}} \sqrt {3787} \sqrt {217} \sqrt {256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 5640925850 \, x + 564092585 \, \sqrt {35} + 2820462925} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} - \frac {1}{53405465484875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (39 \, \sqrt {35} - 140\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 357492 \cdot 256095875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{14576916328302298621875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {-852075 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 4806491893638750 \, x + 480649189363875 \, \sqrt {35} + 2403245946819375} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} - \frac {1}{53405465484875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (39 \, \sqrt {35} - 140\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 3 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (\frac {852075}{31} \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 155048125601250 \, x + 15504812560125 \, \sqrt {35} + 77524062800625\right ) + 3 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (-\frac {852075}{31} \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 155048125601250 \, x + 15504812560125 \, \sqrt {35} + 77524062800625\right ) - 1224080909450 \, {\left (600 \, x^{3} + 695 \, x^{2} + 565 \, x - 21\right )} \sqrt {2 \, x + 1}}{2352683507962900 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")

[Out]

-1/2352683507962900*(357492*256095875^(1/4)*sqrt(217)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(8159
3620*sqrt(35) + 512191750)*arctan(1/971794421886819908125*256095875^(3/4)*sqrt(3787)*sqrt(217)*sqrt(256095875^
(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 564092
5850*x + 564092585*sqrt(35) + 2820462925)*(39*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(81593620*sqrt(35) + 51219
1750) - 1/53405465484875*256095875^(3/4)*sqrt(217)*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750)*(39*sqrt(
35) - 140) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 357492*256095875^(1/4)*sqrt(217)*sqrt(35)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*sqrt(81593620*sqrt(35) + 512191750)*arctan(1/14576916328302298621875*256095875^(3/4)*
sqrt(217)*sqrt(-852075*256095875^(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(815936
20*sqrt(35) + 512191750) + 4806491893638750*x + 480649189363875*sqrt(35) + 2403245946819375)*(39*sqrt(35)*sqrt
(31) - 140*sqrt(31))*sqrt(81593620*sqrt(35) + 512191750) - 1/53405465484875*256095875^(3/4)*sqrt(217)*sqrt(2*x
+ 1)*sqrt(81593620*sqrt(35) + 512191750)*(39*sqrt(35) - 140) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 3*25
6095875^(1/4)*sqrt(217)*(15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 94675*sqrt(31)*(25*x^
4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(81593620*sqrt(35) + 512191750)*log(852075/31*256095875^(1/4)*sqrt(217)*(
4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 155048125601250*x + 155
04812560125*sqrt(35) + 77524062800625) + 3*256095875^(1/4)*sqrt(217)*(15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3
+ 29*x^2 + 12*x + 4) - 94675*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(81593620*sqrt(35) + 5121917
50)*log(-852075/31*256095875^(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*s
qrt(35) + 512191750) + 155048125601250*x + 15504812560125*sqrt(35) + 77524062800625) - 1224080909450*(600*x^3
+ 695*x^2 + 565*x - 21)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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giac [B]  time = 1.43, size = 640, normalized size = 2.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")

[Out]

3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^
(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt
(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 19110*(7/5)^(1/4)*sqrt
(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/sqrt(-1/
35*sqrt(35) + 1/2)) + 3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2
450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/
5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
19110*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) -
sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)
^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) +
35)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(1
40*sqrt(35) + 2450) - 19110*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*
sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/
2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*
sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(140*s
qrt(35) + 2450) - 19110*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqr
t(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 2/961*(300*(2*x + 1)^(7/2) - 205*(2*x + 1)^(5/2) + 640*(2*x + 1)^(3/2) -
819*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

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maple [B]  time = 0.31, size = 662, normalized size = 2.21 \begin {gather*} -\frac {705 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {705 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {141 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{119164}+\frac {327 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{417074}+\frac {141 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{119164}-\frac {327 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{417074}+\frac {\frac {600 \left (2 x +1\right )^{\frac {7}{2}}}{961}-\frac {410 \left (2 x +1\right )^{\frac {5}{2}}}{961}+\frac {1280 \left (2 x +1\right )^{\frac {3}{2}}}{961}-\frac {1638 \sqrt {2 x +1}}{961}}{\left (-8 x +5 \left (2 x +1\right )^{2}+3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)^(3/2)/(5*x^2+3*x+2)^3,x)

[Out]

1600*(3/7688*(2*x+1)^(7/2)-41/153760*(2*x+1)^(5/2)+4/4805*(2*x+1)^(3/2)-819/768800*(2*x+1)^(1/2))/(-8*x+5*(2*x
+1)^2+3)^2+141/119164*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*
5^(1/2)*(2*x+1)^(1/2)+5)-327/417074*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(
1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)-705/59582/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((5
^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+327/208537/(10*5^(1/2)*7^(
1/2)-20)^(1/2)*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1
/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+234/6727/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*7^(1/2)*arctan((5^(1/2)*(2*
5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-141/119164*5^(1/2)*(2*5^(1/2)*7^(1/2
)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)+327/417074*7^(1/2)*(2*
5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)-705/5958
2/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^
(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+327/208537/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7
^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+234/6727/
(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*7^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(
10*5^(1/2)*7^(1/2)-20)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3, x)

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mupad [B]  time = 1.04, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1638\,\sqrt {2\,x+1}}{24025}-\frac {256\,{\left (2\,x+1\right )}^{3/2}}{4805}+\frac {82\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {24\,{\left (2\,x+1\right )}^{7/2}}{961}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}+\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}-\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(3/2)/(3*x + 5*x^2 + 2)^3,x)

[Out]

((1638*(2*x + 1)^(1/2))/24025 - (256*(2*x + 1)^(3/2))/4805 + (82*(2*x + 1)^(5/2))/4805 - (24*(2*x + 1)^(7/2))/
961)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) + (217^(1/2)*atan((217^(1/2)*
(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 + 94176/8080808
75)) + (864*31^(1/2)*217^(1/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848
i)/808080875 + 94176/808080875)))*(- 31^(1/2)*961i - 15082)^(1/2)*3i)/208537 - (217^(1/2)*atan((217^(1/2)*(31^
(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 - 94176/808080875)) -
(864*31^(1/2)*217^(1/2)*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848i)/80808
0875 - 94176/808080875)))*(31^(1/2)*961i - 15082)^(1/2)*3i)/208537

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sympy [B]  time = 167.23, size = 527, normalized size = 1.76 \begin {gather*} \frac {1145600 \left (2 x + 1\right )^{\frac {7}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {8870400 \left (2 x + 1\right )^{\frac {7}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} - \frac {1295360 \left (2 x + 1\right )^{\frac {5}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {4701760 \left (2 x + 1\right )^{\frac {5}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {3017984 \left (2 x + 1\right )^{\frac {3}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {6868736 \left (2 x + 1\right )^{\frac {3}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {640 \left (2 x + 1\right )^{\frac {3}{2}}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} - \frac {974848 \sqrt {2 x + 1}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {27016640 \sqrt {2 x + 1}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {1728 \sqrt {2 x + 1}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} + 64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left (t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {448 \operatorname {RootSum} {\left (3697830642882758349886062592 t^{4} + 2111968303753265086464 t^{2} + 705698730253125, \left (t \mapsto t \log {\left (- \frac {3459438283411209322496 t^{3}}{1377792122625} + \frac {251494140770688 t}{357205365125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left (t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {64 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left (t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)

[Out]

1145600*(2*x + 1)**(7/2)/(-120547840*x + 26908000*(2*x + 1)**4 - 43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2
- 7534240) - 8870400*(2*x + 1)**(7/2)/(-843834880*x + 188356000*(2*x + 1)**4 - 301369600*(2*x + 1)**3 + 64794
4640*(2*x + 1)**2 - 52739680) - 1295360*(2*x + 1)**(5/2)/(-120547840*x + 26908000*(2*x + 1)**4 - 43052800*(2*x
+ 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 4701760*(2*x + 1)**(5/2)/(-843834880*x + 188356000*(2*x + 1)**4
- 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 3017984*(2*x + 1)**(3/2)/(-120547840*x + 26908
000*(2*x + 1)**4 - 43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 6868736*(2*x + 1)**(3/2)/(-84383
4880*x + 188356000*(2*x + 1)**4 - 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 640*(2*x + 1)*
*(3/2)/(-34720*x + 21700*(2*x + 1)**2 + 13020) - 974848*sqrt(2*x + 1)/(-120547840*x + 26908000*(2*x + 1)**4 -
43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 27016640*sqrt(2*x + 1)/(-843834880*x + 188356000*(2
*x + 1)**4 - 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 1728*sqrt(2*x + 1)/(-34720*x + 2170
0*(2*x + 1)**2 + 13020) + 64*RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 3331425781
25, Lambda(_t, _t*log(21632117045402271744*_t**3/158378125 + 10865340674816*_t/1108646875 + sqrt(2*x + 1)))) -
448*RootSum(3697830642882758349886062592*_t**4 + 2111968303753265086464*_t**2 + 705698730253125, Lambda(_t, _
t*log(-3459438283411209322496*_t**3/1377792122625 + 251494140770688*_t/357205365125 + sqrt(2*x + 1))))/5 - 64*
RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 333142578125, Lambda(_t, _t*log(2163211
7045402271744*_t**3/158378125 + 10865340674816*_t/1108646875 + sqrt(2*x + 1))))/5 + 64*RootSum(199500603446394
88*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/22051
25 + sqrt(2*x + 1))))/5

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