### 3.2318 $$\int \frac{(1+2 x)^{3/2}}{(2+3 x+5 x^2)^2} \, dx$$

Optimal. Leaf size=270 $-\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

[Out]

-((5 - 4*x)*Sqrt[1 + 2*x])/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[1
0*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 - (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 + (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt[35]
+ Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31

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Rubi [A]  time = 0.363569, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.318, Rules used = {738, 826, 1169, 634, 618, 204, 628} $-\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((5 - 4*x)*Sqrt[1 + 2*x])/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[1
0*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 - (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 + (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt[35]
+ Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31

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 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]

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{9+4 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{2}{31} \operatorname{Subst}\left (\int \frac{14+4 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (14-4 \sqrt{\frac{7}{5}}\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 )}{31 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (14-4 \sqrt{\frac{7}{5}}\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 )}{31 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{155} \left (2+\sqrt{35}\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 )+\frac{1}{155} \left (2+\sqrt{35}\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 )-\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\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 )+\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\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 )\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\right )} \log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )-\frac{1}{155} \left (2 \left (2+\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 )-\frac{1}{155} \left (2 \left (2+\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 )\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \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{1}{31} \sqrt{\frac{2}{155} \left (218+47 \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{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\right )} \log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )\\ \end{align*}

Mathematica [C]  time = 0.718012, size = 141, normalized size = 0.52 $\frac{\frac{155 \sqrt{2 x+1} (4 x-5)}{5 x^2+3 x+2}+2 \left (31-4 i \sqrt{31}\right ) \sqrt{10-5 i \sqrt{31}} \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )+2 \left (31+4 i \sqrt{31}\right ) \sqrt{10+5 i \sqrt{31}} \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )}{4805}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((155*Sqrt[1 + 2*x]*(-5 + 4*x))/(2 + 3*x + 5*x^2) + 2*(31 - (4*I)*Sqrt[31])*Sqrt[10 - (5*I)*Sqrt[31]]*ArcTanh[
Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*(31 + (4*I)*Sqrt[31])*Sqrt[10 + (5*I)*Sqrt[31]]*ArcTanh[Sqrt[5 + 10*x
]/Sqrt[2 + I*Sqrt[31]]])/4805

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Maple [B]  time = 0.082, size = 642, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*(1/310*(1+2*x)^(3/2)-7/620*(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)+39/9610*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2
)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-2/961*ln(5^(1/2)*7^(1/2)+10*x+5+
(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-39/961/(10*5^(1/2)*7^(1
/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*
5^(1/2)*7^(1/2)+4)+4/961/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^
(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)+4/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)
*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)-
39/9610*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*5^(1/2)*(2*5^(1/2)*7^(1/
2)+4)^(1/2)+2/961*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*7^(1/2)*(2*5^(
1/2)*7^(1/2)+4)^(1/2)-39/961/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+
2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+4/961/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-
5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*(2*5^(1/2)*7^(1/2
)+4)*7^(1/2)+4/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))
/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.19425, size = 2195, normalized size = 8.13 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15191920450*(3844*77315^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(20492*sqrt(35) + 154630)*arctan(1/3
788913966425*77315^(3/4)*sqrt(155)*sqrt(47)*sqrt(77315^(1/4)*sqrt(155)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqr
t(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 15808450*x + 1580845*sqrt(35) + 7904225)*(sqrt(35)*sqrt(7) - 2*sqrt
(7))*sqrt(20492*sqrt(35) + 154630) - 1/74299715*77315^(3/4)*sqrt(155)*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 1546
30)*(sqrt(35) - 2) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 3844*77315^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*
x + 2)*sqrt(20492*sqrt(35) + 154630)*arctan(1/7577827932850*77315^(3/4)*sqrt(155)*sqrt(-188*77315^(1/4)*sqrt(1
55)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 2971988600*x + 297198860
*sqrt(35) + 1485994300)*(sqrt(35)*sqrt(7) - 2*sqrt(7))*sqrt(20492*sqrt(35) + 154630) - 1/74299715*77315^(3/4)*
sqrt(155)*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630)*(sqrt(35) - 2) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31))
- 77315^(1/4)*sqrt(155)*(218*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) - 1645*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(2049
2*sqrt(35) + 154630)*log(188/7*77315^(1/4)*sqrt(155)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(2*x + 1)*sqrt(20
492*sqrt(35) + 154630) + 424569800*x + 42456980*sqrt(35) + 212284900) + 77315^(1/4)*sqrt(155)*(218*sqrt(35)*sq
rt(31)*(5*x^2 + 3*x + 2) - 1645*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(20492*sqrt(35) + 154630)*log(-188/7*77315^(1/
4)*sqrt(155)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 424569800*x + 4
2456980*sqrt(35) + 212284900) - 490061950*(4*x - 5)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

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Sympy [A]  time = 58.9013, size = 211, normalized size = 0.78 \begin{align*} \frac{64 \left (2 x + 1\right )^{\frac{3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac{224 \left (2 x + 1\right )^{\frac{3}{2}}}{- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604} - \frac{128 \sqrt{2 x + 1}}{5 \left (- 992 x + 620 \left (2 x + 1\right )^{2} + 372\right )} - \frac{3024 \sqrt{2 x + 1}}{5 \left (- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604\right )} + \frac{64 \operatorname{RootSum}{\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log{\left (\frac{33312534528 t^{3}}{235} + \frac{166784 t}{235} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} - \frac{112 \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} + \frac{16 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

64*(2*x + 1)**(3/2)/(-992*x + 620*(2*x + 1)**2 + 372) - 224*(2*x + 1)**(3/2)/(-6944*x + 4340*(2*x + 1)**2 + 26
04) - 128*sqrt(2*x + 1)/(5*(-992*x + 620*(2*x + 1)**2 + 372)) - 3024*sqrt(2*x + 1)/(5*(-6944*x + 4340*(2*x + 1
)**2 + 2604)) + 64*RootSum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t*
*3/235 + 166784*_t/235 + sqrt(2*x + 1))))/5 - 112*RootSum(19950060344639488*_t**4 + 498437272576*_t**2 + 10878
125, Lambda(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/5 + 16*RootSum(
1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/5

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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