### 3.2320 $$\int \frac{1}{\sqrt{1+2 x} (2+3 x+5 x^2)^2} \, dx$$

Optimal. Leaf size=270 $\frac{\sqrt{2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(217*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*ArcTan[(Sqrt[10*
(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*A
rcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 - (Sqrt[(-32678 + 10325*Sqrt
[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217 + (Sqrt[(-32678 + 10325*Sq
rt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217

________________________________________________________________________________________

Rubi [A]  time = 0.323751, 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 = {740, 826, 1169, 634, 618, 204, 628} $\frac{\sqrt{2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(217*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*ArcTan[(Sqrt[10*
(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*A
rcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 - (Sqrt[(-32678 + 10325*Sqrt
[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217 + (Sqrt[(-32678 + 10325*Sq
rt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && 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}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac{1}{217} \int \frac{107+20 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac{2}{217} \operatorname{Subst}\left (\int \frac{194+20 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{194 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (194-4 \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 )}{217 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{194 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (194-4 \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 )}{217 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=\frac{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac{\left (70+97 \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 )}{7595}+\frac{\left (70+97 \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 )}{7595}-\frac{1}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \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}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \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{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac{1}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \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}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )-\frac{\left (2 \left (70+97 \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 )}{7595}-\frac{\left (2 \left (70+97 \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 )}{7595}\\ &=\frac{\sqrt{1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \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}{217} \sqrt{\frac{2}{217} \left (32678+10325 \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}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \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}{217} \sqrt{\frac{1}{434} \left (-32678+10325 \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 = 0.308927, size = 166, normalized size = 0.61 $\frac{1}{217} \left (\frac{\sqrt{2 x+1} (20 x+37)}{5 x^2+3 x+2}+\frac{2 \sqrt{10-5 i \sqrt{31}} \left (101 \sqrt{31}-62 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )}{31 \left (\sqrt{31}+2 i\right )}+\frac{2 \sqrt{10+5 i \sqrt{31}} \left (101 \sqrt{31}+62 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )}{31 \left (\sqrt{31}-2 i\right )}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((Sqrt[1 + 2*x]*(37 + 20*x))/(2 + 3*x + 5*x^2) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(-62*I + 101*Sqrt[31])*ArcTanh[S
qrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]])/(31*(2*I + Sqrt[31])) + (2*Sqrt[10 + (5*I)*Sqrt[31]]*(62*I + 101*Sqrt[31]
)*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/(31*(-2*I + Sqrt[31])))/217

________________________________________________________________________________________

Maple [B]  time = 0.223, size = 968, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/47089*(2/37375*(-3244150*5^(1/2)*7^(1/2)+6488300)/(2*5^(1/2)-5*7^(1/2))*5^(1/2)*(1+2*x)^(1/2)+1/7475/(2*5^(1
/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1946490*5^(1/2)*7^(1/2)+13949845))/(1/5*5^(1/2)*7^(1/2)+2*x+1+1/5
*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))-4063/94178/(2*5^(1/2)-5*7^(1/2))*ln(5+10*x+35^(1/2)+(1+2*x
)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)*35^(1/2)+1165/6727/(2*5^(1/2)-5*7^(1/2))*ln(5+10*x+35^(1/
2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)+4063/47089/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2
))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*(2*3
5^(1/2)+4)^(1/2)*35^(1/2)-2330/6727/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20
+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*(2*35^(1/2)+4)^(1/2)-1940/217/(2*5^(1/2)-
5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*5
^(1/2)+3880/1519/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2)
)/(-20+10*35^(1/2))^(1/2))*7^(1/2)+5/47089*(2/37375*(-3244150*5^(1/2)*7^(1/2)+6488300)/(2*5^(1/2)-5*7^(1/2))*5
^(1/2)*(1+2*x)^(1/2)-1/7475/(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1946490*5^(1/2)*7^(1/2)+139498
45))/(1/5*5^(1/2)*7^(1/2)+2*x+1-1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))+4063/94178/(2*5^(1/2)-5
*7^(1/2))*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)*35^(1/2)-1165/6727/(2*
5^(1/2)-5*7^(1/2))*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)+4063/47089/(2
*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*35^(1/2)
)^(1/2))*(20+10*35^(1/2))^(1/2)*(2*35^(1/2)+4)^(1/2)*35^(1/2)-2330/6727/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2)
)^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*(2*3
5^(1/2)+4)^(1/2)-1940/217/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(1+
2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))*5^(1/2)+3880/1519/(2*5^(1/2)-5*7^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((-
(20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))*7^(1/2)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 3.06592, size = 2591, normalized size = 9.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/391166933403610*(1149356*5969915^(1/4)*sqrt(826)*sqrt(217)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(32678*sqrt(35) +
361375)*arctan(1/229833696387700298075*5969915^(3/4)*sqrt(826)*sqrt(299)*sqrt(295)*sqrt(217)*sqrt(5969915^(1/
4)*sqrt(826)*sqrt(217)*(2*sqrt(35)*sqrt(31) - 97*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375) + 83069
70490*x + 830697049*sqrt(35) + 4153485245)*sqrt(32678*sqrt(35) + 361375)*(97*sqrt(35) - 70) - 1/12007725843295
*5969915^(3/4)*sqrt(826)*sqrt(217)*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375)*(97*sqrt(35) - 70) + 1/31*sqrt(
35)*sqrt(31) + 2/31*sqrt(31)) + 1149356*5969915^(1/4)*sqrt(826)*sqrt(217)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(3267
8*sqrt(35) + 361375)*arctan(1/80441793735695104326250*5969915^(3/4)*sqrt(826)*sqrt(299)*sqrt(217)*sqrt(-361375
00*5969915^(1/4)*sqrt(826)*sqrt(217)*(2*sqrt(35)*sqrt(31) - 97*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 3
61375) + 300193146082375000*x + 30019314608237500*sqrt(35) + 150096573041187500)*sqrt(32678*sqrt(35) + 361375)
*(97*sqrt(35) - 70) - 1/12007725843295*5969915^(3/4)*sqrt(826)*sqrt(217)*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 3
61375)*(97*sqrt(35) - 70) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 5969915^(1/4)*sqrt(826)*sqrt(217)*(32678
*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) - 361375*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(32678*sqrt(35) + 361375)*log(36
137500/299*5969915^(1/4)*sqrt(826)*sqrt(217)*(2*sqrt(35)*sqrt(31) - 97*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt
(35) + 361375) + 1003990455125000*x + 100399045512500*sqrt(35) + 501995227562500) + 5969915^(1/4)*sqrt(826)*sq
rt(217)*(32678*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) - 361375*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(32678*sqrt(35) +
361375)*log(-36137500/299*5969915^(1/4)*sqrt(826)*sqrt(217)*(2*sqrt(35)*sqrt(31) - 97*sqrt(31))*sqrt(2*x + 1)*
sqrt(32678*sqrt(35) + 361375) + 1003990455125000*x + 100399045512500*sqrt(35) + 501995227562500) - 18026125963
30*(20*x + 37)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**2), x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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