### 3.2316 $$\int \frac{(1+2 x)^{7/2}}{(2+3 x+5 x^2)^2} \, dx$$

Optimal. Leaf size=296 $-\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (Sqrt
[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[
35])]])/775 - (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775

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Rubi [A]  time = 0.445667, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {738, 824, 826, 1169, 634, 618, 204, 628} $-\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (Sqrt
[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[
35])]])/775 - (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775

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 824

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

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)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{(29-12 x) (1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\\ &=-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{155} \int \frac{\sqrt{1+2 x} (193+302 x)}{2+3 x+5 x^2} \, dx\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \int \frac{-243+1628 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{2}{775} \operatorname{Subst}\left (\int \frac{-2114+1628 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-2114 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (-2114-1628 \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 )}{775 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{-2114 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (-2114-1628 \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 )}{775 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{\sqrt{1460631-245828 \sqrt{35}} \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 )}{3875}-\frac{\sqrt{1460631-245828 \sqrt{35}} \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 )}{3875}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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 \sqrt{1460631-245828 \sqrt{35}}\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 )}{3875}+\frac{\left (2 \sqrt{1460631-245828 \sqrt{35}}\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 )}{3875}\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \sqrt{\frac{2}{155} \left (-5682718+968975 \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}{775} \sqrt{\frac{2}{155} \left (-5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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 = 0.433634, size = 199, normalized size = 0.67 $\frac{1}{217} \left (\frac{(20 x+37) (2 x+1)^{9/2}}{5 x^2+3 x+2}-8 (2 x+1)^{7/2}-28 (2 x+1)^{5/2}-\frac{56}{5} (2 x+1)^{3/2}+\frac{4228}{25} \sqrt{2 x+1}-\frac{14 i \left (\sqrt{2-i \sqrt{31}} \left (512 \sqrt{31}-4681 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )-\sqrt{2+i \sqrt{31}} \left (512 \sqrt{31}+4681 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )\right )}{775 \sqrt{5}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

((4228*Sqrt[1 + 2*x])/25 - (56*(1 + 2*x)^(3/2))/5 - 28*(1 + 2*x)^(5/2) - 8*(1 + 2*x)^(7/2) + ((1 + 2*x)^(9/2)*
(37 + 20*x))/(2 + 3*x + 5*x^2) - (((14*I)/775)*(Sqrt[2 - I*Sqrt[31]]*(-4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 +
10*x]/Sqrt[2 - I*Sqrt[31]]] - Sqrt[2 + I*Sqrt[31]]*(4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*
Sqrt[31]]]))/Sqrt[5])/217

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Maple [B]  time = 0.078, size = 651, normalized size = 2.2 \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)^(7/2)/(5*x^2+3*x+2)^2,x)

[Out]

16/25*(1+2*x)^(1/2)+16/25*(-89/310*(1+2*x)^(3/2)+189/620*(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)-3657/240250*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
56/24025*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)+3657/24025/(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)+512/24025/(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)-604/775/(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)+3657/240250*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)+256/24025*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)+3657/24025/(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)+512/24025/(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)-604/775/(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{7}{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)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)

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Fricas [B]  time = 3.00175, size = 2880, normalized size = 9.73 \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)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")

[Out]

1/977420404174943750*(16794436*21898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(3
5) + 65723878543750)*arctan(1/60332699662225359002939375*21898835^(3/4)*sqrt(4369)*sqrt(3955)*sqrt(155)*sqrt(2
1898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) +
65723878543750) + 40683471557750*x + 4068347155775*sqrt(35) + 20341735778875)*(151*sqrt(35) + 814)*sqrt(-11012
823348100*sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35)
+ 814)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 16794436*2
1898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(35) + 65723878543750)*arctan(1/42
23288976355775130205756250*21898835^(3/4)*sqrt(4369)*sqrt(155)*sqrt(-19379500*21898835^(1/4)*sqrt(155)*(814*sq
rt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 7884253370534
16125000*x + 78842533705341612500*sqrt(35) + 394212668526708062500)*(151*sqrt(35) + 814)*sqrt(-11012823348100*
sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35) + 814)*sq
rt(-11012823348100*sqrt(35) + 65723878543750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) - 21898835^(1/4)*sqrt(
155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*
sqrt(35) + 65723878543750)*log(19379500/4369*21898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*
sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 180458992230125000*x + 18045899223012500*sqrt(
35) + 90229496115062500) + 21898835^(1/4)*sqrt(155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sq
rt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*sqrt(35) + 65723878543750)*log(-19379500/4369*21898835^(1/4)*sq
rt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750)
+ 180458992230125000*x + 18045899223012500*sqrt(35) + 90229496115062500) + 1261187618290250*(2480*x^2 + 1132*x
+ 1003)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{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)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)