3.2321 \(\int \frac{1}{(1+2 x)^{3/2} (2+3 x+5 x^2)^2} \, dx\)

Optimal. Leaf size=283 \[ \frac{20 x+37}{217 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{604}{1519 \sqrt{2 x+1}}-\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519}-\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519} \]

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(
-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]
])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/1519 + (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5
*(1 + 2*x)])/1519

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Rubi [A]  time = 0.366829, antiderivative size = 283, normalized size of antiderivative = 1., 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 = {740, 828, 826, 1169, 634, 618, 204, 628} \[ \frac{20 x+37}{217 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{604}{1519 \sqrt{2 x+1}}-\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519}-\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(
-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]
])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/1519 + (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5
*(1 + 2*x)])/1519

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 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 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}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{1}{217} \int \frac{181+60 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{\int \frac{59-1510 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1519}\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{1628-1510 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )}{1519}\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{1628 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (1628+302 \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 )}{1519 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{1628 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (1628+302 \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 )}{1519 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{\left (-5285+814 \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 )}{53165}+\frac{\left (-5285+814 \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 )}{53165}-\frac{\sqrt{\frac{1}{434} \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 )}{1519}+\frac{\sqrt{\frac{1}{434} \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 )}{1519}\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}-\frac{\sqrt{\frac{1}{434} \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 )}{1519}+\frac{\sqrt{\frac{1}{434} \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 )}{1519}+\frac{\left (2 \left (5285-814 \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 )}{53165}+\frac{\left (2 \left (5285-814 \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 )}{53165}\\ &=-\frac{604}{1519 \sqrt{1+2 x}}+\frac{37+20 x}{217 \sqrt{1+2 x} \left (2+3 x+5 x^2\right )}+\frac{\sqrt{\frac{2}{217} \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 )}{1519}-\frac{\sqrt{\frac{2}{217} \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 )}{1519}-\frac{\sqrt{\frac{1}{434} \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 )}{1519}+\frac{\sqrt{\frac{1}{434} \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 )}{1519}\\ \end{align*}

Mathematica [C]  time = 0.392974, size = 158, normalized size = 0.56 \[ \frac{1}{217} \left (\frac{20 x+37}{\sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{604}{7 \sqrt{2 x+1}}+\frac{2 \sqrt{10-5 i \sqrt{31}} \left (25234+3657 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )+2 \sqrt{10+5 i \sqrt{31}} \left (25234-3657 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )}{7595}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-604/(7*Sqrt[1 + 2*x]) + (37 + 20*x)/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(25234
+ (3657*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(25234 - (3657
*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/7595)/217

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Maple [B]  time = 0.081, size = 651, normalized size = 2.3 \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)^(3/2)/(5*x^2+3*x+2)^2,x)

[Out]

-16/49*(27/124*(1+2*x)^(3/2)-89/310*(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)+256/47089*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)+3657/659246*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)-2560/4708
9/(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)-3657/329623/(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)+3256/10633
/(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)-256/47089*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)-3657/659246*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)-2560/47089/(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)-365
7/329623/(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)+3256/10633/(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)-16/49/(1
+2*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.92854, size = 3001, normalized size = 10.6 \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)^(3/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")

[Out]

1/2682041589056045650*(16794436*21898835^(1/4)*sqrt(217)*sqrt(35)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-1101282334
8100*sqrt(35) + 65723878543750)*arctan(1/84465779527115502604115125*21898835^(3/4)*sqrt(4369)*sqrt(791)*sqrt(2
17)*sqrt(21898835^(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sq
rt(35) + 65723878543750) + 8136694311550*x + 813669431155*sqrt(35) + 4068347155775)*(814*sqrt(35) + 5285)*sqrt
(-11012823348100*sqrt(35) + 65723878543750) - 1/22526438201526175*21898835^(3/4)*sqrt(217)*sqrt(2*x + 1)*(814*
sqrt(35) + 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 1
6794436*21898835^(1/4)*sqrt(217)*sqrt(35)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-11012823348100*sqrt(35) + 65723878
543750)*arctan(1/206941159841432981380082056250*21898835^(3/4)*sqrt(4369)*sqrt(217)*sqrt(-4747977500*21898835^
(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 657238785
43750) + 38632841515617390125000*x + 3863284151561739012500*sqrt(35) + 19316420757808695062500)*(814*sqrt(35)
+ 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/22526438201526175*21898835^(3/4)*sqrt(217)*sqrt(2*
x + 1)*(814*sqrt(35) + 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 1/31*sqrt(35)*sqrt(31) + 2/31*s
qrt(31)) + 21898835^(1/4)*sqrt(217)*(5682718*sqrt(35)*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2) + 33914125*sqrt(31)
*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(-11012823348100*sqrt(35) + 65723878543750)*log(4747977500/4369*21898835^(1/
4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 657238785437
50) + 8842490619276125000*x + 884249061927612500*sqrt(35) + 4421245309638062500) - 21898835^(1/4)*sqrt(217)*(5
682718*sqrt(35)*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2) + 33914125*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(-11
012823348100*sqrt(35) + 65723878543750)*log(-4747977500/4369*21898835^(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) +
814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 8842490619276125000*x + 8842490
61927612500*sqrt(35) + 4421245309638062500) - 1765662665606350*(3020*x^2 + 1672*x + 949)*sqrt(2*x + 1))/(10*x^
3 + 11*x^2 + 7*x + 2)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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