3.24.19 \(\int \frac {\sqrt {1+2 x}}{(2+3 x+5 x^2)^2} \, dx\) [2319]

Optimal. Leaf size=270 \[ \frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

1/31*(3+10*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)+1/13454*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-9
4612+20398*35^(1/2))^(1/2)-1/13454*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-94612+20398*35^(
1/2))^(1/2)-1/6727*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(94612+20398*35^
(1/2))^(1/2)+1/6727*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(94612+20398*35^
(1/2))^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 270, normalized size of antiderivative = 1.00, 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 = {750, 840, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \text {ArcTan}\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}{217} \left (218+47 \sqrt {35}\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}+\frac {1}{31} \sqrt {\frac {1}{434} \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}{434} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + 2*x]*(3 + 10*x))/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/217]*ArcTan[(Sqrt[1
0*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(-218 + 47*Sqrt[35])/434]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 - (Sqrt[(-218 + 47*Sqrt[35])/434]*Log[Sqrt[35]
+ Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31

Rule 210

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

Rule 632

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 642

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 648

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 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*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
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*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
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 840

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 1183

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 {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \int \frac {-7-10 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{31} \text {Subst}\left (\int \frac {-4-10 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-4+2 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\text {Subst}\left (\int \frac {-4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-4+2 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}+\frac {\left (35+2 \sqrt {35}\right ) \text {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 )}{1085}+\frac {\left (35+2 \sqrt {35}\right ) \text {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 )}{1085}+\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \text {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}{434} \left (-218+47 \sqrt {35}\right )} \text {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} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \sqrt {\frac {1}{434} \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}{434} \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 {\left (2 \left (35+2 \sqrt {35}\right )\right ) \text {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 )}{1085}-\frac {\left (2 \left (35+2 \sqrt {35}\right )\right ) \text {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 )}{1085}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{217} \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}{217} \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}{434} \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}{434} \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.55, size = 129, normalized size = 0.48 \begin {gather*} \frac {2 \left (\frac {217 \sqrt {1+2 x} (3+10 x)}{4+6 x+10 x^2}+\sqrt {217 \left (218-31 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (218+31 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6727} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((217*Sqrt[1 + 2*x]*(3 + 10*x))/(4 + 6*x + 10*x^2) + Sqrt[217*(218 - (31*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*
Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[217*(218 + (31*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2
*x]]))/6727

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(184)=368\).
time = 2.14, size = 555, normalized size = 2.06 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/6727*(2/25*(-5425*7^(1/2)+2170*5^(1/2))/(2*5^(1/2)-5*7^(1/2))*(2*x+1)^(1/2)+1/25*7^(1/2)*(2*5^(1/2)*7^(1/2)+
4)^(1/2)*(-1085*5^(1/2)+310*7^(1/2))/(2*5^(1/2)-5*7^(1/2)))/(1/5*5^(1/2)*7^(1/2)+1/5*(2*5^(1/2)*7^(1/2)+4)^(1/
2)*5^(1/2)*(2*x+1)^(1/2)+2*x+1)+5/6727/(2*5^(1/2)-5*7^(1/2))*(1/10*(-218*(2*35^(1/2)+4)^(1/2)*35^(1/2)+1645*(2
*35^(1/2)+4)^(1/2))*ln(10*x+5+35^(1/2)+(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))+2*(-868*5^(1/2)+248*7^(1/2)-1/10*
(-218*(2*35^(1/2)+4)^(1/2)*35^(1/2)+1645*(2*35^(1/2)+4)^(1/2))*(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2)
*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2)))-5/6727*(-2/25*(-5425*7^(1/2)+2170*
5^(1/2))/(2*5^(1/2)-5*7^(1/2))*(2*x+1)^(1/2)+1/25*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1085*5^(1/2)+310*7^(1/
2))/(2*5^(1/2)-5*7^(1/2)))/(1/5*5^(1/2)*7^(1/2)-1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+2*x+1)-5
/6727/(2*5^(1/2)-5*7^(1/2))*(-1/10*(218*(2*35^(1/2)+4)^(1/2)*35^(1/2)-1645*(2*35^(1/2)+4)^(1/2))*ln(10*x+5+35^
(1/2)-(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))-2*(-868*5^(1/2)+248*7^(1/2)+1/10*(218*(2*35^(1/2)+4)^(1/2)*35^(1/2
)-1645*(2*35^(1/2)+4)^(1/2))*(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+1
0*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (187) = 374\).
time = 2.65, size = 526, normalized size = 1.95 \begin {gather*} \frac {3844 \cdot 77315^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{26522397764975} \cdot 77315^{\frac {3}{4}} \sqrt {1645} \sqrt {217} \sqrt {77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 3161690 \, x + 316169 \, \sqrt {35} + 1580845} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{520098005} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 3844 \cdot 77315^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{53044795529950} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {-6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{520098005} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 77315^{\frac {1}{4}} \sqrt {217} {\left (218 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 1645 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {20492 \, \sqrt {35} + 154630} \log \left (6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100\right ) - 77315^{\frac {1}{4}} \sqrt {217} {\left (218 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 1645 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {20492 \, \sqrt {35} + 154630} \log \left (-6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100\right ) + 686086730 \, {\left (10 \, x + 3\right )} \sqrt {2 \, x + 1}}{21268688630 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21268688630*(3844*77315^(1/4)*sqrt(217)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(20492*sqrt(35) + 154630)*arctan(1/26
522397764975*77315^(3/4)*sqrt(1645)*sqrt(217)*sqrt(77315^(1/4)*sqrt(217)*(sqrt(35)*sqrt(31) - 2*sqrt(31))*sqrt
(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 3161690*x + 316169*sqrt(35) + 1580845)*sqrt(20492*sqrt(35) + 154630)
*(2*sqrt(35) - 35) - 1/520098005*77315^(3/4)*sqrt(217)*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630)*(2*sqrt(35)
- 35) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 3844*77315^(1/4)*sqrt(217)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(
20492*sqrt(35) + 154630)*arctan(1/53044795529950*77315^(3/4)*sqrt(217)*sqrt(-6580*77315^(1/4)*sqrt(217)*(sqrt(
35)*sqrt(31) - 2*sqrt(31))*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 20803920200*x + 2080392020*sqrt(35) +
10401960100)*sqrt(20492*sqrt(35) + 154630)*(2*sqrt(35) - 35) - 1/520098005*77315^(3/4)*sqrt(217)*sqrt(2*x + 1
)*sqrt(20492*sqrt(35) + 154630)*(2*sqrt(35) - 35) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 77315^(1/4)*sqrt
(217)*(218*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) - 1645*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(20492*sqrt(35) + 154630
)*log(6580*77315^(1/4)*sqrt(217)*(sqrt(35)*sqrt(31) - 2*sqrt(31))*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630)
+ 20803920200*x + 2080392020*sqrt(35) + 10401960100) - 77315^(1/4)*sqrt(217)*(218*sqrt(35)*sqrt(31)*(5*x^2 + 3
*x + 2) - 1645*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(20492*sqrt(35) + 154630)*log(-6580*77315^(1/4)*sqrt(217)*(sqrt
(35)*sqrt(31) - 2*sqrt(31))*sqrt(2*x + 1)*sqrt(20492*sqrt(35) + 154630) + 20803920200*x + 2080392020*sqrt(35)
+ 10401960100) + 686086730*(10*x + 3)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

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Sympy [A]
time = 3.44, size = 83, normalized size = 0.31 \begin {gather*} \frac {80 \left (2 x + 1\right )^{\frac {3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac {32 \sqrt {2 x + 1}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} + 16 \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

80*(2*x + 1)**(3/2)/(-992*x + 620*(2*x + 1)**2 + 372) - 32*sqrt(2*x + 1)/(-992*x + 620*(2*x + 1)**2 + 372) + 1
6*RootSum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_
t/235 + sqrt(2*x + 1))))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (187) = 374\).
time = 1.65, size = 622, normalized size = 2.30 \begin {gather*} \frac {1}{230736100} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1960 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3920 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{230736100} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1960 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3920 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{461472200} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1960 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3920 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {1}{461472200} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1960 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3920 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) + \frac {4 \, {\left (5 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {2 \, x + 1}\right )}}{31 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/230736100*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) + 1960*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3920*(7/5)^(1/4)*sqrt(1
40*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)) + 1/230736100*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) + 1960*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 392
0*(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)) + 1/461472200*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)*s
qrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1960*sqrt(31)*(7/5)^(1/4)*sqrt(140*sq
rt(35) + 2450) - 3920*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(3
5) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/461472200*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 21
0*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(-1
40*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1960*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35)
+ 2450) - 3920*(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) + 4/31*(5*(2*x + 1)^(3/2) - 2*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

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Mupad [B]
time = 1.03, size = 208, normalized size = 0.77 \begin {gather*} -\frac {\frac {8\,\sqrt {2\,x+1}}{155}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{31}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}-\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}+\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2)*128i)/(5886125*((31^(1/2)*256i)/840875 +
4992/840875)) + (256*31^(1/2)*217^(1/2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(182469875*((31^(1/2)*256
i)/840875 + 4992/840875)))*(31^(1/2)*31i - 218)^(1/2)*2i)/6727 - (217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*31i -
218)^(1/2)*(2*x + 1)^(1/2)*128i)/(5886125*((31^(1/2)*256i)/840875 - 4992/840875)) - (256*31^(1/2)*217^(1/2)*(-
31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(182469875*((31^(1/2)*256i)/840875 - 4992/840875)))*(- 31^(1/2)*31
i - 218)^(1/2)*2i)/6727 - ((8*(2*x + 1)^(1/2))/155 - (4*(2*x + 1)^(3/2))/31)/((2*x + 1)^2 - (8*x)/5 + 3/5)

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