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]

-1/31*(5-4*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-1/9610*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-67
580+14570*35^(1/2))^(1/2)+1/9610*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-67580+14570*35^(1/
2))^(1/2)-1/4805*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(67580+14570*35^(1
/2))^(1/2)+1/4805*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(67580+14570*35^(1
/2))^(1/2)

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Rubi [A]  time = 0.36, 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 = {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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.71, 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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fricas [B]  time = 0.78, size = 538, normalized size = 1.99 \[ -\frac {3844 \cdot 77315^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{3788913966425} \cdot 77315^{\frac {3}{4}} \sqrt {155} \sqrt {47} \sqrt {77315^{\frac {1}{4}} \sqrt {155} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 15808450 \, x + 1580845 \, \sqrt {35} + 7904225} {\left (\sqrt {35} \sqrt {7} - 2 \, \sqrt {7}\right )} \sqrt {20492 \, \sqrt {35} + 154630} - \frac {1}{74299715} \cdot 77315^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (\sqrt {35} - 2\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 3844 \cdot 77315^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{7577827932850} \cdot 77315^{\frac {3}{4}} \sqrt {155} \sqrt {-188 \cdot 77315^{\frac {1}{4}} \sqrt {155} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 2971988600 \, x + 297198860 \, \sqrt {35} + 1485994300} {\left (\sqrt {35} \sqrt {7} - 2 \, \sqrt {7}\right )} \sqrt {20492 \, \sqrt {35} + 154630} - \frac {1}{74299715} \cdot 77315^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (\sqrt {35} - 2\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 77315^{\frac {1}{4}} \sqrt {155} {\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 (\frac {188}{7} \cdot 77315^{\frac {1}{4}} \sqrt {155} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 424569800 \, x + 42456980 \, \sqrt {35} + 212284900\right ) + 77315^{\frac {1}{4}} \sqrt {155} {\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 (-\frac {188}{7} \cdot 77315^{\frac {1}{4}} \sqrt {155} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 424569800 \, x + 42456980 \, \sqrt {35} + 212284900\right ) - 490061950 \, {\left (4 \, x - 5\right )} \sqrt {2 \, x + 1}}{15191920450 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

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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giac [B]  time = 1.32, size = 622, normalized size = 2.30 \[ \text {result too large to display} \]

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]

1/576840250*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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 34300*(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)) + 1/576840250*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 24
50) - 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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
34300*(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)) + 1/1153680500*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)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(
140*sqrt(35) + 2450) - 34300*(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) - 1/1153680500*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)
*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(140
*sqrt(35) + 2450) - 34300*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*s
qrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 4/31*(2*(2*x + 1)^(3/2) - 7*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

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maple [B]  time = 0.29, size = 642, normalized size = 2.38 \[ -\frac {39 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {39 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{9610}+\frac {2 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{961}+\frac {39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{9610}-\frac {2 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{961}+\frac {\frac {8 \left (2 x +1\right )^{\frac {3}{2}}}{155}-\frac {28 \sqrt {2 x +1}}{155}}{-\frac {8 x}{5}+\left (2 x +1\right )^{2}+\frac {3}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

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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mupad [B]  time = 1.04, size = 208, normalized size = 0.77 \[ -\frac {\frac {28\,\sqrt {2\,x+1}}{155}-\frac {8\,{\left (2\,x+1\right )}^{3/2}}{155}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{3003125\,\left (\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}+\frac {256\,\sqrt {31}\,\sqrt {155}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{93096875\,\left (\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{4805}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{3003125\,\left (-\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}-\frac {256\,\sqrt {31}\,\sqrt {155}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{93096875\,\left (-\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{4805} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(155^(1/2)*atan((155^(1/2)*(- 31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2)*128i)/(3003125*((31^(1/2)*896i)/600625
+ 3584/600625)) + (256*31^(1/2)*155^(1/2)*(- 31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(93096875*((31^(1/2)*
896i)/600625 + 3584/600625)))*(- 31^(1/2)*31i - 218)^(1/2)*2i)/4805 - ((28*(2*x + 1)^(1/2))/155 - (8*(2*x + 1)
^(3/2))/155)/((2*x + 1)^2 - (8*x)/5 + 3/5) - (155^(1/2)*atan((155^(1/2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(
1/2)*128i)/(3003125*((31^(1/2)*896i)/600625 - 3584/600625)) - (256*31^(1/2)*155^(1/2)*(31^(1/2)*31i - 218)^(1/
2)*(2*x + 1)^(1/2))/(93096875*((31^(1/2)*896i)/600625 - 3584/600625)))*(31^(1/2)*31i - 218)^(1/2)*2i)/4805

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sympy [A]  time = 111.05, size = 248, normalized size = 0.92 \[ \frac {320 \left (2 x + 1\right )^{\frac {3}{2}}}{- 4960 x + 3100 \left (2 x + 1\right )^{2} + 1860} - \frac {1120 \left (2 x + 1\right )^{\frac {3}{2}}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} - \frac {128 \sqrt {2 x + 1}}{- 4960 x + 3100 \left (2 x + 1\right )^{2} + 1860} - \frac {3024 \sqrt {2 x + 1}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} + 16 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left (t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {112 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left (t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {16 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left (t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {16 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} \]

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]

320*(2*x + 1)**(3/2)/(-4960*x + 3100*(2*x + 1)**2 + 1860) - 1120*(2*x + 1)**(3/2)/(-34720*x + 21700*(2*x + 1)*
*2 + 13020) - 128*sqrt(2*x + 1)/(-4960*x + 3100*(2*x + 1)**2 + 1860) - 3024*sqrt(2*x + 1)/(-34720*x + 21700*(2
*x + 1)**2 + 13020) + 16*RootSum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(333125345
28*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1)))) - 112*RootSum(19950060344639488*_t**4 + 498437272576*_t**2 + 1
0878125, Lambda(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/5 - 16*Root
Sum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235
+ 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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