3.24.11 \(\int \frac {(1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\) [2311]

Optimal. Leaf size=253 \[ \frac {4}{5} \sqrt {1+2 x}+\frac {1}{5} \sqrt {\frac {2}{155} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{5} \sqrt {\frac {2}{155} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{5} \sqrt {\frac {1}{310} \left (178+35 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{5} \sqrt {\frac {1}{310} \left (178+35 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

4/5*(1+2*x)^(1/2)+1/775*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-55180+108
50*35^(1/2))^(1/2)-1/775*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-55180+108
50*35^(1/2))^(1/2)+1/1550*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(55180+10850*35^(1/2))^(1/2
)-1/1550*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(55180+10850*35^(1/2))^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 253, 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 = {717, 840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{5} \sqrt {\frac {2}{155} \left (35 \sqrt {35}-178\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}{5} \sqrt {\frac {2}{155} \left (35 \sqrt {35}-178\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 {4}{5} \sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {1}{310} \left (178+35 \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}{5} \sqrt {\frac {1}{310} \left (178+35 \sqrt {35}\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[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2),x]

[Out]

(4*Sqrt[1 + 2*x])/5 + (Sqrt[(2*(-178 + 35*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/5 - (Sqrt[(2*(-178 + 35*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1
+ 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 + (Sqrt[(178 + 35*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*
Sqrt[1 + 2*x] + 5*(1 + 2*x)])/5 - (Sqrt[(178 + 35*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1
+ 2*x] + 5*(1 + 2*x)])/5

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 717

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

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 {(1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx &=\frac {4}{5} \sqrt {1+2 x}+\frac {1}{5} \int \frac {-3+8 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {4}{5} \sqrt {1+2 x}+\frac {2}{5} \text {Subst}\left (\int \frac {-14+8 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {4}{5} \sqrt {1+2 x}+\frac {\text {Subst}\left (\int \frac {-14 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-14-8 \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 )}{5 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-14 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-14-8 \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 )}{5 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {4}{5} \sqrt {1+2 x}+\frac {1}{25} \left (4-\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 )+\frac {1}{25} \left (4-\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 )+\frac {1}{5} \sqrt {\frac {1}{310} \left (178+35 \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}{5} \sqrt {\frac {1}{310} \left (178+35 \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 {4}{5} \sqrt {1+2 x}+\frac {1}{5} \sqrt {\frac {1}{310} \left (178+35 \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}{5} \sqrt {\frac {1}{310} \left (178+35 \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}{25} \left (2 \left (4-\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 )-\frac {1}{25} \left (2 \left (4-\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 )\\ &=\frac {4}{5} \sqrt {1+2 x}+\frac {1}{5} \sqrt {\frac {2}{155} \left (-178+35 \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}{5} \sqrt {\frac {2}{155} \left (-178+35 \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}{5} \sqrt {\frac {1}{310} \left (178+35 \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}{5} \sqrt {\frac {1}{310} \left (178+35 \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 = 0.38, size = 114, normalized size = 0.45 \begin {gather*} \frac {2}{775} \left (310 \sqrt {1+2 x}-\sqrt {155 \left (-178+19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {155 \left (-178-19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(310*Sqrt[1 + 2*x] - Sqrt[155*(-178 + (19*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - S
qrt[155*(-178 - (19*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/775

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(167)=334\).
time = 1.88, size = 389, normalized size = 1.54 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(2*x+1)^(1/2)+1/1550*(-27*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-10*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*ln(5
^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2))+2/155*(-62*5^(1/2)*7^(1/2)-1/10*(-27*
(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-10*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2
))/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(2*x+1)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(
1/2)-20)^(1/2))+1/1550*(27*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*ln(-(2*
5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)+2/155*(-62*5^(1/2)*7^(1/2)+1/10*(27*(2*
5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/
(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (170) = 340\).
time = 2.86, size = 454, normalized size = 1.79 \begin {gather*} -\frac {1}{1118363750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (178 \, \sqrt {35} \sqrt {31} + 1225 \, \sqrt {31}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (\frac {620}{19} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (4 \, \sqrt {35} \sqrt {31} + 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 235445000 \, x + 23544500 \, \sqrt {35} + 117722500\right ) + \frac {1}{1118363750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (178 \, \sqrt {35} \sqrt {31} + 1225 \, \sqrt {31}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (-\frac {620}{19} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (4 \, \sqrt {35} \sqrt {31} + 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 235445000 \, x + 23544500 \, \sqrt {35} + 117722500\right ) + \frac {2}{949375} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (\frac {1}{5205983256250} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {-620 \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (4 \, \sqrt {35} \sqrt {31} + 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 4473455000 \, x + 447345500 \, \sqrt {35} + 2236727500} {\left (\sqrt {35} \sqrt {19} + 4 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{25253375} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (\sqrt {35} + 4\right )} \sqrt {-12460 \, \sqrt {35} + 85750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + \frac {2}{949375} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (-\frac {1}{25253375} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (\sqrt {35} + 4\right )} \sqrt {-12460 \, \sqrt {35} + 85750} + \frac {1}{16793494375} \cdot 42875^{\frac {3}{4}} \sqrt {42875^{\frac {1}{4}} \sqrt {155} {\left (4 \, \sqrt {35} \sqrt {31} + 35 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 7215250 \, x + 721525 \, \sqrt {35} + 3607625} {\left (\sqrt {35} \sqrt {19} + 4 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + \frac {4}{5} \, \sqrt {2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1118363750*42875^(1/4)*sqrt(155)*(178*sqrt(35)*sqrt(31) + 1225*sqrt(31))*sqrt(-12460*sqrt(35) + 85750)*log(
620/19*42875^(1/4)*sqrt(155)*(4*sqrt(35)*sqrt(31) + 35*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) +
235445000*x + 23544500*sqrt(35) + 117722500) + 1/1118363750*42875^(1/4)*sqrt(155)*(178*sqrt(35)*sqrt(31) + 12
25*sqrt(31))*sqrt(-12460*sqrt(35) + 85750)*log(-620/19*42875^(1/4)*sqrt(155)*(4*sqrt(35)*sqrt(31) + 35*sqrt(31
))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) + 235445000*x + 23544500*sqrt(35) + 117722500) + 2/949375*42875
^(1/4)*sqrt(155)*sqrt(35)*sqrt(-12460*sqrt(35) + 85750)*arctan(1/5205983256250*42875^(3/4)*sqrt(155)*sqrt(-620
*42875^(1/4)*sqrt(155)*(4*sqrt(35)*sqrt(31) + 35*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) + 44734
55000*x + 447345500*sqrt(35) + 2236727500)*(sqrt(35)*sqrt(19) + 4*sqrt(19))*sqrt(-12460*sqrt(35) + 85750) - 1/
25253375*42875^(3/4)*sqrt(155)*sqrt(2*x + 1)*(sqrt(35) + 4)*sqrt(-12460*sqrt(35) + 85750) + 1/31*sqrt(35)*sqrt
(31) + 2/31*sqrt(31)) + 2/949375*42875^(1/4)*sqrt(155)*sqrt(35)*sqrt(-12460*sqrt(35) + 85750)*arctan(-1/252533
75*42875^(3/4)*sqrt(155)*sqrt(2*x + 1)*(sqrt(35) + 4)*sqrt(-12460*sqrt(35) + 85750) + 1/16793494375*42875^(3/4
)*sqrt(42875^(1/4)*sqrt(155)*(4*sqrt(35)*sqrt(31) + 35*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) +
7215250*x + 721525*sqrt(35) + 3607625)*(sqrt(35)*sqrt(19) + 4*sqrt(19))*sqrt(-12460*sqrt(35) + 85750) - 1/31*
sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 4/5*sqrt(2*x + 1)

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Sympy [A]
time = 14.06, size = 119, normalized size = 0.47 \begin {gather*} \frac {4 \sqrt {2 x + 1}}{5} + 4 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {28 \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} - \frac {4 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(2*x + 1)/5 + 4*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x +
1)))) - 28*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1
))))/5 - 4*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (170) = 340\).
time = 2.56, size = 594, normalized size = 2.35 \begin {gather*} \frac {1}{46519375} \, \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 )} - 8575 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 17150 \, \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}{46519375} \, \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 )} - 8575 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 17150 \, \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}{93038750} \, \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}} - 8575 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 17150 \, \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}{93038750} \, \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}} - 8575 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 17150 \, \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}{5} \, \sqrt {2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/46519375*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(3
5) + 2450)*(2*sqrt(35) - 35) - 8575*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 17150*(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)) + 1/46519375*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) - 8575*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 1715
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)) + 1/93038750*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)*sq
rt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 8575*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqr
t(35) + 2450) + 17150*(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) - 1/93038750*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(-14
0*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 8575*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35)
+ 2450) + 17150*(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) + 4/5*sqrt(2*x + 1)

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Mupad [B]
time = 1.04, size = 182, normalized size = 0.72 \begin {gather*} \frac {4\,\sqrt {2\,x+1}}{5}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{390625\,\left (-\frac {34048}{78125}+\frac {\sqrt {31}\,17024{}\mathrm {i}}{78125}\right )}+\frac {4864\,\sqrt {31}\,\sqrt {155}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{12109375\,\left (-\frac {34048}{78125}+\frac {\sqrt {31}\,17024{}\mathrm {i}}{78125}\right )}\right )\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{775}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{390625\,\left (\frac {34048}{78125}+\frac {\sqrt {31}\,17024{}\mathrm {i}}{78125}\right )}-\frac {4864\,\sqrt {31}\,\sqrt {155}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{12109375\,\left (\frac {34048}{78125}+\frac {\sqrt {31}\,17024{}\mathrm {i}}{78125}\right )}\right )\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{775} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(2*x + 1)^(1/2))/5 - (155^(1/2)*atan((155^(1/2)*(178 - 31^(1/2)*19i)^(1/2)*(2*x + 1)^(1/2)*2432i)/(390625*(
(31^(1/2)*17024i)/78125 - 34048/78125)) + (4864*31^(1/2)*155^(1/2)*(178 - 31^(1/2)*19i)^(1/2)*(2*x + 1)^(1/2))
/(12109375*((31^(1/2)*17024i)/78125 - 34048/78125)))*(178 - 31^(1/2)*19i)^(1/2)*2i)/775 + (155^(1/2)*atan((155
^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2)*2432i)/(390625*((31^(1/2)*17024i)/78125 + 34048/78125)) - (4
864*31^(1/2)*155^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2))/(12109375*((31^(1/2)*17024i)/78125 + 34048/
78125)))*(31^(1/2)*19i + 178)^(1/2)*2i)/775

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