3.2310 \(\int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=266 \[ \frac {4}{15} (2 x+1)^{3/2}+\frac {16}{25} \sqrt {2 x+1}+\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

[Out]

4/15*(1+2*x)^(3/2)+16/25*(1+2*x)^(1/2)+1/7750*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-22202
20+379750*35^(1/2))^(1/2)-1/7750*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-2220220+379750*35^
(1/2))^(1/2)+1/3875*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2220220+379750
*35^(1/2))^(1/2)-1/3875*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2220220+379
750*35^(1/2))^(1/2)

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Rubi [A]  time = 0.44, antiderivative size = 266, normalized size of antiderivative = 1.00, 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 = {703, 824, 826, 1169, 634, 618, 204, 628} \[ \frac {4}{15} (2 x+1)^{3/2}+\frac {16}{25} \sqrt {2 x+1}+\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*Sqrt[1 + 2*x])/25 + (4*(1 + 2*x)^(3/2))/15 + (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqr
t[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 + (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Lo
g[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25 - (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Log
[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25

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 703

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 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]

Rubi steps

\begin {align*} \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx &=\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{5} \int \frac {\sqrt {1+2 x} (-3+8 x)}{2+3 x+5 x^2} \, dx\\ &=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \int \frac {-47-38 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {2}{25} \operatorname {Subst}\left (\int \frac {-56-38 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {-56 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-56+38 \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 )}{25 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {-56 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-56+38 \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 )}{25 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}-\frac {1}{125} \sqrt {921+152 \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 )-\frac {1}{125} \sqrt {921+152 \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 )+\frac {1}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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 {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{125} \left (2 \sqrt {921+152 \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 )+\frac {1}{125} \left (2 \sqrt {921+152 \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 )\\ &=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \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}{25} \sqrt {\frac {2}{155} \left (7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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} \sqrt {\frac {1}{310} \left (-7162+1225 \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 = 0.26, size = 133, normalized size = 0.50 \[ \frac {2 \left (310 \sqrt {2 x+1} (10 x+17)+3 i \sqrt {10-5 i \sqrt {31}} \left (27 \sqrt {31}+124 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )-3 i \sqrt {10+5 i \sqrt {31}} \left (27 \sqrt {31}-124 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{11625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(310*Sqrt[1 + 2*x]*(17 + 10*x) + (3*I)*Sqrt[10 - (5*I)*Sqrt[31]]*(124*I + 27*Sqrt[31])*ArcTanh[Sqrt[5 + 10*
x]/Sqrt[2 - I*Sqrt[31]]] - (3*I)*Sqrt[10 + (5*I)*Sqrt[31]]*(-124*I + 27*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[
2 + I*Sqrt[31]]]))/11625

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fricas [B]  time = 0.97, size = 493, normalized size = 1.85 \[ -\frac {1}{1673341250} \, \sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} - 42875 \, \sqrt {31}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (\frac {26582500}{199} \, \sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (19 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 288420125000 \, x + 28842012500 \, \sqrt {35} + 144210062500\right ) + \frac {1}{1673341250} \, \sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} - 42875 \, \sqrt {31}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (-\frac {26582500}{199} \, \sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (19 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 288420125000 \, x + 28842012500 \, \sqrt {35} + 144210062500\right ) + \frac {2}{135625} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{46619287225} \, \sqrt {217} \sqrt {199} \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {\sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (19 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 2159150 \, x + 215915 \, \sqrt {35} + 1079575} \sqrt {7162 \, \sqrt {35} + 42875} {\left (4 \, \sqrt {35} - 19\right )} - \frac {1}{215915} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (4 \, \sqrt {35} - 19\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + \frac {2}{135625} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{16316750528750} \, \sqrt {199} \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {-26582500 \, \sqrt {155} 35^{\frac {1}{4}} \sqrt {2} {\left (19 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 57395604875000 \, x + 5739560487500 \, \sqrt {35} + 28697802437500} \sqrt {7162 \, \sqrt {35} + 42875} {\left (4 \, \sqrt {35} - 19\right )} - \frac {1}{215915} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (4 \, \sqrt {35} - 19\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + \frac {4}{75} \, {\left (10 \, x + 17\right )} \sqrt {2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1673341250*sqrt(155)*35^(1/4)*sqrt(2)*(7162*sqrt(35)*sqrt(31) - 42875*sqrt(31))*sqrt(7162*sqrt(35) + 42875)
*log(26582500/199*sqrt(155)*35^(1/4)*sqrt(2)*(19*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqr
t(35) + 42875) + 288420125000*x + 28842012500*sqrt(35) + 144210062500) + 1/1673341250*sqrt(155)*35^(1/4)*sqrt(
2)*(7162*sqrt(35)*sqrt(31) - 42875*sqrt(31))*sqrt(7162*sqrt(35) + 42875)*log(-26582500/199*sqrt(155)*35^(1/4)*
sqrt(2)*(19*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 288420125000*x + 288
42012500*sqrt(35) + 144210062500) + 2/135625*sqrt(155)*35^(3/4)*sqrt(2)*sqrt(7162*sqrt(35) + 42875)*arctan(1/4
6619287225*sqrt(217)*sqrt(199)*sqrt(155)*35^(3/4)*sqrt(2)*sqrt(sqrt(155)*35^(1/4)*sqrt(2)*(19*sqrt(35)*sqrt(31
) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 2159150*x + 215915*sqrt(35) + 1079575)*sqrt(7162
*sqrt(35) + 42875)*(4*sqrt(35) - 19) - 1/215915*sqrt(155)*35^(3/4)*sqrt(2)*sqrt(2*x + 1)*sqrt(7162*sqrt(35) +
42875)*(4*sqrt(35) - 19) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 2/135625*sqrt(155)*35^(3/4)*sqrt(2)*sqrt(
7162*sqrt(35) + 42875)*arctan(1/16316750528750*sqrt(199)*sqrt(155)*35^(3/4)*sqrt(2)*sqrt(-26582500*sqrt(155)*3
5^(1/4)*sqrt(2)*(19*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 573956048750
00*x + 5739560487500*sqrt(35) + 28697802437500)*sqrt(7162*sqrt(35) + 42875)*(4*sqrt(35) - 19) - 1/215915*sqrt(
155)*35^(3/4)*sqrt(2)*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875)*(4*sqrt(35) - 19) - 1/31*sqrt(35)*sqrt(31) - 2
/31*sqrt(31)) + 4/75*(10*x + 17)*sqrt(2*x + 1)

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giac [B]  time = 1.21, size = 605, normalized size = 2.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/930387500*sqrt(31)*(3990*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 19*sqrt(31)*(7
/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 38*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 7980*(7/5)^(3/4)*sqrt(14
0*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 274400*(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)) - 1/930387500*sqrt(31)*(3990*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqr
t(35) + 2450) - 19*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 38*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3
/2) + 7980*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqr
t(35) + 2450) + 274400*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*s
qrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/1860775000*sqrt(31)*(19*sqrt(31)*(7/5)^(3/4)*(
140*sqrt(35) + 2450)^(3/2) + 3990*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 7980*(7/5
)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 38*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 137200*sq
rt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 1/1860775000*sqrt(31)*(19*sqrt(31)*(7/5)^(3
/4)*(140*sqrt(35) + 2450)^(3/2) + 3990*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 7980
*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 38*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1372
00*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 4/15*(2*x + 1)^(3/2) + 16/25*sqrt(2*x
+ 1)

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maple [B]  time = 0.32, size = 625, normalized size = 2.35 \[ \frac {178 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {16 \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 )}{25 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {178 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {16 \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 )}{25 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {89 \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 )}{3875}-\frac {27 \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 )}{1550}-\frac {89 \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 )}{3875}+\frac {27 \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 )}{1550}+\frac {4 \left (2 x +1\right )^{\frac {3}{2}}}{15}+\frac {16 \sqrt {2 x +1}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*(2*x+1)^(3/2)+16/25*(2*x+1)^(1/2)-89/3875*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)+27/1550*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)+178/775/(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))-27/775
/(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))-16/25/(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))+89/3875*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)-27/1550*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)+1
78/775/(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))-27/775/(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))-16/25/(1
0*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 \[ \int \frac {{\left (2 \, x + 1\right )}^{\frac {5}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 191, normalized size = 0.72 \[ \frac {16\,\sqrt {2\,x+1}}{25}+\frac {4\,{\left (2\,x+1\right )}^{3/2}}{15}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{48828125\,\left (\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {155}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{1513671875\,\left (\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{3875}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{48828125\,\left (-\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {155}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{1513671875\,\left (-\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{3875} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(16*(2*x + 1)^(1/2))/25 + (4*(2*x + 1)^(3/2))/15 - (155^(1/2)*atan((155^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(
2*x + 1)^(1/2)*25472i)/(48828125*((31^(1/2)*713216i)/9765625 + 4814208/9765625)) + (50944*31^(1/2)*155^(1/2)*(
- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))/(1513671875*((31^(1/2)*713216i)/9765625 + 4814208/9765625)))*(-
31^(1/2)*199i - 7162)^(1/2)*2i)/3875 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2
)*25472i)/(48828125*((31^(1/2)*713216i)/9765625 - 4814208/9765625)) - (50944*31^(1/2)*155^(1/2)*(31^(1/2)*199i
- 7162)^(1/2)*(2*x + 1)^(1/2))/(1513671875*((31^(1/2)*713216i)/9765625 - 4814208/9765625)))*(31^(1/2)*199i -
7162)^(1/2)*2i)/3875

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sympy [A]  time = 56.21, size = 206, normalized size = 0.77 \[ \frac {4 \left (2 x + 1\right )^{\frac {3}{2}}}{15} + \frac {16 \sqrt {2 x + 1}}{25} + 4 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {136 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {56 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {8 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {168 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(2*x + 1)**(3/2)/15 + 16*sqrt(2*x + 1)/25 + 4*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920
*_t**3 + 8*_t + sqrt(2*x + 1)))) - 136*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 +
8*_t + sqrt(2*x + 1))))/25 - 56*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108
*_t/5 + sqrt(2*x + 1))))/5 - 8*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + s
qrt(2*x + 1))))/5 + 168*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 +
sqrt(2*x + 1))))/25

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