3.2315 \(\int \frac {1}{(1+2 x)^{5/2} (2+3 x+5 x^2)} \, dx\)

Optimal. Leaf size=266 \[ -\frac {16}{49 \sqrt {2 x+1}}-\frac {4}{21 (2 x+1)^{3/2}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

[Out]

-4/21/(1+2*x)^(3/2)-16/49/(1+2*x)^(1/2)-1/21266*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-310
8308+531650*35^(1/2))^(1/2)+1/21266*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-3108308+531650*
35^(1/2))^(1/2)+1/10633*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(3108308+53
1650*35^(1/2))^(1/2)-1/10633*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(310830
8+531650*35^(1/2))^(1/2)

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Rubi [A]  time = 0.38, 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 = {709, 828, 826, 1169, 634, 618, 204, 628} \[ -\frac {16}{49 \sqrt {2 x+1}}-\frac {4}{21 (2 x+1)^{3/2}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(21*(1 + 2*x)^(3/2)) - 16/(49*Sqrt[1 + 2*x]) + (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + S
qrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sq
rt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(-7162 + 1225*Sqrt[35])/434]*L
og[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49 + (Sqrt[(-7162 + 1225*Sqrt[35])/434]*Lo
g[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49

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 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 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}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac {4}{21 (1+2 x)^{3/2}}+\frac {1}{7} \int \frac {-1-10 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \int \frac {-39-40 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {2}{49} \operatorname {Subst}\left (\int \frac {-38-40 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {\operatorname {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-38+8 \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 )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-38+8 \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 )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {\left (140+19 \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 )}{1715}-\frac {\left (140+19 \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 )}{1715}-\frac {1}{49} \sqrt {\frac {1}{434} \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}{49} \sqrt {\frac {1}{434} \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 {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {1}{49} \sqrt {\frac {1}{434} \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}{49} \sqrt {\frac {1}{434} \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 {\left (2 \left (140+19 \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 )}{1715}+\frac {\left (2 \left (140+19 \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 )}{1715}\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \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}{49} \sqrt {\frac {2}{217} \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}{49} \sqrt {\frac {1}{434} \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}{49} \sqrt {\frac {1}{434} \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 = 0.51, size = 133, normalized size = 0.50 \[ \frac {2 \left (-\frac {2170 (24 x+19)}{(2 x+1)^{3/2}}+3 i \sqrt {10-5 i \sqrt {31}} \left (178 \sqrt {31}+589 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )-3 i \sqrt {10+5 i \sqrt {31}} \left (178 \sqrt {31}-589 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{159495} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-2170*(19 + 24*x))/(1 + 2*x)^(3/2) + (3*I)*Sqrt[10 - (5*I)*Sqrt[31]]*(589*I + 178*Sqrt[31])*ArcTanh[Sqrt[
5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] - (3*I)*Sqrt[10 + (5*I)*Sqrt[31]]*(-589*I + 178*Sqrt[31])*ArcTanh[Sqrt[5 + 10*
x]/Sqrt[2 + I*Sqrt[31]]]))/159495

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fricas [B]  time = 1.01, size = 567, normalized size = 2.13 \[ -\frac {74028 \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{326335010575} \, \sqrt {1085} \sqrt {217} \sqrt {199} 35^{\frac {3}{4}} \sqrt {2} \sqrt {\sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 431830 \, x + 43183 \, \sqrt {35} + 215915} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{1511405} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 74028 \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{799520775908750} \, \sqrt {217} \sqrt {199} 35^{\frac {3}{4}} \sqrt {2} \sqrt {-6512712500 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 2812384638875000 \, x + 281238463887500 \, \sqrt {35} + 1406192319437500} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{1511405} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 3 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )} - 42875 \, \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (\frac {6512712500}{199} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 14132586125000 \, x + 1413258612500 \, \sqrt {35} + 7066293062500\right ) - 3 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )} - 42875 \, \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (-\frac {6512712500}{199} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 14132586125000 \, x + 1413258612500 \, \sqrt {35} + 7066293062500\right ) + 374828440 \, {\left (24 \, x + 19\right )} \sqrt {2 \, x + 1}}{13774945170 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13774945170*(74028*sqrt(217)*35^(3/4)*sqrt(2)*(4*x^2 + 4*x + 1)*sqrt(7162*sqrt(35) + 42875)*arctan(1/326335
010575*sqrt(1085)*sqrt(217)*sqrt(199)*35^(3/4)*sqrt(2)*sqrt(sqrt(217)*35^(1/4)*sqrt(2)*(4*sqrt(35)*sqrt(31) -
19*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 431830*x + 43183*sqrt(35) + 215915)*sqrt(7162*sqrt(35
) + 42875)*(19*sqrt(35) - 140) - 1/1511405*sqrt(217)*35^(3/4)*sqrt(2)*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875
)*(19*sqrt(35) - 140) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 74028*sqrt(217)*35^(3/4)*sqrt(2)*(4*x^2 + 4*
x + 1)*sqrt(7162*sqrt(35) + 42875)*arctan(1/799520775908750*sqrt(217)*sqrt(199)*35^(3/4)*sqrt(2)*sqrt(-6512712
500*sqrt(217)*35^(1/4)*sqrt(2)*(4*sqrt(35)*sqrt(31) - 19*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) +
2812384638875000*x + 281238463887500*sqrt(35) + 1406192319437500)*sqrt(7162*sqrt(35) + 42875)*(19*sqrt(35) -
140) - 1/1511405*sqrt(217)*35^(3/4)*sqrt(2)*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875)*(19*sqrt(35) - 140) - 1/
31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 3*sqrt(217)*35^(1/4)*sqrt(2)*(7162*sqrt(35)*sqrt(31)*(4*x^2 + 4*x + 1)
- 42875*sqrt(31)*(4*x^2 + 4*x + 1))*sqrt(7162*sqrt(35) + 42875)*log(6512712500/199*sqrt(217)*35^(1/4)*sqrt(2)
*(4*sqrt(35)*sqrt(31) - 19*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 14132586125000*x + 1413258612
500*sqrt(35) + 7066293062500) - 3*sqrt(217)*35^(1/4)*sqrt(2)*(7162*sqrt(35)*sqrt(31)*(4*x^2 + 4*x + 1) - 42875
*sqrt(31)*(4*x^2 + 4*x + 1))*sqrt(7162*sqrt(35) + 42875)*log(-6512712500/199*sqrt(217)*35^(1/4)*sqrt(2)*(4*sqr
t(35)*sqrt(31) - 19*sqrt(31))*sqrt(2*x + 1)*sqrt(7162*sqrt(35) + 42875) + 14132586125000*x + 1413258612500*sqr
t(35) + 7066293062500) + 374828440*(24*x + 19)*sqrt(2*x + 1))/(4*x^2 + 4*x + 1)

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giac [B]  time = 1.15, size = 599, normalized size = 2.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/91177975*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 9310*(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)) - 1/91177975*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 9310
*(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)) - 1/182355950*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqr
t(35) + 2450) + 9310*(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) + 1/182355950*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35)
+ 2450) + 9310*(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) - 4/147*(24*x + 19)/(2*x + 1)^(3/2)

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maple [B]  time = 0.16, size = 625, normalized size = 2.35 \[ \frac {135 \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 )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {178 \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 )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {76 \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 )}{343 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {135 \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 )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {178 \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 )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {76 \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 )}{343 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {27 \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 )}{3038}-\frac {89 \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 )}{10633}-\frac {27 \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 )}{3038}+\frac {89 \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 )}{10633}-\frac {4}{21 \left (2 x +1\right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/3038*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)+89/10633*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)+135/1519/(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))-178/10633/(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))-76/343/(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))+27/3038*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)-89/10633*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)+135/1519/(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))-178/10633/(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))-76/343/(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)
)-4/21/(2*x+1)^(3/2)-16/49/(2*x+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 187, normalized size = 0.70 \[ -\frac {\frac {32\,x}{49}+\frac {76}{147}}{{\left (2\,x+1\right )}^{3/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)*25472i)/(720600125*((31^(1/2)*483968
i)/102942875 - 4534016/102942875)) - (50944*31^(1/2)*217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))
/(22338603875*((31^(1/2)*483968i)/102942875 - 4534016/102942875)))*(- 31^(1/2)*199i - 7162)^(1/2)*2i)/10633 -
((32*x)/49 + 76/147)/(2*x + 1)^(3/2) - (217^(1/2)*atan((217^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)
*25472i)/(720600125*((31^(1/2)*483968i)/102942875 + 4534016/102942875)) + (50944*31^(1/2)*217^(1/2)*(31^(1/2)*
199i - 7162)^(1/2)*(2*x + 1)^(1/2))/(22338603875*((31^(1/2)*483968i)/102942875 + 4534016/102942875)))*(31^(1/2
)*199i - 7162)^(1/2)*2i)/10633

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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