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

Optimal. Leaf size=300 \[ \frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{6727} \]

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Rubi [A]  time = 0.38, antiderivative size = 300, 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 = {736, 822, 826, 1169, 634, 618, 204, 628} \begin {gather*} \frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + 2*x]*(3 + 10*x))/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(599 + 1790*x))/(13454*(2 + 3*x + 5*x^2))
- (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + S
qrt[35])]])/6727 + (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454 - (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2
+ Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454

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 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], 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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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 {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {-27-50 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1439-1790 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-1088-1790 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\left (6265+544 \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 )}{470890}+\frac {\left (6265+544 \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 )}{470890}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\left (6265+544 \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 )}{235445}-\frac {\left (6265+544 \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 )}{235445}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}\\ \end {align*}

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Mathematica [C]  time = 0.64, size = 151, normalized size = 0.50 \begin {gather*} \frac {\frac {1085 \sqrt {2 x+1} \left (8950 x^3+8365 x^2+7547 x+1849\right )}{\left (5 x^2+3 x+2\right )^2}+2 \sqrt {10-5 i \sqrt {31}} \left (16864-7353 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+2 \sqrt {10+5 i \sqrt {31}} \left (16864+7353 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{14597590} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1085*Sqrt[1 + 2*x]*(1849 + 7547*x + 8365*x^2 + 8950*x^3))/(2 + 3*x + 5*x^2)^2 + 2*Sqrt[10 - (5*I)*Sqrt[31]]*
(16864 - (7353*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(16864
+ (7353*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/14597590

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IntegrateAlgebraic [C]  time = 0.97, size = 167, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {2 x+1} \left (4475 (2 x+1)^3-5060 (2 x+1)^2+11789 (2 x+1)-3808\right )}{6727 \left (5 (2 x+1)^2-4 (2 x+1)+7\right )^2}+\frac {\sqrt {\frac {1}{217} \left (9651062-825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )}{6727}+\frac {\sqrt {\frac {1}{217} \left (9651062+825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )}{6727} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + 2*x]*(-3808 + 11789*(1 + 2*x) - 5060*(1 + 2*x)^2 + 4475*(1 + 2*x)^3))/(6727*(7 - 4*(1 + 2*x) + 5*(
1 + 2*x)^2)^2) + (Sqrt[(9651062 - (825499*I)*Sqrt[31])/217]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])
/6727 + (Sqrt[(9651062 + (825499*I)*Sqrt[31])/217]*ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/6727

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fricas [B]  time = 0.45, size = 652, normalized size = 2.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/308559878029380460*(102361876*121835^(1/4)*sqrt(217)*sqrt(118)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4
)*sqrt(9651062*sqrt(35) + 63240625)*arctan(1/37203599497016727197675*121835^(3/4)*sqrt(26629)*sqrt(2065)*sqrt(
217)*sqrt(118)*sqrt(121835^(1/4)*sqrt(217)*sqrt(118)*(179*sqrt(35)*sqrt(31) - 544*sqrt(31))*sqrt(2*x + 1)*sqrt
(9651062*sqrt(35) + 63240625) + 105688636970*x + 10568863697*sqrt(35) + 52844318485)*sqrt(9651062*sqrt(35) + 6
3240625)*(544*sqrt(35) - 6265) - 1/21824703534305*121835^(3/4)*sqrt(217)*sqrt(118)*sqrt(2*x + 1)*sqrt(9651062*
sqrt(35) + 63240625)*(544*sqrt(35) - 6265) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 102361876*121835^(1/4)*
sqrt(217)*sqrt(118)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(9651062*sqrt(35) + 63240625)*arctan(1/
1139360234596137270428796875*121835^(3/4)*sqrt(26629)*sqrt(217)*sqrt(118)*sqrt(-1936744140625*121835^(1/4)*sqr
t(217)*sqrt(118)*(179*sqrt(35)*sqrt(31) - 544*sqrt(31))*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 2046
91848382290253906250*x + 20469184838229025390625*sqrt(35) + 102345924191145126953125)*sqrt(9651062*sqrt(35) +
63240625)*(544*sqrt(35) - 6265) - 1/21824703534305*121835^(3/4)*sqrt(217)*sqrt(118)*sqrt(2*x + 1)*sqrt(9651062
*sqrt(35) + 63240625)*(544*sqrt(35) - 6265) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 121835^(1/4)*sqrt(217)
*sqrt(118)*(9651062*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 63240625*sqrt(31)*(25*x^4 + 30*x
^3 + 29*x^2 + 12*x + 4))*sqrt(9651062*sqrt(35) + 63240625)*log(1936744140625/26629*121835^(1/4)*sqrt(217)*sqrt
(118)*(179*sqrt(35)*sqrt(31) - 544*sqrt(31))*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 768680192205078
1250*x + 768680192205078125*sqrt(35) + 3843400961025390625) - 121835^(1/4)*sqrt(217)*sqrt(118)*(9651062*sqrt(3
5)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 63240625*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*s
qrt(9651062*sqrt(35) + 63240625)*log(-1936744140625/26629*121835^(1/4)*sqrt(217)*sqrt(118)*(179*sqrt(35)*sqrt(
31) - 544*sqrt(31))*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 7686801922050781250*x + 7686801922050781
25*sqrt(35) + 3843400961025390625) + 22934434222490*(8950*x^3 + 8365*x^2 + 7547*x + 1849)*sqrt(2*x + 1))/(25*x
^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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giac [B]  time = 1.43, size = 642, normalized size = 2.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/100139467400*sqrt(31)*(37590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 179*sqrt(31
)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 358*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 75180*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 1066240*
(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)) + 1/100139467400*sqrt(31)*(37590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 179*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 358*(7/5)^(3/4)*(140*sqrt(3
5) + 2450)^(3/2) + 75180*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)
*sqrt(-140*sqrt(35) + 2450) + 1066240*(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)) + 1/200278934800*sqrt(31)*(179*sqrt
(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 37590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35
) - 35) - 75180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 358*(7/5)^(3/4)*(-140*sqrt(35) + 24
50)^(3/2) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 1066240*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2
450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/200278934800*sqrt(3
1)*(179*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 37590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450
)*(2*sqrt(35) - 35) - 75180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 358*(7/5)^(3/4)*(-140*s
qrt(35) + 2450)^(3/2) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 1066240*(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) + 2/6727*(
4475*(2*x + 1)^(7/2) - 5060*(2*x + 1)^(5/2) + 11789*(2*x + 1)^(3/2) - 3808*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x
+ 3)^2

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maple [B]  time = 1.45, size = 1108, normalized size = 3.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2919518*(2/21475*5^(1/2)*(-13012793430*5^(1/2)[phone]*7^(1/2))/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(3/2)+1
/107375/(-390+40*5^(1/2)*7^(1/2))*(-214587133600*5^(1/2)[phone]*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(2*
x+1)+2/107375*(-141628999400*5^(1/2)*7^(1/2)[phone])/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(1/2)+1/107375*(-
76332028500*7^(1/2)[phone]*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)/(-390+40*5^(1/2)*7^(1/2)))/(2*x+1/5*5^(1/
2)*7^(1/2)+1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+1)^2-161475/417074/(20*5^(1/2)*7^(1/2)-195)*l
n(10*x+5+35^(1/2)+(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))*5^(1/2)*(2*35^(1/2)+4)^(1/2)+2065235/5839036/(20*5^(1/
2)*7^(1/2)-195)*ln(10*x+5+35^(1/2)+(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)*7^(1/2)+161475/2
08537/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+1
0*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*5^(1/2)*(2*35^(1/2)+4)^(1/2)-2065235/2919518/(20*5^(1/2)*7^(1/2)-195
)/(-20+10*35^(1/2))^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^
(1/2))^(1/2)*(2*35^(1/2)+4)^(1/2)*7^(1/2)-212160/47089/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan
((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)+108800/6727/(20*5^(1/2)*7^(1/2)-1
95)/(-20+10*35^(1/2))^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))+5/291951
8*(2/21475*5^(1/2)*(-13012793430*5^(1/2)[phone]*7^(1/2))/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(3/2)-1/107375/
(-390+40*5^(1/2)*7^(1/2))*(-214587133600*5^(1/2)[phone]*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(2*x+1)+2/1
07375*(-141628999400*5^(1/2)*7^(1/2)[phone])/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(1/2)-1/107375*(-76332028
500*7^(1/2)[phone]*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)/(-390+40*5^(1/2)*7^(1/2)))/(2*x+1/5*5^(1/2)*7^(1/
2)-1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+1)^2+161475/417074/(20*5^(1/2)*7^(1/2)-195)*ln(10*x+5
+35^(1/2)-(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))*5^(1/2)*(2*35^(1/2)+4)^(1/2)-2065235/5839036/(20*5^(1/2)*7^(1/
2)-195)*ln(10*x+5+35^(1/2)-(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))*(2*35^(1/2)+4)^(1/2)*7^(1/2)+161475/208537/(2
0*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1
/2))^(1/2))*(20+10*35^(1/2))^(1/2)*5^(1/2)*(2*35^(1/2)+4)^(1/2)-2065235/2919518/(20*5^(1/2)*7^(1/2)-195)/(-20+
10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))
^(1/2)*(2*35^(1/2)+4)^(1/2)*7^(1/2)-212160/47089/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan((-(20
+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)+108800/6727/(20*5^(1/2)*7^(1/2)-195)/(
-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.06, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1088\,\sqrt {2\,x+1}}{24025}-\frac {23578\,{\left (2\,x+1\right )}^{3/2}}{168175}+\frac {2024\,{\left (2\,x+1\right )}^{5/2}}{33635}-\frac {358\,{\left (2\,x+1\right )}^{7/2}}{6727}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1088*(2*x + 1)^(1/2))/24025 - (23578*(2*x + 1)^(3/2))/168175 + (2024*(2*x + 1)^(5/2))/33635 - (358*(2*x + 1)
^(7/2))/6727)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (217^(1/2)*atan((2
17^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744i)/(1940202180875*((31^(1/2)*7476736i)/2771
71740125 - 101059632/277171740125)) - (27488*31^(1/2)*217^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)
^(1/2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 - 101059632/277171740125)))*(- 31^(1/2)*825499i - 96
51062)^(1/2)*1i)/1459759 + (217^(1/2)*atan((217^(1/2)*(31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744
i)/(1940202180875*((31^(1/2)*7476736i)/277171740125 + 101059632/277171740125)) + (27488*31^(1/2)*217^(1/2)*(31
^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 + 101059632
/277171740125)))*(31^(1/2)*825499i - 9651062)^(1/2)*1i)/1459759

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sympy [A]  time = 11.72, size = 199, normalized size = 0.66 \begin {gather*} \frac {286400 \left (2 x + 1\right )^{\frac {7}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac {323840 \left (2 x + 1\right )^{\frac {5}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + \frac {754496 \left (2 x + 1\right )^{\frac {3}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac {243712 \sqrt {2 x + 1}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + 64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left (t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

286400*(2*x + 1)**(7/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)**3 + 18512704*(2*x + 1)**2 - 1
506848) - 323840*(2*x + 1)**(5/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)**3 + 18512704*(2*x +
1)**2 - 1506848) + 754496*(2*x + 1)**(3/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)**3 + 18512
704*(2*x + 1)**2 - 1506848) - 243712*sqrt(2*x + 1)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)**3
+ 18512704*(2*x + 1)**2 - 1506848) + 64*RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 +
333142578125, Lambda(_t, _t*log(21632117045402271744*_t**3/158378125 + 10865340674816*_t/1108646875 + sqrt(2*
x + 1))))

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