### 3.257 $$\int \frac{1+3 x+4 x^2}{(1+2 x)^3 (2-x+3 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=135 $\frac{2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}-\frac{144 \sqrt{3 x^2-x+2}}{28561 (2 x+1)}-\frac{8 \sqrt{3 x^2-x+2}}{2197 (2 x+1)^2}+\frac{12 (103526 x+25771)}{15108769 \sqrt{3 x^2-x+2}}-\frac{2084 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{28561 \sqrt{13}}$

[Out]

(2*(2363 + 3693*x))/(151593*(2 - x + 3*x^2)^(3/2)) + (12*(25771 + 103526*x))/(15108769*Sqrt[2 - x + 3*x^2]) -
(8*Sqrt[2 - x + 3*x^2])/(2197*(1 + 2*x)^2) - (144*Sqrt[2 - x + 3*x^2])/(28561*(1 + 2*x)) - (2084*ArcTanh[(9 -
8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(28561*Sqrt[13])

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Rubi [A]  time = 0.212556, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.156, Rules used = {1646, 1650, 806, 724, 206} $\frac{2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}-\frac{144 \sqrt{3 x^2-x+2}}{28561 (2 x+1)}-\frac{8 \sqrt{3 x^2-x+2}}{2197 (2 x+1)^2}+\frac{12 (103526 x+25771)}{15108769 \sqrt{3 x^2-x+2}}-\frac{2084 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{28561 \sqrt{13}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 - x + 3*x^2)^(5/2)),x]

[Out]

(2*(2363 + 3693*x))/(151593*(2 - x + 3*x^2)^(3/2)) + (12*(25771 + 103526*x))/(15108769*Sqrt[2 - x + 3*x^2]) -
(8*Sqrt[2 - x + 3*x^2])/(2197*(1 + 2*x)^2) - (144*Sqrt[2 - x + 3*x^2])/(28561*(1 + 2*x)) - (2084*ArcTanh[(9 -
8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(28561*Sqrt[13])

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*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*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{32433}{2197}+\frac{106830 x}{2197}+\frac{160116 x^2}{2197}+\frac{59088 x^3}{2197}}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{12 (25771+103526 x)}{15108769 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{\frac{1434648}{28561}+\frac{3345396 x}{28561}+\frac{3097824 x^2}{28561}}{(1+2 x)^3 \sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{12 (25771+103526 x)}{15108769 \sqrt{2-x+3 x^2}}-\frac{8 \sqrt{2-x+3 x^2}}{2197 (1+2 x)^2}-\frac{2 \int \frac{-\frac{2167842}{2197}-\frac{2850252 x}{2197}}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx}{20631}\\ &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{12 (25771+103526 x)}{15108769 \sqrt{2-x+3 x^2}}-\frac{8 \sqrt{2-x+3 x^2}}{2197 (1+2 x)^2}-\frac{144 \sqrt{2-x+3 x^2}}{28561 (1+2 x)}+\frac{2084 \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{28561}\\ &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{12 (25771+103526 x)}{15108769 \sqrt{2-x+3 x^2}}-\frac{8 \sqrt{2-x+3 x^2}}{2197 (1+2 x)^2}-\frac{144 \sqrt{2-x+3 x^2}}{28561 (1+2 x)}-\frac{4168 \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )}{28561}\\ &=\frac{2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac{12 (25771+103526 x)}{15108769 \sqrt{2-x+3 x^2}}-\frac{8 \sqrt{2-x+3 x^2}}{2197 (1+2 x)^2}-\frac{144 \sqrt{2-x+3 x^2}}{28561 (1+2 x)}-\frac{2084 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{28561 \sqrt{13}}\\ \end{align*}

Mathematica [A]  time = 0.0764865, size = 89, normalized size = 0.66 $\frac{2 \left (20304864 x^5+20074356 x^4+19381992 x^3+21890266 x^2+10777477 x+847141\right )}{45326307 (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}}-\frac{2084 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{28561 \sqrt{13}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 - x + 3*x^2)^(5/2)),x]

[Out]

(2*(847141 + 10777477*x + 21890266*x^2 + 19381992*x^3 + 20074356*x^4 + 20304864*x^5))/(45326307*(1 + 2*x)^2*(2
- x + 3*x^2)^(3/2)) - (2084*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(28561*Sqrt[13])

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Maple [A]  time = 0.061, size = 148, normalized size = 1.1 \begin{align*} -{\frac{1}{104} \left ( x+{\frac{1}{2}} \right ) ^{-2} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{338} \left ( x+{\frac{1}{2}} \right ) ^{-1} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{521}{13182} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-886+5316\,x}{151593} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-188008+1128048\,x}{15108769}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{1042}{28561}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{2084\,\sqrt{13}}{371293}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(5/2),x)

[Out]

-1/104/(x+1/2)^2/(3*(x+1/2)^2-4*x+5/4)^(3/2)-1/338/(x+1/2)/(3*(x+1/2)^2-4*x+5/4)^(3/2)+521/13182/(3*(x+1/2)^2-
4*x+5/4)^(3/2)+886/151593*(-1+6*x)/(3*(x+1/2)^2-4*x+5/4)^(3/2)+188008/15108769*(-1+6*x)/(3*(x+1/2)^2-4*x+5/4)^
(1/2)+1042/28561/(3*(x+1/2)^2-4*x+5/4)^(1/2)-2084/371293*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^
2-16*x+5)^(1/2))

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Maxima [A]  time = 1.49437, size = 235, normalized size = 1.74 \begin{align*} \frac{2084}{371293} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{1128048 \, x}{15108769 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{363210}{15108769 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{1772 \, x}{50531 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{1}{26 \,{\left (4 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x^{2} + 4 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{1}{169 \,{\left (2 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}\right )}} + \frac{10211}{303186 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(5/2),x, algorithm="maxima")

[Out]

2084/371293*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 1128048/15108769*x/s
qrt(3*x^2 - x + 2) + 363210/15108769/sqrt(3*x^2 - x + 2) + 1772/50531*x/(3*x^2 - x + 2)^(3/2) - 1/26/(4*(3*x^2
- x + 2)^(3/2)*x^2 + 4*(3*x^2 - x + 2)^(3/2)*x + (3*x^2 - x + 2)^(3/2)) - 1/169/(2*(3*x^2 - x + 2)^(3/2)*x +
(3*x^2 - x + 2)^(3/2)) + 10211/303186/(3*x^2 - x + 2)^(3/2)

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Fricas [A]  time = 1.19942, size = 460, normalized size = 3.41 \begin{align*} \frac{2 \,{\left (826827 \, \sqrt{13}{\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 13 \,{\left (20304864 \, x^{5} + 20074356 \, x^{4} + 19381992 \, x^{3} + 21890266 \, x^{2} + 10777477 \, x + 847141\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{589241991 \,{\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(5/2),x, algorithm="fricas")

[Out]

2/589241991*(826827*sqrt(13)*(36*x^6 + 12*x^5 + 37*x^4 + 30*x^3 + 13*x^2 + 12*x + 4)*log(-(4*sqrt(13)*sqrt(3*x
^2 - x + 2)*(8*x - 9) + 220*x^2 - 196*x + 185)/(4*x^2 + 4*x + 1)) + 13*(20304864*x^5 + 20074356*x^4 + 19381992
*x^3 + 21890266*x^2 + 10777477*x + 847141)*sqrt(3*x^2 - x + 2))/(36*x^6 + 12*x^5 + 37*x^4 + 30*x^3 + 13*x^2 +
12*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{3} \left (3 x^{2} - x + 2\right )^{\frac{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2+3*x+1)/(1+2*x)**3/(3*x**2-x+2)**(5/2),x)

[Out]

Integral((4*x**2 + 3*x + 1)/((2*x + 1)**3*(3*x**2 - x + 2)**(5/2)), x)

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Giac [B]  time = 1.23155, size = 315, normalized size = 2.33 \begin{align*} \frac{2084}{371293} \, \sqrt{13} \log \left (-\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{13} - 2 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} - x + 2} \right |}}{2 \,{\left (2 \, \sqrt{3} x - \sqrt{13} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} - x + 2}\right )}}\right ) + \frac{2 \,{\left (3 \,{\left (6 \,{\left (310578 \, x - 26213\right )} x + 1455755\right )} x + 1634293\right )}}{45326307 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (66 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{3} + 21 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{2} - 1015 \, \sqrt{3} x + 431 \, \sqrt{3} + 1015 \, \sqrt{3 \, x^{2} - x + 2}\right )}}{28561 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(5/2),x, algorithm="giac")

[Out]

2084/371293*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x
- sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2))) + 2/45326307*(3*(6*(310578*x - 26213)*x + 1455755)*x + 1634293)
/(3*x^2 - x + 2)^(3/2) - 8/28561*(66*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^3 + 21*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
- x + 2))^2 - 1015*sqrt(3)*x + 431*sqrt(3) + 1015*sqrt(3*x^2 - x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2
+ 2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) - 5)^2