### 3.256 $$\int \frac{1+3 x+4 x^2}{(1+2 x)^2 (2-x+3 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=110 $-\frac{24 (841-6633 x)}{1162213 \sqrt{3 x^2-x+2}}-\frac{16 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}}$

[Out]

(-2*(197 - 837*x))/(11661*(2 - x + 3*x^2)^(3/2)) - (24*(841 - 6633*x))/(1162213*Sqrt[2 - x + 3*x^2]) - (16*Sqr
t[2 - x + 3*x^2])/(2197*(1 + 2*x)) - (56*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

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Rubi [A]  time = 0.150562, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {1646, 806, 724, 206} $-\frac{24 (841-6633 x)}{1162213 \sqrt{3 x^2-x+2}}-\frac{16 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(197 - 837*x))/(11661*(2 - x + 3*x^2)^(3/2)) - (24*(841 - 6633*x))/(1162213*Sqrt[2 - x + 3*x^2]) - (16*Sqr
t[2 - x + 3*x^2])/(2197*(1 + 2*x)) - (56*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*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 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)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{2226}{169}+\frac{462 x}{13}+\frac{6696 x^2}{169}}{(1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{\frac{50784}{2197}+\frac{19044 x}{2197}}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}+\frac{56 \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{2197}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{112 \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )}{2197}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{2197 \sqrt{13}}\\ \end{align*}

Mathematica [A]  time = 0.0616853, size = 111, normalized size = 1.01 $\frac{26 \left (1318464 x^4+133308 x^3+1021566 x^2+569989 x-170239\right )-88872 \sqrt{13} \sqrt{3 x^2-x+2} \left (6 x^3+x^2+3 x+2\right ) \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{45326307 (2 x+1) \left (3 x^2-x+2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(26*(-170239 + 569989*x + 1021566*x^2 + 133308*x^3 + 1318464*x^4) - 88872*Sqrt[13]*Sqrt[2 - x + 3*x^2]*(2 + 3*
x + x^2 + 6*x^3)*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(45326307*(1 + 2*x)*(2 - x + 3*x^2)^(3/2
))

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Maple [A]  time = 0.057, size = 165, normalized size = 1.5 \begin{align*}{\frac{-2+12\,x}{69} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-16+96\,x}{529}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{7}{507} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-128+768\,x}{11661} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-10736+64416\,x}{1162213}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{28}{2197}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{56\,\sqrt{13}}{28561}{\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 ) }-{\frac{1}{26} \left ( x+{\frac{1}{2}} \right ) ^{-1} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/69*(-1+6*x)/(3*x^2-x+2)^(3/2)+16/529*(-1+6*x)/(3*x^2-x+2)^(1/2)+7/507/(3*(x+1/2)^2-4*x+5/4)^(3/2)-128/11661*
(-1+6*x)/(3*(x+1/2)^2-4*x+5/4)^(3/2)-10736/1162213*(-1+6*x)/(3*(x+1/2)^2-4*x+5/4)^(1/2)+28/2197/(3*(x+1/2)^2-4
*x+5/4)^(1/2)-56/28561*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2))-1/26/(x+1/2)/(3*(
x+1/2)^2-4*x+5/4)^(3/2)

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Maxima [A]  time = 1.48197, size = 169, normalized size = 1.54 \begin{align*} \frac{56}{28561} \, \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{146496 \, x}{1162213 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{9604}{1162213 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{420 \, x}{3887 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{1}{13 \,{\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{49}{11661 \,{\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)^2/(3*x^2-x+2)^(5/2),x, algorithm="maxima")

[Out]

56/28561*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 146496/1162213*x/sqrt(3
*x^2 - x + 2) - 9604/1162213/sqrt(3*x^2 - x + 2) + 420/3887*x/(3*x^2 - x + 2)^(3/2) - 1/13/(2*(3*x^2 - x + 2)^
(3/2)*x + (3*x^2 - x + 2)^(3/2)) - 49/11661/(3*x^2 - x + 2)^(3/2)

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Fricas [A]  time = 1.08474, size = 397, normalized size = 3.61 \begin{align*} \frac{2 \,{\left (22218 \, \sqrt{13}{\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, x^{2} + 4 \, 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 (1318464 \, x^{4} + 133308 \, x^{3} + 1021566 \, x^{2} + 569989 \, x - 170239\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{45326307 \,{\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, x^{2} + 4 \, 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)^2/(3*x^2-x+2)^(5/2),x, algorithm="fricas")

[Out]

2/45326307*(22218*sqrt(13)*(18*x^5 - 3*x^4 + 20*x^3 + 5*x^2 + 4*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*(1318464*x^4 + 133308*x^3 + 1021566*x^2 + 569989*x -
170239)*sqrt(3*x^2 - x + 2))/(18*x^5 - 3*x^4 + 20*x^3 + 5*x^2 + 4*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 )^{2} \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)**2/(3*x**2-x+2)**(5/2),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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