3.251 \(\int \frac {1+3 x+4 x^2}{(1+2 x)^3 (2-x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=112 \[ \frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}-\frac {4 \sqrt {3 x^2-x+2}}{2197 (2 x+1)}-\frac {2 \sqrt {3 x^2-x+2}}{169 (2 x+1)^2}-\frac {487 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2197 \sqrt {13}} \]

[Out]

-487/28561*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1/2))*13^(1/2)+2/50531*(2363+3693*x)/(3*x^2-x+2)^(1/2)-2
/169*(3*x^2-x+2)^(1/2)/(1+2*x)^2-4/2197*(3*x^2-x+2)^(1/2)/(1+2*x)

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Rubi [A]  time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {2 (3693 x+2363)}{50531 \sqrt {3 x^2-x+2}}-\frac {4 \sqrt {3 x^2-x+2}}{2197 (2 x+1)}-\frac {2 \sqrt {3 x^2-x+2}}{169 (2 x+1)^2}-\frac {487 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2197 \sqrt {13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2363 + 3693*x))/(50531*Sqrt[2 - x + 3*x^2]) - (2*Sqrt[2 - x + 3*x^2])/(169*(1 + 2*x)^2) - (4*Sqrt[2 - x +
3*x^2])/(2197*(1 + 2*x)) - (487*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

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])

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 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 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]

Rubi steps

\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac {2 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}+\frac {2}{23} \int \frac {\frac {8349}{2197}+\frac {20838 x}{2197}+\frac {23828 x^2}{2197}}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}-\frac {2 \sqrt {2-x+3 x^2}}{169 (1+2 x)^2}-\frac {1}{299} \int \frac {-\frac {11615}{169}-\frac {22034 x}{169}}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}-\frac {2 \sqrt {2-x+3 x^2}}{169 (1+2 x)^2}-\frac {4 \sqrt {2-x+3 x^2}}{2197 (1+2 x)}+\frac {487 \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx}{2197}\\ &=\frac {2 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}-\frac {2 \sqrt {2-x+3 x^2}}{169 (1+2 x)^2}-\frac {4 \sqrt {2-x+3 x^2}}{2197 (1+2 x)}-\frac {974 \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 (2363+3693 x)}{50531 \sqrt {2-x+3 x^2}}-\frac {2 \sqrt {2-x+3 x^2}}{169 (1+2 x)^2}-\frac {4 \sqrt {2-x+3 x^2}}{2197 (1+2 x)}-\frac {487 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{2197 \sqrt {13}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.71 \[ \frac {2 \left (14496 x^3+23281 x^2+13306 x+1673\right )}{50531 (2 x+1)^2 \sqrt {3 x^2-x+2}}-\frac {487 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2197 \sqrt {13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1673 + 13306*x + 23281*x^2 + 14496*x^3))/(50531*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2]) - (487*ArcTanh[(9 - 8*x)/
(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])

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fricas [A]  time = 0.64, size = 126, normalized size = 1.12 \[ \frac {11201 \, \sqrt {13} {\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 2\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 52 \, {\left (14496 \, x^{3} + 23281 \, x^{2} + 13306 \, x + 1673\right )} \sqrt {3 \, x^{2} - x + 2}}{1313806 \, {\left (12 \, x^{4} + 8 \, x^{3} + 7 \, x^{2} + 7 \, x + 2\right )}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

1/1313806*(11201*sqrt(13)*(12*x^4 + 8*x^3 + 7*x^2 + 7*x + 2)*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)) + 52*(14496*x^3 + 23281*x^2 + 13306*x + 1673)*sqrt(3*x^2 - x + 2))/(
12*x^4 + 8*x^3 + 7*x^2 + 7*x + 2)

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giac [B]  time = 0.31, size = 223, normalized size = 1.99 \[ \frac {487}{28561} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2 \, {\left (3693 \, x + 2363\right )}}{50531 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2 \, {\left (62 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} - 37 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 263 \, \sqrt {3} x - 71 \, \sqrt {3} - 263 \, \sqrt {3 \, x^{2} - x + 2}\right )}}{2197 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \]

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)^(3/2),x, algorithm="giac")

[Out]

487/28561*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))) + 2/50531*(3693*x + 2363)/sqrt(3*x^2 - x + 2) + 2/2197*(62*(sqrt(
3)*x - sqrt(3*x^2 - x + 2))^3 - 37*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 263*sqrt(3)*x - 71*sqrt(3) -
263*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

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maple [A]  time = 0.01, size = 111, normalized size = 0.99 \[ -\frac {487 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{28561}+\frac {487}{4394 \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}+\frac {\frac {7248 x}{50531}-\frac {1208}{50531}}{\sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}+\frac {3}{338 \left (x +\frac {1}{2}\right ) \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}-\frac {1}{104 \left (x +\frac {1}{2}\right )^{2} \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

487/4394/(-4*x+3*(x+1/2)^2+5/4)^(1/2)+1208/50531*(6*x-1)/(-4*x+3*(x+1/2)^2+5/4)^(1/2)-487/28561*13^(1/2)*arcta
nh(2/13*(-4*x+9/2)*13^(1/2)/(-16*x+12*(x+1/2)^2+5)^(1/2))+3/338/(x+1/2)/(-4*x+3*(x+1/2)^2+5/4)^(1/2)-1/104/(x+
1/2)^2/(-4*x+3*(x+1/2)^2+5/4)^(1/2)

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maxima [A]  time = 0.97, size = 145, normalized size = 1.29 \[ \frac {487}{28561} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {7248 \, x}{50531 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {8785}{101062 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {1}{26 \, {\left (4 \, \sqrt {3 \, x^{2} - x + 2} x^{2} + 4 \, \sqrt {3 \, x^{2} - x + 2} x + \sqrt {3 \, x^{2} - x + 2}\right )}} + \frac {3}{169 \, {\left (2 \, \sqrt {3 \, x^{2} - x + 2} x + \sqrt {3 \, x^{2} - x + 2}\right )}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

487/28561*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 7248/50531*x/sqrt(3*x^
2 - x + 2) + 8785/101062/sqrt(3*x^2 - x + 2) - 1/26/(4*sqrt(3*x^2 - x + 2)*x^2 + 4*sqrt(3*x^2 - x + 2)*x + sqr
t(3*x^2 - x + 2)) + 3/169/(2*sqrt(3*x^2 - x + 2)*x + sqrt(3*x^2 - x + 2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^3\,{\left (3\,x^2-x+2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{3} \left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

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

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