### 3.129 $$\int \frac{1+3 x+4 x^2}{(1+2 x)^3 (2+3 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=97 $\frac{351 x+358}{2662 \sqrt{3 x^2+2}}+\frac{2 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{2 \sqrt{3 x^2+2}}{121 (2 x+1)^2}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}}$

[Out]

(358 + 351*x)/(2662*Sqrt[2 + 3*x^2]) - (2*Sqrt[2 + 3*x^2])/(121*(1 + 2*x)^2) + (2*Sqrt[2 + 3*x^2])/(1331*(1 +
2*x)) - (322*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(1331*Sqrt[11])

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Rubi [A]  time = 0.122966, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.172, Rules used = {1647, 1651, 807, 725, 206} $\frac{351 x+358}{2662 \sqrt{3 x^2+2}}+\frac{2 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{2 \sqrt{3 x^2+2}}{121 (2 x+1)^2}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(358 + 351*x)/(2662*Sqrt[2 + 3*x^2]) - (2*Sqrt[2 + 3*x^2])/(121*(1 + 2*x)^2) + (2*Sqrt[2 + 3*x^2])/(1331*(1 +
2*x)) - (322*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(1331*Sqrt[11])

Rule 1647

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

Rule 1651

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

Rule 807

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

Rule 725

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

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+3 x^2\right )^{3/2}} \, dx &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{1}{6} \int \frac{-\frac{2940}{1331}-\frac{7272 x}{1331}-\frac{8592 x^2}{1331}}{(1+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{1}{132} \int \frac{\frac{3768}{121}+\frac{7800 x}{121}}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}+\frac{322 \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx}{1331}\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{322 \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )}{1331}\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{1331 \sqrt{11}}\\ \end{align*}

Mathematica [A]  time = 0.088082, size = 78, normalized size = 0.8 $\frac{11 \left (1428 x^3+2716 x^2+1799 x+278\right )-644 (2 x+1)^2 \sqrt{33 x^2+22} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{29282 (2 x+1)^2 \sqrt{3 x^2+2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(11*(278 + 1799*x + 2716*x^2 + 1428*x^3) - 644*(1 + 2*x)^2*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^
2]])/(29282*(1 + 2*x)^2*Sqrt[2 + 3*x^2])

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Maple [A]  time = 0.059, size = 107, normalized size = 1.1 \begin{align*}{\frac{7}{484} \left ( x+{\frac{1}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{161}{1331}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{357\,x}{2662}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{322\,\sqrt{11}}{14641}{\it Artanh} \left ({\frac{ \left ( 8-6\,x \right ) \sqrt{11}}{11}{\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-12\,x+5}}}} \right ) }-{\frac{1}{88} \left ( x+{\frac{1}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}} \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+2)^(3/2),x)

[Out]

7/484/(x+1/2)/(3*(x+1/2)^2-3*x+5/4)^(1/2)+161/1331/(3*(x+1/2)^2-3*x+5/4)^(1/2)+357/2662*x/(3*(x+1/2)^2-3*x+5/4
)^(1/2)-322/14641*11^(1/2)*arctanh(2/11*(4-3*x)*11^(1/2)/(12*(x+1/2)^2-12*x+5)^(1/2))-1/88/(x+1/2)^2/(3*(x+1/2
)^2-3*x+5/4)^(1/2)

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Maxima [A]  time = 1.53963, size = 167, normalized size = 1.72 \begin{align*} \frac{322}{14641} \, \sqrt{11} \operatorname{arsinh}\left (\frac{\sqrt{6} x}{2 \,{\left | 2 \, x + 1 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{357 \, x}{2662 \, \sqrt{3 \, x^{2} + 2}} + \frac{161}{1331 \, \sqrt{3 \, x^{2} + 2}} - \frac{1}{22 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 4 \, \sqrt{3 \, x^{2} + 2} x + \sqrt{3 \, x^{2} + 2}\right )}} + \frac{7}{242 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + \sqrt{3 \, x^{2} + 2}\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+2)^(3/2),x, algorithm="maxima")

[Out]

322/14641*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 357/2662*x/sqrt(3*x^2 + 2)
+ 161/1331/sqrt(3*x^2 + 2) - 1/22/(4*sqrt(3*x^2 + 2)*x^2 + 4*sqrt(3*x^2 + 2)*x + sqrt(3*x^2 + 2)) + 7/242/(2*
sqrt(3*x^2 + 2)*x + sqrt(3*x^2 + 2))

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Fricas [A]  time = 1.65576, size = 321, normalized size = 3.31 \begin{align*} \frac{322 \, \sqrt{11}{\left (12 \, x^{4} + 12 \, x^{3} + 11 \, x^{2} + 8 \, x + 2\right )} \log \left (-\frac{\sqrt{11} \sqrt{3 \, x^{2} + 2}{\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \,{\left (1428 \, x^{3} + 2716 \, x^{2} + 1799 \, x + 278\right )} \sqrt{3 \, x^{2} + 2}}{29282 \,{\left (12 \, x^{4} + 12 \, x^{3} + 11 \, x^{2} + 8 \, x + 2\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+2)^(3/2),x, algorithm="fricas")

[Out]

1/29282*(322*sqrt(11)*(12*x^4 + 12*x^3 + 11*x^2 + 8*x + 2)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 -
12*x + 19)/(4*x^2 + 4*x + 1)) + 11*(1428*x^3 + 2716*x^2 + 1799*x + 278)*sqrt(3*x^2 + 2))/(12*x^4 + 12*x^3 + 1
1*x^2 + 8*x + 2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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+2)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.30109, size = 265, normalized size = 2.73 \begin{align*} \frac{322}{14641} \, \sqrt{11} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{11} - \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{11} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{351 \, x + 358}{2662 \, \sqrt{3 \, x^{2} + 2}} + \frac{36 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 48 \, \sqrt{3} x + 8 \, \sqrt{3} - 48 \, \sqrt{3 \, x^{2} + 2}}{1331 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\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+2)^(3/2),x, algorithm="giac")

[Out]

322/14641*sqrt(11)*log(-abs(-2*sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(11) + s
qrt(3) - 2*sqrt(3*x^2 + 2))) + 1/2662*(351*x + 358)/sqrt(3*x^2 + 2) + 1/1331*(36*(sqrt(3)*x - sqrt(3*x^2 + 2))
^3 - sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 48*sqrt(3)*x + 8*sqrt(3) - 48*sqrt(3*x^2 + 2))/((sqrt(3)*x - sq
rt(3*x^2 + 2))^2 + sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2