3.135 \(\int \frac{1+3 x+4 x^2}{(1+2 x)^3 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)^2}+\frac{2133 x+1216}{29282 \sqrt{3 x^2+2}}-\frac{1216 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{14641 \sqrt{11}} \]

[Out]

(358 + 351*x)/(7986*(2 + 3*x^2)^(3/2)) + (1216 + 2133*x)/(29282*Sqrt[2 + 3*x^2]) - (8*Sqrt[2 + 3*x^2])/(1331*(
1 + 2*x)^2) - (8*Sqrt[2 + 3*x^2])/(1331*(1 + 2*x)) - (1216*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(146
41*Sqrt[11])

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Rubi [A]  time = 0.20628, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)^2}+\frac{2133 x+1216}{29282 \sqrt{3 x^2+2}}-\frac{1216 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{14641 \sqrt{11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(358 + 351*x)/(7986*(2 + 3*x^2)^(3/2)) + (1216 + 2133*x)/(29282*Sqrt[2 + 3*x^2]) - (8*Sqrt[2 + 3*x^2])/(1331*(
1 + 2*x)^2) - (8*Sqrt[2 + 3*x^2])/(1331*(1 + 2*x)) - (1216*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(146
41*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 )^{5/2}} \, dx &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}-\frac{1}{18} \int \frac{-\frac{10926}{1331}-\frac{3132 x}{121}-\frac{51048 x^2}{1331}-\frac{16848 x^3}{1331}}{(1+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac{1216+2133 x}{29282 \sqrt{2+3 x^2}}+\frac{1}{108} \int \frac{\frac{245376}{14641}+\frac{544320 x}{14641}+\frac{525312 x^2}{14641}}{(1+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac{1216+2133 x}{29282 \sqrt{2+3 x^2}}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)^2}-\frac{\int \frac{-\frac{338688}{1331}-\frac{468288 x}{1331}}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx}{2376}\\ &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac{1216+2133 x}{29282 \sqrt{2+3 x^2}}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)^2}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)}+\frac{1216 \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx}{14641}\\ &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac{1216+2133 x}{29282 \sqrt{2+3 x^2}}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)^2}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{1216 \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )}{14641}\\ &=\frac{358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac{1216+2133 x}{29282 \sqrt{2+3 x^2}}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)^2}-\frac{8 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{1216 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{14641 \sqrt{11}}\\ \end{align*}

Mathematica [A]  time = 0.105104, size = 75, normalized size = 0.64 \[ \frac{\frac{11 \left (67284 x^5+111060 x^4+116937 x^3+109844 x^2+57371 x+7010\right )}{(2 x+1)^2 \left (3 x^2+2\right )^{3/2}}-7296 \sqrt{11} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{966306} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(7010 + 57371*x + 109844*x^2 + 116937*x^3 + 111060*x^4 + 67284*x^5))/((1 + 2*x)^2*(2 + 3*x^2)^(3/2)) - 72
96*Sqrt[11]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/966306

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Maple [A]  time = 0.063, size = 140, normalized size = 1.2 \begin{align*}{\frac{1}{484} \left ( x+{\frac{1}{2}} \right ) ^{-1} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{152}{3993} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{87\,x}{2662} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1869\,x}{29282}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{608}{14641}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{1216\,\sqrt{11}}{161051}{\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} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,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)^3/(3*x^2+2)^(5/2),x)

[Out]

1/484/(x+1/2)/(3*(x+1/2)^2-3*x+5/4)^(3/2)+152/3993/(3*(x+1/2)^2-3*x+5/4)^(3/2)+87/2662*x/(3*(x+1/2)^2-3*x+5/4)
^(3/2)+1869/29282*x/(3*(x+1/2)^2-3*x+5/4)^(1/2)+608/14641/(3*(x+1/2)^2-3*x+5/4)^(1/2)-1216/161051*11^(1/2)*arc
tanh(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)^(3/2)

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Maxima [A]  time = 1.52314, size = 198, normalized size = 1.69 \begin{align*} \frac{1216}{161051} \, \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{1869 \, x}{29282 \, \sqrt{3 \, x^{2} + 2}} + \frac{608}{14641 \, \sqrt{3 \, x^{2} + 2}} + \frac{87 \, x}{2662 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{1}{22 \,{\left (4 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + 4 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{1}{242 \,{\left (2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{152}{3993 \,{\left (3 \, x^{2} + 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+2)^(5/2),x, algorithm="maxima")

[Out]

1216/161051*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 1869/29282*x/sqrt(3*x^2
+ 2) + 608/14641/sqrt(3*x^2 + 2) + 87/2662*x/(3*x^2 + 2)^(3/2) - 1/22/(4*(3*x^2 + 2)^(3/2)*x^2 + 4*(3*x^2 + 2)
^(3/2)*x + (3*x^2 + 2)^(3/2)) + 1/242/(2*(3*x^2 + 2)^(3/2)*x + (3*x^2 + 2)^(3/2)) + 152/3993/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.70003, size = 417, normalized size = 3.56 \begin{align*} \frac{3648 \, \sqrt{11}{\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, x + 4\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 (67284 \, x^{5} + 111060 \, x^{4} + 116937 \, x^{3} + 109844 \, x^{2} + 57371 \, x + 7010\right )} \sqrt{3 \, x^{2} + 2}}{966306 \,{\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, 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+2)^(5/2),x, algorithm="fricas")

[Out]

1/966306*(3648*sqrt(11)*(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 + 16*x + 4)*log(-(sqrt(11)*sqrt(3*x^2 + 2)
*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)) + 11*(67284*x^5 + 111060*x^4 + 116937*x^3 + 109844*x^2 + 5
7371*x + 7010)*sqrt(3*x^2 + 2))/(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 + 16*x + 4)

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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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.32507, size = 247, normalized size = 2.11 \begin{align*} \frac{1216}{161051} \, \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{9 \,{\left ({\left (2133 \, x + 1216\right )} x + 1851\right )} x + 11234}{87846 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4 \,{\left (\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 24 \, \sqrt{3} x - 8 \, \sqrt{3} - 24 \, \sqrt{3 \, x^{2} + 2}\right )}}{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)^(5/2),x, algorithm="giac")

[Out]

1216/161051*sqrt(11)*log(-abs(-2*sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(11) +
sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/87846*(9*((2133*x + 1216)*x + 1851)*x + 11234)/(3*x^2 + 2)^(3/2) + 4/1331*(
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 24*sqrt(3)*x - 8*sqrt(3) - 24*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*
x^2 + 2))^2 + sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2