\(\int \frac {1+3 x+4 x^2}{(1+2 x) (2+3 x^2)^{5/2}} \, dx\) [133]

Optimal result

Integrand size = 29, antiderivative size = 73 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {-38+21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{121 \sqrt {11}} \]

[Out]

1/198*(-38+21*x)/(3*x^2+2)^(3/2)-8/1331*arctanh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)+1/726*(24+95*x
)/(3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1661, 837, 12, 739, 212} \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{121 \sqrt {11}}-\frac {38-21 x}{198 \left (3 x^2+2\right )^{3/2}}+\frac {95 x+24}{726 \sqrt {3 x^2+2}} \]

[In]

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

[Out]

-1/198*(38 - 21*x)/(2 + 3*x^2)^(3/2) + (24 + 95*x)/(726*Sqrt[2 + 3*x^2]) - (8*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt
[2 + 3*x^2])])/(121*Sqrt[11])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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

Rule 739

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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1661

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}-\frac {1}{18} \int \frac {-\frac {78}{11}-\frac {84 x}{11}}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}+\frac {\int \frac {864}{11 (1+2 x) \sqrt {2+3 x^2}} \, dx}{1188} \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}+\frac {8}{121} \int \frac {1}{(1+2 x) \sqrt {2+3 x^2}} \, dx \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8}{121} \text {Subst}\left (\int \frac {1}{11-x^2} \, dx,x,\frac {4-3 x}{\sqrt {2+3 x^2}}\right ) \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{121 \sqrt {11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {-274+801 x+216 x^2+855 x^3}{2178 \left (2+3 x^2\right )^{3/2}}-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{121 \sqrt {11}} \]

[In]

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

[Out]

(-274 + 801*x + 216*x^2 + 855*x^3)/(2178*(2 + 3*x^2)^(3/2)) - (8*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(121*Sq
rt[11])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.56 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01

method	result	size
trager	\(\frac {855 x^{3}+216 x^{2}+801 x -274}{2178 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{1331}\)	\(74\)
default	\(\frac {x}{12 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {x}{12 \sqrt {3 x^{2}+2}}-\frac {2}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {1}{33 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {x}{44 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {23 x}{484 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}+\frac {4}{121 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}-\frac {8 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{1331}\)	\(133\)

[In]

int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/2178*(855*x^3+216*x^2+801*x-274)/(3*x^2+2)^(3/2)+8/1331*RootOf(_Z^2-11)*ln((3*RootOf(_Z^2-11)*x+11*(3*x^2+2)
^(1/2)-4*RootOf(_Z^2-11))/(1+2*x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {72 \, \sqrt {11} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (855 \, x^{3} + 216 \, x^{2} + 801 \, x - 274\right )} \sqrt {3 \, x^{2} + 2}}{23958 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

[In]

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

[Out]

1/23958*(72*sqrt(11)*(9*x^4 + 12*x^2 + 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*(855*x^3 + 216*x^2 + 801*x - 274)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

Sympy [F(-1)]

Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{1331} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {95 \, x}{726 \, \sqrt {3 \, x^{2} + 2}} + \frac {4}{121 \, \sqrt {3 \, x^{2} + 2}} + \frac {7 \, x}{66 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {19}{99 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

8/1331*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 95/726*x/sqrt(3*x^2 + 2) + 4/
121/sqrt(3*x^2 + 2) + 7/66*x/(3*x^2 + 2)^(3/2) - 19/99/(3*x^2 + 2)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{1331} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {9 \, {\left ({\left (95 \, x + 24\right )} x + 89\right )} x - 274}{2178 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

8/1331*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))) + 1/2178*(9*((95*x + 24)*x + 89)*x - 274)/(3*x^2 + 2)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.99 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {11}\,\left (8\,\ln \left (x+\frac {1}{2}\right )-8\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{1331}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {21}{176}+\frac {\sqrt {6}\,19{}\mathrm {i}}{176}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {7}{88}+\frac {\sqrt {6}\,19{}\mathrm {i}}{264}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {21}{176}+\frac {\sqrt {6}\,19{}\mathrm {i}}{176}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {7}{88}+\frac {\sqrt {6}\,19{}\mathrm {i}}{264}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-288+\sqrt {6}\,303{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{104544\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (288+\sqrt {6}\,303{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{104544\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]

[In]

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

[Out]

(11^(1/2)*(8*log(x + 1/2) - 8*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3)))/1331 - (3^(1/2)*(x^2 + 2
/3)^(1/2)*(((6^(1/2)*19i)/176 - 21/176)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*19i)/264 - 7/88)*1i)/(2*(x +
(6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*19i)/176 + 21/176)/(x - (6^(1/2)*1i)/3) - (6^
(1/2)*((6^(1/2)*19i)/264 + 7/88)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*303i - 288)*(
x^2 + 2/3)^(1/2)*1i)/(104544*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*303i + 288)*(x^2 + 2/3)^(1/2)*1
i)/(104544*(x - (6^(1/2)*1i)/3))