\(\int \frac {(1+2 x)^2 (1+3 x+4 x^2)}{(2+3 x^2)^{5/2}} \, dx\) [131]

Optimal result

Integrand size = 29, antiderivative size = 82 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {35}{27 \left (2+3 x^2\right )^{3/2}}-\frac {47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {28}{9 \sqrt {2+3 x^2}}-\frac {59 x}{54 \sqrt {2+3 x^2}}+\frac {16 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \] Output:

35/27/(3*x^2+2)^(3/2)-47/54*x/(3*x^2+2)^(3/2)-28/9/(3*x^2+2)^(1/2)-59/54*x 
/(3*x^2+2)^(1/2)+16/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {-266-165 x-504 x^2-177 x^3}{54 \left (2+3 x^2\right )^{3/2}}-\frac {16 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \] Input:

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

Output:

(-266 - 165*x - 504*x^2 - 177*x^3)/(54*(2 + 3*x^2)^(3/2)) - (16*Log[-(Sqrt 
[3]*x) + Sqrt[2 + 3*x^2]])/(9*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2345, 27, 2345, 27, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {70-47 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {1}{6} \int -\frac {2 \left (144 x^2+252 x+37\right )}{9 \left (3 x^2+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \int \frac {144 x^2+252 x+37}{\left (3 x^2+2\right )^{3/2}}dx+\frac {70-47 x}{54 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{27} \left (-\frac {1}{2} \int -\frac {96}{\sqrt {3 x^2+2}}dx-\frac {59 x+168}{2 \sqrt {3 x^2+2}}\right )+\frac {70-47 x}{54 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (48 \int \frac {1}{\sqrt {3 x^2+2}}dx-\frac {59 x+168}{2 \sqrt {3 x^2+2}}\right )+\frac {70-47 x}{54 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{27} \left (16 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {59 x+168}{2 \sqrt {3 x^2+2}}\right )+\frac {70-47 x}{54 \left (3 x^2+2\right )^{3/2}}\)

Input:

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

Output:

(70 - 47*x)/(54*(2 + 3*x^2)^(3/2)) + (-1/2*(168 + 59*x)/Sqrt[2 + 3*x^2] + 
16*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/27
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49

method	result	size
risch	\(-\frac {177 x^{3}+504 x^{2}+165 x +266}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {16 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}\)	\(40\)
trager	\(-\frac {177 x^{3}+504 x^{2}+165 x +266}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\)	\(57\)
default	\(-\frac {37 x}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {x}{2 \sqrt {3 x^{2}+2}}-\frac {133}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {28 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {16 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}\)	\(77\)
meijerg	\(\frac {\sqrt {2}\, x \left (3 x^{2}+3\right )}{24 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, x^{3}}{6 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {7 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{18 \sqrt {\pi }}+\frac {28 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}+\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{81 \sqrt {\pi }}\)	\(154\)

Input:

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

Output:

-1/54*(177*x^3+504*x^2+165*x+266)/(3*x^2+2)^(3/2)+16/27*arcsinh(1/2*6^(1/2 
)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {16 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (177 \, x^{3} + 504 \, x^{2} + 165 \, x + 266\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:

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

Output:

1/54*(16*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x 
^2 - 1) - (177*x^3 + 504*x^2 + 165*x + 266)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x 
^2 + 4)
 

Sympy [F]

\[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\int \frac {\left (2 x + 1\right )^{2} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} + 2\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.11 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {16}{27} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} + \frac {16}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {37 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {28 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {37 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {133}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-16/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 16/27*sqrt(3)*a 
rcsinh(1/2*sqrt(6)*x) + 37/54*x/sqrt(3*x^2 + 2) - 28/3*x^2/(3*x^2 + 2)^(3/ 
2) - 37/18*x/(3*x^2 + 2)^(3/2) - 133/27/(3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {16}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left ({\left (59 \, x + 168\right )} x + 55\right )} x + 266}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-16/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/54*(3*((59*x + 168)*x 
 + 55)*x + 266)/(3*x^2 + 2)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.44 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {16\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {47}{48}+\frac {\sqrt {6}\,35{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {47}{72}+\frac {\sqrt {6}\,35{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {47}{48}+\frac {\sqrt {6}\,35{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {47}{72}+\frac {\sqrt {6}\,35{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-672+\sqrt {6}\,63{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (672+\sqrt {6}\,63{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(16*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)* 
(((6^(1/2)*35i)/48 - 47/48)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*35i) 
/72 - 47/72)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1 
/2)*(((6^(1/2)*35i)/48 + 47/48)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)* 
35i)/72 + 47/72)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*6^(1/2)*(6 
^(1/2)*63i - 672)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x + (6^(1/2)*1i)/3)) + (3^( 
1/2)*6^(1/2)*(6^(1/2)*63i + 672)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2) 
*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.90 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {-177 \sqrt {3 x^{2}+2}\, x^{3}-504 \sqrt {3 x^{2}+2}\, x^{2}-165 \sqrt {3 x^{2}+2}\, x -266 \sqrt {3 x^{2}+2}+288 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{4}+384 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}+128 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )+153 \sqrt {3}\, x^{4}+204 \sqrt {3}\, x^{2}+68 \sqrt {3}}{486 x^{4}+648 x^{2}+216} \] Input:

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

Output:

( - 177*sqrt(3*x**2 + 2)*x**3 - 504*sqrt(3*x**2 + 2)*x**2 - 165*sqrt(3*x** 
2 + 2)*x - 266*sqrt(3*x**2 + 2) + 288*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt 
(3)*x)/sqrt(2))*x**4 + 384*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt 
(2))*x**2 + 128*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)) + 153* 
sqrt(3)*x**4 + 204*sqrt(3)*x**2 + 68*sqrt(3))/(54*(9*x**4 + 12*x**2 + 4))