Integrand size = 29, antiderivative size = 95 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {199}{27 \sqrt {2+3 x^2}}+\frac {31 x}{6 \sqrt {2+3 x^2}}+\frac {76}{27} \sqrt {2+3 x^2}+4 x \sqrt {2+3 x^2}+\frac {32}{81} \left (2+3 x^2\right )^{3/2}-\frac {38 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \] Output:
199/27/(3*x^2+2)^(1/2)+31/6*x/(3*x^2+2)^(1/2)+76/27*(3*x^2+2)^(1/2)+4*x*(3 *x^2+2)^(1/2)+32/81*(3*x^2+2)^(3/2)-38/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {2362+2133 x+2136 x^2+1944 x^3+576 x^4}{162 \sqrt {2+3 x^2}}+\frac {38 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \] Input:
Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(3/2),x]
Output:
(2362 + 2133*x + 2136*x^2 + 1944*x^3 + 576*x^4)/(162*Sqrt[2 + 3*x^2]) + (3 8*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2345, 27, 2346, 27, 2346, 27, 455, 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.
| \(\displaystyle \int \frac {(2 x+1)^3 \left (4 x^2+3 x+1\right )}{\left (3 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {1}{2} \int \frac {4 \left (-48 x^3-108 x^2-70 x+21\right )}{9 \sqrt {3 x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \int \frac {-48 x^3-108 x^2-70 x+21}{\sqrt {3 x^2+2}}dx\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{9} \int \frac {3 \left (-324 x^2-146 x+63\right )}{\sqrt {3 x^2+2}}dx-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{3} \int \frac {-324 x^2-146 x+63}{\sqrt {3 x^2+2}}dx-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {1}{6} \int \frac {6 (171-146 x)}{\sqrt {3 x^2+2}}dx-54 x \sqrt {3 x^2+2}\right )-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{3} \left (\int \frac {171-146 x}{\sqrt {3 x^2+2}}dx-54 x \sqrt {3 x^2+2}\right )-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{3} \left (171 \int \frac {1}{\sqrt {3 x^2+2}}dx-54 \sqrt {3 x^2+2} x-\frac {146}{3} \sqrt {3 x^2+2}\right )-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {279 x+398}{54 \sqrt {3 x^2+2}}-\frac {2}{9} \left (\frac {1}{3} \left (57 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-54 \sqrt {3 x^2+2} x-\frac {146}{3} \sqrt {3 x^2+2}\right )-\frac {16}{3} x^2 \sqrt {3 x^2+2}\right )\) |
Input:
Int[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(3/2),x]
Output:
(398 + 279*x)/(54*Sqrt[2 + 3*x^2]) - (2*((-16*x^2*Sqrt[2 + 3*x^2])/3 + ((- 146*Sqrt[2 + 3*x^2])/3 - 54*x*Sqrt[2 + 3*x^2] + 57*Sqrt[3]*ArcSinh[Sqrt[3/ 2]*x])/3))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.90 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(\frac {576 x^{4}+1944 x^{3}+2136 x^{2}+2133 x +2362}{162 \sqrt {3 x^{2}+2}}-\frac {38 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}\) | \(45\) |
| trager | \(\frac {576 x^{4}+1944 x^{3}+2136 x^{2}+2133 x +2362}{162 \sqrt {3 x^{2}+2}}+\frac {38 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(63\) |
| default | \(\frac {79 x}{6 \sqrt {3 x^{2}+2}}-\frac {38 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}+\frac {356 x^{2}}{27 \sqrt {3 x^{2}+2}}+\frac {1181}{81 \sqrt {3 x^{2}+2}}+\frac {12 x^{3}}{\sqrt {3 x^{2}+2}}+\frac {32 x^{4}}{9 \sqrt {3 x^{2}+2}}\) | \(79\) |
| meijerg | \(\frac {\sqrt {2}\, x}{4 \sqrt {\frac {3 x^{2}}{2}+1}}+\frac {34 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}}{2 \sqrt {\frac {3 x^{2}}{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {3 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {\frac {3 x^{2}}{2}+1}}\right )}{2 \sqrt {\pi }}+\frac {68 \sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (6 x^{2}+8\right )}{4 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{9 \sqrt {\pi }}+\frac {16 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {15 x^{2}}{2}+15\right )}{20 \sqrt {\frac {3 x^{2}}{2}+1}}-\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{3 \sqrt {\pi }}+\frac {64 \sqrt {2}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {9}{2} x^{4}+12 x^{2}+16\right )}{6 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{27 \sqrt {\pi }}\) | \(214\) |
Input:
int((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/162*(576*x^4+1944*x^3+2136*x^2+2133*x+2362)/(3*x^2+2)^(1/2)-38/9*arcsinh (1/2*6^(1/2)*x)*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.80 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {342 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (576 \, x^{4} + 1944 \, x^{3} + 2136 \, x^{2} + 2133 \, x + 2362\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (3 \, x^{2} + 2\right )}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2+2)^(3/2),x, algorithm="fricas")
Output:
1/162*(342*sqrt(3)*(3*x^2 + 2)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + (576*x^4 + 1944*x^3 + 2136*x^2 + 2133*x + 2362)*sqrt(3*x^2 + 2))/(3*x^2 + 2)
\[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\int \frac {\left (2 x + 1\right )^{3} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((1+2*x)**3*(4*x**2+3*x+1)/(3*x**2+2)**(3/2),x)
Output:
Integral((2*x + 1)**3*(4*x**2 + 3*x + 1)/(3*x**2 + 2)**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {32 \, x^{4}}{9 \, \sqrt {3 \, x^{2} + 2}} + \frac {12 \, x^{3}}{\sqrt {3 \, x^{2} + 2}} + \frac {356 \, x^{2}}{27 \, \sqrt {3 \, x^{2} + 2}} - \frac {38}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {79 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} + \frac {1181}{81 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2+2)^(3/2),x, algorithm="maxima")
Output:
32/9*x^4/sqrt(3*x^2 + 2) + 12*x^3/sqrt(3*x^2 + 2) + 356/27*x^2/sqrt(3*x^2 + 2) - 38/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 79/6*x/sqrt(3*x^2 + 2) + 1181 /81/sqrt(3*x^2 + 2)
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {38}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left (8 \, {\left (3 \, {\left (8 \, x + 27\right )} x + 89\right )} x + 711\right )} x + 2362}{162 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2+2)^(3/2),x, algorithm="giac")
Output:
38/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/162*(3*(8*(3*(8*x + 27) *x + 89)*x + 711)*x + 2362)/sqrt(3*x^2 + 2)
Time = 16.77 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {32\,x^2}{9}+12\,x+\frac {292}{27}\right )}{3}-\frac {38\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-1194+\sqrt {6}\,279{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (1194+\sqrt {6}\,279{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int(((2*x + 1)^3*(3*x + 4*x^2 + 1))/(3*x^2 + 2)^(3/2),x)
Output:
(3^(1/2)*(x^2 + 2/3)^(1/2)*(12*x + (32*x^2)/9 + 292/27))/3 - (38*3^(1/2)*a sinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*6^(1/2)*(6^(1/2)*279i - 1194)*(x ^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2 )*279i + 1194)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3))
Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.38 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {576 \sqrt {3 x^{2}+2}\, x^{4}+1944 \sqrt {3 x^{2}+2}\, x^{3}+2136 \sqrt {3 x^{2}+2}\, x^{2}+2133 \sqrt {3 x^{2}+2}\, x +2362 \sqrt {3 x^{2}+2}-2052 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}-1368 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )+1161 \sqrt {3}\, x^{2}+774 \sqrt {3}}{486 x^{2}+324} \] Input:
int((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2+2)^(3/2),x)
Output:
(576*sqrt(3*x**2 + 2)*x**4 + 1944*sqrt(3*x**2 + 2)*x**3 + 2136*sqrt(3*x**2 + 2)*x**2 + 2133*sqrt(3*x**2 + 2)*x + 2362*sqrt(3*x**2 + 2) - 2052*sqrt(3 )*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**2 - 1368*sqrt(3)*log((sqr t(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)) + 1161*sqrt(3)*x**2 + 774*sqrt(3))/(16 2*(3*x**2 + 2))