Integrand size = 27, antiderivative size = 63 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {1}{27 \left (2+3 x^2\right )^{3/2}}-\frac {17 x}{18 \left (2+3 x^2\right )^{3/2}}-\frac {8}{9 \sqrt {2+3 x^2}}+\frac {13 x}{18 \sqrt {2+3 x^2}} \] Output:
1/27/(3*x^2+2)^(3/2)-17/18*x/(3*x^2+2)^(3/2)-8/9/(3*x^2+2)^(1/2)+13/18*x/( 3*x^2+2)^(1/2)
Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.48 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {-94+27 x-144 x^2+117 x^3}{54 \left (2+3 x^2\right )^{3/2}} \] Input:
Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]
Output:
(-94 + 27*x - 144*x^2 + 117*x^3)/(54*(2 + 3*x^2)^(3/2))
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2345, 27, 453}
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.
\(\displaystyle \int \frac {(2 x+1) \left (4 x^2+3 x+1\right )}{\left (3 x^2+2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {1}{6} \int -\frac {2 (24 x+13)}{3 \left (3 x^2+2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {24 x+13}{\left (3 x^2+2\right )^{3/2}}dx+\frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}\) |
\(\Big \downarrow \) 453 |
\(\displaystyle \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {3 x^2+2}}\) |
Input:
Int[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]
Output:
(2 - 51*x)/(54*(2 + 3*x^2)^(3/2)) - (16 - 13*x)/(18*Sqrt[2 + 3*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
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]
Time = 0.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
trager | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
risch | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
orering | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
default | \(-\frac {17 x}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {13 x}{18 \sqrt {3 x^{2}+2}}-\frac {47}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(51\) |
meijerg | \(\frac {\sqrt {2}\, x \left (3 x^{2}+3\right )}{24 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, x^{3}}{12 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{18 \sqrt {\pi }}+\frac {8 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}\) | \(102\) |
Input:
int((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/54*(117*x^3-144*x^2+27*x-94)/(3*x^2+2)^(3/2)
Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.63 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {{\left (117 \, x^{3} - 144 \, x^{2} + 27 \, x - 94\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="fricas")
Output:
1/54*(117*x^3 - 144*x^2 + 27*x - 94)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (56) = 112\).
Time = 20.34 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.86 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {10 x^{3}}{18 x^{2} \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}} + \frac {x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {72 x^{2}}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} + \frac {x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {32}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} - \frac {5}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \] Input:
integrate((1+2*x)*(4*x**2+3*x+1)/(3*x**2+2)**(5/2),x)
Output:
10*x**3/(18*x**2*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2 + 2)) + x**3/(6*x**2*sq rt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) - 72*x**2/(81*x**2*sqrt(3*x**2 + 2) + 54*sqrt(3*x**2 + 2)) + x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) - 32/(81*x**2*sqrt(3*x**2 + 2) + 54*sqrt(3*x**2 + 2)) - 5/(27*x**2*sqrt(3*x **2 + 2) + 18*sqrt(3*x**2 + 2))
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {13 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {17 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {47}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="maxima")
Output:
13/18*x/sqrt(3*x^2 + 2) - 8/3*x^2/(3*x^2 + 2)^(3/2) - 17/18*x/(3*x^2 + 2)^ (3/2) - 47/27/(3*x^2 + 2)^(3/2)
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.40 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {9 \, {\left ({\left (13 \, x - 16\right )} x + 3\right )} x - 94}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="giac")
Output:
1/54*(9*((13*x - 16)*x + 3)*x - 94)/(3*x^2 + 2)^(3/2)
Time = 16.71 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.94 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int(((2*x + 1)*(3*x + 4*x^2 + 1))/(3*x^2 + 2)^(5/2),x)
Output:
(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/48 - 17/16)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*1i)/72 - 17/24)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/48 + 17/16)/(x - (6^(1/2)*1i)/ 3) - (6^(1/2)*((6^(1/2)*1i)/72 + 17/24)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/2 7 - (3^(1/2)*6^(1/2)*(6^(1/2)*69i - 192)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*69i + 192)*(x^2 + 2/3)^(1/2)* 1i)/(2592*(x + (6^(1/2)*1i)/3))
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.29 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {117 \sqrt {3 x^{2}+2}\, x^{3}-144 \sqrt {3 x^{2}+2}\, x^{2}+27 \sqrt {3 x^{2}+2}\, x -94 \sqrt {3 x^{2}+2}+63 \sqrt {3}\, x^{4}+84 \sqrt {3}\, x^{2}+28 \sqrt {3}}{486 x^{4}+648 x^{2}+216} \] Input:
int((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x)
Output:
(117*sqrt(3*x**2 + 2)*x**3 - 144*sqrt(3*x**2 + 2)*x**2 + 27*sqrt(3*x**2 + 2)*x - 94*sqrt(3*x**2 + 2) + 63*sqrt(3)*x**4 + 84*sqrt(3)*x**2 + 28*sqrt(3 ))/(54*(9*x**4 + 12*x**2 + 4))