Integrand size = 24, antiderivative size = 104 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=-\frac {33 (4-9 x) \sqrt {2+3 x^2}}{8575 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {89 \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^3}-\frac {198 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8575 \sqrt {35}} \]
-13/140*(3*x^2+2)^(3/2)/(3+2*x)^4-89/2940*(3*x^2+2)^(3/2)/(3+2*x)^3-198/30 0125*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-33/8575*(4-9* x)*(3*x^2+2)^(1/2)/(3+2*x)^2
Time = 0.70 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=-\frac {\sqrt {2+3 x^2} \left (26028-304 x+10134 x^2+2217 x^3\right )}{51450 (3+2 x)^4}+\frac {396 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{8575 \sqrt {35}} \]
-1/51450*(Sqrt[2 + 3*x^2]*(26028 - 304*x + 10134*x^2 + 2217*x^3))/(3 + 2*x )^4 + (396*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] )/(8575*Sqrt[35])
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {688, 25, 679, 486, 488, 219}
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.
\(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+2}}{(2 x+3)^5} \, dx\) |
\(\Big \downarrow \) 688 |
\(\displaystyle -\frac {1}{140} \int -\frac {(164-39 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{140} \int \frac {(164-39 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {1}{140} \left (\frac {264}{7} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {89 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {1}{140} \left (\frac {264}{7} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {89 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{140} \left (\frac {264}{7} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {89 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{140} \left (\frac {264}{7} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {89 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\) |
(-13*(2 + 3*x^2)^(3/2))/(140*(3 + 2*x)^4) + ((-89*(2 + 3*x^2)^(3/2))/(21*( 3 + 2*x)^3) + (264*(-1/70*((4 - 9*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - (3*Arc Tanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])))/7)/140
3.14.65.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
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] + Simp[(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] && EqQ[Simplify[m + 2*p + 3], 0]
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)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.73 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {6651 x^{5}+30402 x^{4}+3522 x^{3}+98352 x^{2}-608 x +52056}{51450 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+2}}-\frac {198 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}\) | \(75\) |
trager | \(-\frac {\left (2217 x^{3}+10134 x^{2}-304 x +26028\right ) \sqrt {3 x^{2}+2}}{51450 \left (3+2 x \right )^{4}}+\frac {198 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{300125}\) | \(81\) |
default | \(-\frac {89 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{23520 \left (x +\frac {3}{2}\right )^{3}}-\frac {33 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{17150 \left (x +\frac {3}{2}\right )^{2}}-\frac {297 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{300125 \left (x +\frac {3}{2}\right )}+\frac {198 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{300125}-\frac {198 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}+\frac {891 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{300125}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}}\) | \(149\) |
-1/51450*(6651*x^5+30402*x^4+3522*x^3+98352*x^2-608*x+52056)/(3+2*x)^4/(3* x^2+2)^(1/2)-198/300125*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2) ^2-36*x-19)^(1/2))
Time = 0.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=\frac {594 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (2217 \, x^{3} + 10134 \, x^{2} - 304 \, x + 26028\right )} \sqrt {3 \, x^{2} + 2}}{1800750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
1/1800750*(594*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqr t(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(2217*x^3 + 10134*x^2 - 304*x + 26028)*sqrt(3*x^2 + 2))/(16*x^4 + 96* x^3 + 216*x^2 + 216*x + 81)
Timed out. \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.42 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=\frac {198}{300125} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {99}{17150} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {89 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{2940 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {66 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{8575 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {297 \, \sqrt {3 \, x^{2} + 2}}{17150 \, {\left (2 \, x + 3\right )}} \]
198/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2 *x + 3)) + 99/17150*sqrt(3*x^2 + 2) - 13/140*(3*x^2 + 2)^(3/2)/(16*x^4 + 9 6*x^3 + 216*x^2 + 216*x + 81) - 89/2940*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 66/8575*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 297/17150*sq rt(3*x^2 + 2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.74 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=\frac {1}{9604000} \, \sqrt {35} {\left (739 \, \sqrt {35} \sqrt {3} + 6336 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {198}{300125} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{823200} \, {\left (\frac {35 \, {\left (\frac {7 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 257 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 9 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 2217 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \]
1/9604000*sqrt(35)*(739*sqrt(35)*sqrt(3) + 6336*log(sqrt(35)*sqrt(3) - 9)) *sgn(1/(2*x + 3)) - 198/300125*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x + 3)) - 1/82320 0*(35*(7*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 257*sgn(1/(2*x + 3)))/(2*x + 3 ) + 9*sgn(1/(2*x + 3)))/(2*x + 3) + 2217*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)
Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.35 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx=\frac {198\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{300125}-\frac {198\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{300125}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{256\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{274400\,\left (x+\frac {3}{2}\right )}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{15680\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {257\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{13440\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(198*35^(1/2)*log(x + 3/2))/300125 - (198*35^(1/2)*log(x - (3^(1/2)*35^(1/ 2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/300125 - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(2 56*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (739*3^(1/2)*(x^2 + 2/ 3)^(1/2))/(274400*(x + 3/2)) - (3*3^(1/2)*(x^2 + 2/3)^(1/2))/(15680*(3*x + x^2 + 9/4)) + (257*3^(1/2)*(x^2 + 2/3)^(1/2))/(13440*((27*x)/4 + (9*x^2)/ 2 + x^3 + 27/8))