Integrand size = 27, antiderivative size = 66 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {\sqrt {3+2 x} (29+35 x)}{2+5 x+3 x^2}-82 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {316 \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{\sqrt {15}} \] Output:
-(3+2*x)^(1/2)*(29+35*x)/(3*x^2+5*x+2)-82*arctanh((3+2*x)^(1/2))+316/15*15 ^(1/2)*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {\sqrt {3+2 x} (29+35 x)}{2+5 x+3 x^2}-82 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {316 \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{\sqrt {15}} \] Input:
Integrate[((5 - x)*Sqrt[3 + 2*x])/(2 + 5*x + 3*x^2)^2,x]
Output:
-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x ]] + (316*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15]
Time = 0.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1234, 1197, 1480, 220}
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.
\(\displaystyle \int \frac {(5-x) \sqrt {2 x+3}}{\left (3 x^2+5 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 1234 |
\(\displaystyle -\int \frac {35 x+76}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )}dx-\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle -2 \int \frac {35 (2 x+3)+47}{3 (2 x+3)^2-8 (2 x+3)+5}d\sqrt {2 x+3}-\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle -2 \left (158 \int \frac {1}{3 (2 x+3)-5}d\sqrt {2 x+3}-123 \int \frac {1}{3 (2 x+3)-3}d\sqrt {2 x+3}\right )-\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -2 \left (41 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {158 \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}}\right )-\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}\) |
Input:
Int[((5 - x)*Sqrt[3 + 2*x])/(2 + 5*x + 3*x^2)^2,x]
Output:
-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 2*(41*ArcTanh[Sqrt[3 + 2*x]] - (158*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 1.42 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06
method | result | size |
risch | \(-\frac {\sqrt {2 x +3}\, \left (29+35 x \right )}{3 x^{2}+5 x +2}-41 \ln \left (\sqrt {2 x +3}+1\right )+41 \ln \left (\sqrt {2 x +3}-1\right )+\frac {316 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15}\) | \(70\) |
derivativedivides | \(-\frac {6}{\sqrt {2 x +3}+1}-41 \ln \left (\sqrt {2 x +3}+1\right )-\frac {34 \sqrt {2 x +3}}{3 \left (2 x +\frac {4}{3}\right )}+\frac {316 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15}-\frac {6}{\sqrt {2 x +3}-1}+41 \ln \left (\sqrt {2 x +3}-1\right )\) | \(86\) |
default | \(-\frac {6}{\sqrt {2 x +3}+1}-41 \ln \left (\sqrt {2 x +3}+1\right )-\frac {34 \sqrt {2 x +3}}{3 \left (2 x +\frac {4}{3}\right )}+\frac {316 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15}-\frac {6}{\sqrt {2 x +3}-1}+41 \ln \left (\sqrt {2 x +3}-1\right )\) | \(86\) |
trager | \(-\frac {\sqrt {2 x +3}\, \left (29+35 x \right )}{3 x^{2}+5 x +2}-41 \ln \left (\frac {\sqrt {2 x +3}+2+x}{x +1}\right )+\frac {158 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {2 x +3}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{3 x +2}\right )}{15}\) | \(91\) |
pseudoelliptic | \(\frac {948 \sqrt {15}\, \left (x +1\right ) \left (x +\frac {2}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )+\left (1845 x^{2}+3075 x +1230\right ) \ln \left (\sqrt {2 x +3}-1\right )+\left (-1845 x^{2}-3075 x -1230\right ) \ln \left (\sqrt {2 x +3}+1\right )+\left (-525 x -435\right ) \sqrt {2 x +3}}{45 x^{2}+75 x +30}\) | \(94\) |
Input:
int((5-x)*(2*x+3)^(1/2)/(3*x^2+5*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-(2*x+3)^(1/2)*(29+35*x)/(3*x^2+5*x+2)-41*ln((2*x+3)^(1/2)+1)+41*ln((2*x+3 )^(1/2)-1)+316/15*15^(1/2)*arctanh(1/5*15^(1/2)*(2*x+3)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (55) = 110\).
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.71 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {158 \, \sqrt {15} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {15} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 615 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 615 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 15 \, {\left (35 \, x + 29\right )} \sqrt {2 \, x + 3}}{15 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \] Input:
integrate((5-x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^2,x, algorithm="fricas")
Output:
1/15*(158*sqrt(15)*(3*x^2 + 5*x + 2)*log((sqrt(15)*sqrt(2*x + 3) + 3*x + 7 )/(3*x + 2)) - 615*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) + 615*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 15*(35*x + 29)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)
Time = 38.67 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.20 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=- \frac {47 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{5} + 340 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right ) + 41 \log {\left (\sqrt {2 x + 3} - 1 \right )} - 41 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {6}{\sqrt {2 x + 3} + 1} - \frac {6}{\sqrt {2 x + 3} - 1} \] Input:
integrate((5-x)*(3+2*x)**(1/2)/(3*x**2+5*x+2)**2,x)
Output:
-47*sqrt(15)*(log(sqrt(2*x + 3) - sqrt(15)/3) - log(sqrt(2*x + 3) + sqrt(1 5)/3))/5 + 340*Piecewise((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 + 1) ) - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 - 1)))/75, (sqrt(2*x + 3) > -sqrt(15)/3 ) & (sqrt(2*x + 3) < sqrt(15)/3))) + 41*log(sqrt(2*x + 3) - 1) - 41*log(sq rt(2*x + 3) + 1) - 6/(sqrt(2*x + 3) + 1) - 6/(sqrt(2*x + 3) - 1)
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.48 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {158}{15} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (35 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 47 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 41 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \] Input:
integrate((5-x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^2,x, algorithm="maxima")
Output:
-158/15*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 2*(35*(2*x + 3)^(3/2) - 47*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 41*log(sqrt(2*x + 3) + 1) + 41*log(sqrt(2*x + 3) - 1)
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.55 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {158}{15} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {2 \, {\left (35 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 47 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 41 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \] Input:
integrate((5-x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^2,x, algorithm="giac")
Output:
-158/15*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3* sqrt(2*x + 3))) - 2*(35*(2*x + 3)^(3/2) - 47*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 41*log(sqrt(2*x + 3) + 1) + 41*log(abs(sqrt(2*x + 3) - 1))
Time = 11.40 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {316\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15}-\frac {\frac {94\,\sqrt {2\,x+3}}{3}-\frac {70\,{\left (2\,x+3\right )}^{3/2}}{3}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-82\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right ) \] Input:
int(-((2*x + 3)^(1/2)*(x - 5))/(5*x + 3*x^2 + 2)^2,x)
Output:
(316*15^(1/2)*atanh((15^(1/2)*(2*x + 3)^(1/2))/5))/15 - ((94*(2*x + 3)^(1/ 2))/3 - (70*(2*x + 3)^(3/2))/3)/((16*x)/3 - (2*x + 3)^2 + 19/3) - 82*atanh ((2*x + 3)^(1/2))
Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 3.26 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {-525 \sqrt {2 x +3}\, x -435 \sqrt {2 x +3}-474 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{2}-790 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x -316 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right )+474 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{2}+790 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x +316 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right )+1845 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{2}+3075 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x +1230 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right )-1845 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{2}-3075 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x -1230 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right )}{45 x^{2}+75 x +30} \] Input:
int((5-x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^2,x)
Output:
( - 525*sqrt(2*x + 3)*x - 435*sqrt(2*x + 3) - 474*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x**2 - 790*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x - 3 16*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15)) + 474*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x**2 + 790*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x + 3 16*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15)) + 1845*log(sqrt(2*x + 3) - 1)* x**2 + 3075*log(sqrt(2*x + 3) - 1)*x + 1230*log(sqrt(2*x + 3) - 1) - 1845* log(sqrt(2*x + 3) + 1)*x**2 - 3075*log(sqrt(2*x + 3) + 1)*x - 1230*log(sqr t(2*x + 3) + 1))/(15*(3*x**2 + 5*x + 2))