Integrand size = 27, antiderivative size = 59 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47 \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{5 \sqrt {5}} \] Output:
-13*(3*x^2+5*x+2)^(1/2)/(15+10*x)+47/25*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x ^2+5*x+2)^(1/2))
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47 \text {arctanh}\left (\frac {\sqrt {2+5 x+3 x^2}}{\sqrt {5} (1+x)}\right )}{5 \sqrt {5}} \] Input:
Integrate[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]),x]
Output:
(-13*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (47*ArcTanh[Sqrt[2 + 5*x + 3*x ^2]/(Sqrt[5]*(1 + x))])/(5*Sqrt[5])
Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1228, 1154, 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.
\(\displaystyle \int \frac {5-x}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}} \, dx\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {47}{10} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -\frac {47}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {47 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10 \sqrt {5}}-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\) |
Input:
Int[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]),x]
Output:
(-13*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (47*ArcTanh[(7 + 8*x)/(2*Sqrt[ 5]*Sqrt[2 + 5*x + 3*x^2])])/(10*Sqrt[5])
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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 1.61 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{10 \left (x +\frac {3}{2}\right )}-\frac {47 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{50}\) | \(53\) |
risch | \(-\frac {13 \sqrt {3 x^{2}+5 x +2}}{5 \left (2 x +3\right )}-\frac {47 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{50}\) | \(53\) |
trager | \(-\frac {13 \sqrt {3 x^{2}+5 x +2}}{5 \left (2 x +3\right )}+\frac {47 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{2 x +3}\right )}{50}\) | \(72\) |
Input:
int((5-x)/(2*x+3)^2/(3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-47/50*5^(1/2)*arctanh(2/5*(-7/ 2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=\frac {47 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 260 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{100 \, {\left (2 \, x + 3\right )}} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
Output:
1/100*(47*sqrt(5)*(2*x + 3)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 260*sqrt(3*x^2 + 5*x + 2))/ (2*x + 3)
\[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=- \int \frac {x}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:
integrate((5-x)/(3+2*x)**2/(3*x**2+5*x+2)**(1/2),x)
Output:
-Integral(x/(4*x**2*sqrt(3*x**2 + 5*x + 2) + 12*x*sqrt(3*x**2 + 5*x + 2) + 9*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(4*x**2*sqrt(3*x**2 + 5*x + 2 ) + 12*x*sqrt(3*x**2 + 5*x + 2) + 9*sqrt(3*x**2 + 5*x + 2)), x)
Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {47}{50} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{5 \, {\left (2 \, x + 3\right )}} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")
Output:
-47/50*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2* x + 3) - 2) - 13/5*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (47) = 94\).
Time = 0.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.15 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=\frac {1}{50} \, \sqrt {5} {\left (13 \, \sqrt {5} \sqrt {3} + 47 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {47 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{50 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {13 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}}{10 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")
Output:
1/50*sqrt(5)*(13*sqrt(5)*sqrt(3) + 47*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2* x + 3)) - 47/50*sqrt(5)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))/sgn(1/(2*x + 3)) - 13/10*sqrt(-8/(2*x + 3 ) + 5/(2*x + 3)^2 + 3)/sgn(1/(2*x + 3))
Timed out. \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^2\,\sqrt {3\,x^2+5\,x+2}} \,d x \] Input:
int(-(x - 5)/((2*x + 3)^2*(5*x + 3*x^2 + 2)^(1/2)),x)
Output:
-int((x - 5)/((2*x + 3)^2*(5*x + 3*x^2 + 2)^(1/2)), x)
Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.31 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx=\frac {-130 \sqrt {3 x^{2}+5 x +2}+94 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x +141 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )-94 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x -141 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )}{100 x +150} \] Input:
int((5-x)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2),x)
Output:
( - 130*sqrt(3*x**2 + 5*x + 2) + 94*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*s qrt(3) - sqrt(15) + 6*x + 9)*x + 141*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)* sqrt(3) - sqrt(15) + 6*x + 9) - 94*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sq rt(3) + sqrt(15) + 6*x + 9)*x - 141*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*s qrt(3) + sqrt(15) + 6*x + 9))/(50*(2*x + 3))