Integrand size = 24, antiderivative size = 53 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 \sqrt {2+3 x^2}}-\frac {26 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{35 \sqrt {35}} \] Output:
1/70*(26+41*x)/(3*x^2+2)^(1/2)-26/1225*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1 /2)/(3*x^2+2)^(1/2))
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 \sqrt {2+3 x^2}}+\frac {52 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{35 \sqrt {35}} \] Input:
Integrate[(5 - x)/((3 + 2*x)*(2 + 3*x^2)^(3/2)),x]
Output:
(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) + (52*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(35*Sqrt[35])
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 27, 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.
\(\displaystyle \int \frac {5-x}{(2 x+3) \left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {41 x+26}{70 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {156}{(2 x+3) \sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {26}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {41 x+26}{70 \sqrt {3 x^2+2}}-\frac {26}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {41 x+26}{70 \sqrt {3 x^2+2}}-\frac {26 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}\) |
Input:
Int[(5 - x)/((3 + 2*x)*(2 + 3*x^2)^(3/2)),x]
Output:
(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) - (26*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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/(((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[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.81 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {26+41 x}{70 \sqrt {3 x^{2}+2}}-\frac {26 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1225}\) | \(48\) |
trager | \(\frac {26+41 x}{70 \sqrt {3 x^{2}+2}}+\frac {26 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{1225}\) | \(64\) |
default | \(-\frac {x}{4 \sqrt {3 x^{2}+2}}+\frac {13}{35 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {117 x}{140 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {26 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1225}\) | \(77\) |
Input:
int((5-x)/(2*x+3)/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/70*(26+41*x)/(3*x^2+2)^(1/2)-26/1225*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1 /2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.57 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26 \, \sqrt {35} {\left (3 \, x^{2} + 2\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, \sqrt {3 \, x^{2} + 2} {\left (41 \, x + 26\right )}}{2450 \, {\left (3 \, x^{2} + 2\right )}} \] Input:
integrate((5-x)/(3+2*x)/(3*x^2+2)^(3/2),x, algorithm="fricas")
Output:
1/2450*(26*sqrt(35)*(3*x^2 + 2)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*sqrt(3*x^2 + 2)*(41*x + 26)) /(3*x^2 + 2)
\[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{6 x^{3} \sqrt {3 x^{2} + 2} + 9 x^{2} \sqrt {3 x^{2} + 2} + 4 x \sqrt {3 x^{2} + 2} + 6 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{6 x^{3} \sqrt {3 x^{2} + 2} + 9 x^{2} \sqrt {3 x^{2} + 2} + 4 x \sqrt {3 x^{2} + 2} + 6 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:
integrate((5-x)/(3+2*x)/(3*x**2+2)**(3/2),x)
Output:
-Integral(x/(6*x**3*sqrt(3*x**2 + 2) + 9*x**2*sqrt(3*x**2 + 2) + 4*x*sqrt( 3*x**2 + 2) + 6*sqrt(3*x**2 + 2)), x) - Integral(-5/(6*x**3*sqrt(3*x**2 + 2) + 9*x**2*sqrt(3*x**2 + 2) + 4*x*sqrt(3*x**2 + 2) + 6*sqrt(3*x**2 + 2)), x)
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26}{1225} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {41 \, x}{70 \, \sqrt {3 \, x^{2} + 2}} + \frac {13}{35 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)/(3+2*x)/(3*x^2+2)^(3/2),x, algorithm="maxima")
Output:
26/1225*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 41/70*x/sqrt(3*x^2 + 2) + 13/35/sqrt(3*x^2 + 2)
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26}{1225} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {41 \, x + 26}{70 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)/(3+2*x)/(3*x^2+2)^(3/2),x, algorithm="giac")
Output:
26/1225*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x ^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/70* (41*x + 26)/sqrt(3*x^2 + 2)
Time = 6.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.00 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\sqrt {35}\,\left (26\,\ln \left (x+\frac {3}{2}\right )-26\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{1225}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-234+\sqrt {6}\,123{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7560\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (234+\sqrt {6}\,123{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7560\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int(-(x - 5)/((2*x + 3)*(3*x^2 + 2)^(3/2)),x)
Output:
(35^(1/2)*(26*log(x + 3/2) - 26*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2 ))/9 - 4/9)))/1225 - (3^(1/2)*6^(1/2)*(6^(1/2)*123i - 234)*(x^2 + 2/3)^(1/ 2)*1i)/(7560*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*123i + 234) *(x^2 + 2/3)^(1/2)*1i)/(7560*(x - (6^(1/2)*1i)/3))
Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 4.57 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {468 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i \,x^{2}+312 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i +4305 \sqrt {3 x^{2}+2}\, x +2730 \sqrt {3 x^{2}+2}+234 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right ) x^{2}+156 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right )-468 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right ) x^{2}-312 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right )+4305 \sqrt {3}\, x^{2}+2870 \sqrt {3}}{22050 x^{2}+14700} \] Input:
int((5-x)/(3+2*x)/(3*x^2+2)^(3/2),x)
Output:
(468*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(35) - 3*sq rt(3)))*i*x**2 + 312*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/ (sqrt(35) - 3*sqrt(3)))*i + 4305*sqrt(3*x**2 + 2)*x + 2730*sqrt(3*x**2 + 2 ) + 234*sqrt(35)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + 3*sqrt(105) + 12*x**2 - 27)*x**2 + 156*sqrt(35)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + 3*sqrt(105) + 12*x**2 - 27) - 468*sqrt(35)*log((2*sqrt(3*x**2 + 2) + sqrt(35) + 2*sqrt( 3)*x + 3*sqrt(3))/sqrt(2))*x**2 - 312*sqrt(35)*log((2*sqrt(3*x**2 + 2) + s qrt(35) + 2*sqrt(3)*x + 3*sqrt(3))/sqrt(2)) + 4305*sqrt(3)*x**2 + 2870*sqr t(3))/(7350*(3*x**2 + 2))