Integrand size = 24, antiderivative size = 119 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {1}{224} (1718-807 x) \left (2+3 x^2\right )^{3/2}+\frac {(151+131 x) \left (2+3 x^2\right )^{5/2}}{28 (3+2 x)^2}-\frac {43995}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {12885}{128} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \] Output:
15/64*(859-267*x)*(3*x^2+2)^(1/2)+1/224*(1718-807*x)*(3*x^2+2)^(3/2)+1/28* (151+131*x)*(3*x^2+2)^(5/2)/(3+2*x)^2-43995/128*arcsinh(1/2*x*6^(1/2))*3^( 1/2)-12885/128*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
Time = 0.89 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {1}{128} \left (-\frac {2 \sqrt {2+3 x^2} \left (-126181-127403 x-19268 x^2+2826 x^3-696 x^4+72 x^5\right )}{(3+2 x)^2}+25770 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+43995 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \] Input:
Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^3,x]
Output:
((-2*Sqrt[2 + 3*x^2]*(-126181 - 127403*x - 19268*x^2 + 2826*x^3 - 696*x^4 + 72*x^5))/(3 + 2*x)^2 + 25770*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 43995*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/128
Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {681, 27, 681, 27, 682, 27, 719, 222, 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.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^3} \, dx\) |
\(\Big \downarrow \) 681 |
\(\displaystyle -\frac {5}{64} \int \frac {4 (4-87 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{16} \int \frac {(4-87 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 681 |
\(\displaystyle -\frac {5}{16} \left (-\frac {1}{8} \int \frac {24 (29-267 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{16} \left (-3 \int \frac {(29-267 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 682 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {1}{24} \int \frac {42 (262-1257 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \int \frac {262-1257 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (\frac {4295}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1257}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (\frac {4295}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (-\frac {4295}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {859}{2} \sqrt {\frac {5}{7}} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\) |
Input:
Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^3,x]
Output:
-1/16*((29 + 2*x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^2 - (5*(-1/2*((178 + 29*x)* (2 + 3*x^2)^(3/2))/(3 + 2*x) - 3*(((859 - 267*x)*Sqrt[2 + 3*x^2])/4 + (7*( (-419*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2 - (859*Sqrt[5/7]*ArcTanh[(4 - 9*x)/( Sqrt[35]*Sqrt[2 + 3*x^2])])/2))/4)))/16
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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] , x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2 *p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.96 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {216 x^{7}-2088 x^{6}+8622 x^{5}-59196 x^{4}-376557 x^{3}-417079 x^{2}-254806 x -252362}{64 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}\) | \(97\) |
trager | \(-\frac {\left (72 x^{5}-696 x^{4}+2826 x^{3}-19268 x^{2}-127403 x -126181\right ) \sqrt {3 x^{2}+2}}{64 \left (2 x +3\right )^{2}}+\frac {43995 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right )-30065 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{128}\) | \(122\) |
default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}+\frac {421 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {2577 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}-\frac {807 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{224}-\frac {4005 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{64}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}+\frac {859 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{112}+\frac {12885 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}-\frac {1263 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}\) | \(185\) |
Input:
int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^3,x,method=_RETURNVERBOSE)
Output:
-1/64*(216*x^7-2088*x^6+8622*x^5-59196*x^4-376557*x^3-417079*x^2-254806*x- 252362)/(2*x+3)^2/(3*x^2+2)^(1/2)-43995/128*arcsinh(1/2*6^(1/2)*x)*3^(1/2) -12885/128*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^( 1/2))
Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.23 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {43995 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12885 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (72 \, x^{5} - 696 \, x^{4} + 2826 \, x^{3} - 19268 \, x^{2} - 127403 \, x - 126181\right )} \sqrt {3 \, x^{2} + 2}}{256 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:
integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^3,x, algorithm="fricas")
Output:
1/256*(43995*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3* x^2 - 1) + 12885*sqrt(35)*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2 )*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 4*(72*x^5 - 696*x^ 4 + 2826*x^3 - 19268*x^2 - 127403*x - 126181)*sqrt(3*x^2 + 2))/(4*x^2 + 12 *x + 9)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\text {Timed out} \] Input:
integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**3,x)
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.22 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {807}{224} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {859}{112} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {421 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{280 \, {\left (2 \, x + 3\right )}} - \frac {4005}{64} \, \sqrt {3 \, x^{2} + 2} x - \frac {43995}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {12885}{128} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {12885}{64} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^3,x, algorithm="maxima")
Output:
39/280*(3*x^2 + 2)^(5/2) - 13/70*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 80 7/224*(3*x^2 + 2)^(3/2)*x + 859/112*(3*x^2 + 2)^(3/2) + 421/280*(3*x^2 + 2 )^(5/2)/(2*x + 3) - 4005/64*sqrt(3*x^2 + 2)*x - 43995/128*sqrt(3)*arcsinh( 1/2*sqrt(6)*x) + 12885/128*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2 /3*sqrt(6)/abs(2*x + 3)) + 12885/64*sqrt(3*x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (93) = 186\).
Time = 0.15 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.93 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\frac {1}{32} \, {\left (3 \, {\left ({\left (3 \, x - 38\right )} x + 225\right )} x - 4177\right )} \sqrt {3 \, x^{2} + 2} + \frac {43995}{128} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {12885}{128} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {35 \, {\left (11472 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 25829 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 57912 \, \sqrt {3} x + 8984 \, \sqrt {3} + 57912 \, \sqrt {3 \, x^{2} + 2}\right )}}{256 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \] Input:
integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^3,x, algorithm="giac")
Output:
-1/32*(3*((3*x - 38)*x + 225)*x - 4177)*sqrt(3*x^2 + 2) + 43995/128*sqrt(3 )*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 12885/128*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))) + 35/256*(11472*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 25829*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 57912*sqrt(3)*x + 8984*sqrt(3) + 57912*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.24 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {12885\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{128}+\frac {4177\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32}-\frac {43995\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {12885\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{128}+\frac {57\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{32}+\frac {39305\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{256\,\left (x+\frac {3}{2}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {675\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \] Input:
int(-((3*x^2 + 2)^(5/2)*(x - 5))/(2*x + 3)^3,x)
Output:
(12885*35^(1/2)*log(x + 3/2))/128 + (4177*3^(1/2)*(x^2 + 2/3)^(1/2))/32 - (43995*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/128 - (12885*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/128 + (57*3^(1/2)*x^2*(x^2 + 2/3)^(1/2))/16 - (9*3^(1/2)*x^3*(x^2 + 2/3)^(1/2))/32 + (39305*3^(1/2)* (x^2 + 2/3)^(1/2))/(256*(x + 3/2)) - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(51 2*(3*x + x^2 + 9/4)) - (675*3^(1/2)*x*(x^2 + 2/3)^(1/2))/32
Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.58 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {-288 \sqrt {3 x^{2}+2}\, x^{5}+2784 \sqrt {3 x^{2}+2}\, x^{4}-11304 \sqrt {3 x^{2}+2}\, x^{3}+77072 \sqrt {3 x^{2}+2}\, x^{2}+509612 \sqrt {3 x^{2}+2}\, x +504724 \sqrt {3 x^{2}+2}+103080 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+309240 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +231930 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-103080 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-309240 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -231930 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )+175980 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{2}+527940 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x +395955 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )-175980 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{2}-527940 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x -395955 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{1024 x^{2}+3072 x +2304} \] Input:
int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^3,x)
Output:
( - 288*sqrt(3*x**2 + 2)*x**5 + 2784*sqrt(3*x**2 + 2)*x**4 - 11304*sqrt(3* x**2 + 2)*x**3 + 77072*sqrt(3*x**2 + 2)*x**2 + 509612*sqrt(3*x**2 + 2)*x + 504724*sqrt(3*x**2 + 2) + 103080*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 309240*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)* x + 231930*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 103080*sqrt (35)*log(2*x + 3)*x**2 - 309240*sqrt(35)*log(2*x + 3)*x - 231930*sqrt(35)* log(2*x + 3) + 175980*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 + 527 940*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x + 395955*sqrt(3)*log(sqrt( 3*x**2 + 2) - sqrt(3)*x) - 175980*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x )*x**2 - 527940*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x - 395955*sqrt( 3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x))/(256*(4*x**2 + 12*x + 9))