Integrand size = 20, antiderivative size = 220 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \]
1/3*a^2*(-11*a*d+6*b*c)*(d*x+c)^(3/2)/b^5+11/9*x^2*(d*x+c)^(5/2)/b^2-x^3*( d*x+c)^(5/2)/b/(b*x+a)-1/315*(d*x+c)^(5/2)*(20*b^2*c^2+180*a*b*c*d-693*a^2 *d^2-5*b*d*(-99*a*d+10*b*c)*x)/b^4/d^2-a^2*(-11*a*d+6*b*c)*(-a*d+b*c)^(3/2 )*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(13/2)+a^2*(-11*a*d+6* b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/b^6
Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} \left (3465 a^5 d^4+10 b^5 x (c+d x)^3 (-2 c+7 d x)+210 a^4 b d^3 (-31 c+11 d x)-10 a b^4 (c+d x)^3 (2 c+11 d x)-21 a^3 b^2 d^2 \left (-153 c^2+214 c d x+22 d^2 x^2\right )+18 a^2 b^3 d \left (-10 c^3+131 c^2 d x+47 c d^2 x^2+11 d^3 x^3\right )\right )}{315 b^6 d^2 (a+b x)}+\frac {a^2 (6 b c-11 a d) (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \]
(Sqrt[c + d*x]*(3465*a^5*d^4 + 10*b^5*x*(c + d*x)^3*(-2*c + 7*d*x) + 210*a ^4*b*d^3*(-31*c + 11*d*x) - 10*a*b^4*(c + d*x)^3*(2*c + 11*d*x) - 21*a^3*b ^2*d^2*(-153*c^2 + 214*c*d*x + 22*d^2*x^2) + 18*a^2*b^3*d*(-10*c^3 + 131*c ^2*d*x + 47*c*d^2*x^2 + 11*d^3*x^3)))/(315*b^6*d^2*(a + b*x)) + (a^2*(6*b* c - 11*a*d)*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c ) + a*d]])/b^(13/2)
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {108, 27, 170, 27, 164, 60, 60, 73, 221}
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.
| \(\displaystyle \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {\int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{2 (a+b x)}dx}{b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{a+b x}dx}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\frac {2 \int -\frac {d x (c+d x)^{3/2} (44 a c-(10 b c-99 a d) x)}{2 (a+b x)}dx}{9 b d}+\frac {22 x^2 (c+d x)^{5/2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\int \frac {x (c+d x)^{3/2} (44 a c-(10 b c-99 a d) x)}{a+b x}dx}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\frac {2 (c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{35 b^2 d^2}-\frac {9 a^2 (6 b c-11 a d) \int \frac {(c+d x)^{3/2}}{a+b x}dx}{b^2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\frac {2 (c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{35 b^2 d^2}-\frac {9 a^2 (6 b c-11 a d) \left (\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{a+b x}dx}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{b^2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\frac {2 (c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{35 b^2 d^2}-\frac {9 a^2 (6 b c-11 a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{b^2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\frac {2 (c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{35 b^2 d^2}-\frac {9 a^2 (6 b c-11 a d) \left (\frac {(b c-a d) \left (\frac {2 (b c-a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b d}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{b^2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {22 x^2 (c+d x)^{5/2}}{9 b}-\frac {\frac {2 (c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{35 b^2 d^2}-\frac {9 a^2 (6 b c-11 a d) \left (\frac {(b c-a d) \left (\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{b^2}}{9 b}}{2 b}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}\) |
-((x^3*(c + d*x)^(5/2))/(b*(a + b*x))) + ((22*x^2*(c + d*x)^(5/2))/(9*b) - ((2*(c + d*x)^(5/2)*(20*b^2*c^2 + 180*a*b*c*d - 693*a^2*d^2 - 5*b*d*(10*b *c - 99*a*d)*x))/(35*b^2*d^2) - (9*a^2*(6*b*c - 11*a*d)*((2*(c + d*x)^(3/2 ))/(3*b) + ((b*c - a*d)*((2*Sqrt[c + d*x])/b - (2*Sqrt[b*c - a*d]*ArcTanh[ (Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(3/2)))/b))/b^2)/(9*b))/(2*b)
3.5.68.3.1 Defintions of rubi rules used
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.66 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(-\frac {11 \left (\left (a d -b c \right )^{2} \left (a d -\frac {6 b c}{11}\right ) d^{2} \left (b x +a \right ) a^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (-\frac {4 x \left (-\frac {7 d x}{2}+c \right ) \left (d x +c \right )^{3} b^{5}}{693}-\frac {4 \left (d x +c \right )^{3} a \left (\frac {11 d x}{2}+c \right ) b^{4}}{693}-\frac {4 \left (-\frac {11}{10} d^{3} x^{3}-\frac {47}{10} c \,d^{2} x^{2}-\frac {131}{10} c^{2} d x +c^{3}\right ) d \,a^{2} b^{3}}{77}+\frac {51 \left (-\frac {22}{153} d^{2} x^{2}-\frac {214}{153} c d x +c^{2}\right ) d^{2} a^{3} b^{2}}{55}-\frac {62 d^{3} \left (-\frac {11 d x}{31}+c \right ) a^{4} b}{33}+a^{5} d^{4}\right ) \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{2} b^{6} \left (b x +a \right )}\) | \(228\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {2 a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {3 a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+5 a^{4} d^{4} \sqrt {d x +c}-8 a^{3} b c \,d^{3} \sqrt {d x +c}+3 a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{6}}-\frac {2 a^{2} d^{2} \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (11 a^{3} d^{3}-28 a^{2} b c \,d^{2}+23 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}}{d^{2}}\) | \(294\) |
| default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {2 a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {3 a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+5 a^{4} d^{4} \sqrt {d x +c}-8 a^{3} b c \,d^{3} \sqrt {d x +c}+3 a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{6}}-\frac {2 a^{2} d^{2} \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (11 a^{3} d^{3}-28 a^{2} b c \,d^{2}+23 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}}{d^{2}}\) | \(294\) |
| risch | \(\frac {2 \left (35 d^{4} x^{4} b^{4}-90 a \,b^{3} d^{4} x^{3}+95 b^{4} c \,d^{3} x^{3}+189 a^{2} b^{2} d^{4} x^{2}-270 a \,b^{3} c \,d^{3} x^{2}+75 b^{4} c^{2} d^{2} x^{2}-420 a^{3} b \,d^{4} x +693 a^{2} b^{2} c \,d^{3} x -270 a \,b^{3} c^{2} d^{2} x +5 b^{4} c^{3} d x +1575 a^{4} d^{4}-2940 a^{3} b c \,d^{3}+1449 a^{2} b^{2} c^{2} d^{2}-90 a \,b^{3} c^{3} d -10 b^{4} c^{4}\right ) \sqrt {d x +c}}{315 d^{2} b^{6}}-\frac {a^{2} \left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {a d \sqrt {d x +c}}{2 \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (11 a d -6 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}\) | \(295\) |
-11/((a*d-b*c)*b)^(1/2)*((a*d-b*c)^2*(a*d-6/11*b*c)*d^2*(b*x+a)*a^2*arctan (b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-(-4/693*x*(-7/2*d*x+c)*(d*x+c)^3*b^5 -4/693*(d*x+c)^3*a*(11/2*d*x+c)*b^4-4/77*(-11/10*d^3*x^3-47/10*c*d^2*x^2-1 31/10*c^2*d*x+c^3)*d*a^2*b^3+51/55*(-22/153*d^2*x^2-214/153*c*d*x+c^2)*d^2 *a^3*b^2-62/33*d^3*(-11/31*d*x+c)*a^4*b+a^5*d^4)*(d*x+c)^(1/2)*((a*d-b*c)* b)^(1/2))/d^2/b^6/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.58 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\left [\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{630 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}, -\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}\right ] \]
[1/630*(315*(6*a^3*b^2*c^2*d^2 - 17*a^4*b*c*d^3 + 11*a^5*d^4 + (6*a^2*b^3* c^2*d^2 - 17*a^3*b^2*c*d^3 + 11*a^4*b*d^4)*x)*sqrt((b*c - a*d)/b)*log((b*d *x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*( 70*b^5*d^4*x^5 - 20*a*b^4*c^4 - 180*a^2*b^3*c^3*d + 3213*a^3*b^2*c^2*d^2 - 6510*a^4*b*c*d^3 + 3465*a^5*d^4 + 10*(19*b^5*c*d^3 - 11*a*b^4*d^4)*x^4 + 2*(75*b^5*c^2*d^2 - 175*a*b^4*c*d^3 + 99*a^2*b^3*d^4)*x^3 + 2*(5*b^5*c^3*d - 195*a*b^4*c^2*d^2 + 423*a^2*b^3*c*d^3 - 231*a^3*b^2*d^4)*x^2 - 2*(10*b^ 5*c^4 + 85*a*b^4*c^3*d - 1179*a^2*b^3*c^2*d^2 + 2247*a^3*b^2*c*d^3 - 1155* a^4*b*d^4)*x)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2), -1/315*(315*(6*a^3*b ^2*c^2*d^2 - 17*a^4*b*c*d^3 + 11*a^5*d^4 + (6*a^2*b^3*c^2*d^2 - 17*a^3*b^2 *c*d^3 + 11*a^4*b*d^4)*x)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqr t(-(b*c - a*d)/b)/(b*c - a*d)) - (70*b^5*d^4*x^5 - 20*a*b^4*c^4 - 180*a^2* b^3*c^3*d + 3213*a^3*b^2*c^2*d^2 - 6510*a^4*b*c*d^3 + 3465*a^5*d^4 + 10*(1 9*b^5*c*d^3 - 11*a*b^4*d^4)*x^4 + 2*(75*b^5*c^2*d^2 - 175*a*b^4*c*d^3 + 99 *a^2*b^3*d^4)*x^3 + 2*(5*b^5*c^3*d - 195*a*b^4*c^2*d^2 + 423*a^2*b^3*c*d^3 - 231*a^3*b^2*d^4)*x^2 - 2*(10*b^5*c^4 + 85*a*b^4*c^3*d - 1179*a^2*b^3*c^ 2*d^2 + 2247*a^3*b^2*c*d^3 - 1155*a^4*b*d^4)*x)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2)]
Timed out. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {{\left (6 \, a^{2} b^{3} c^{3} - 23 \, a^{3} b^{2} c^{2} d + 28 \, a^{4} b c d^{2} - 11 \, a^{5} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {\sqrt {d x + c} a^{3} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{4} b c d^{2} + \sqrt {d x + c} a^{5} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{6}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{16} d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{16} c d^{16} - 90 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{15} d^{17} + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{14} d^{18} + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{14} c d^{18} + 945 \, \sqrt {d x + c} a^{2} b^{14} c^{2} d^{18} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{13} d^{19} - 2520 \, \sqrt {d x + c} a^{3} b^{13} c d^{19} + 1575 \, \sqrt {d x + c} a^{4} b^{12} d^{20}\right )}}{315 \, b^{18} d^{18}} \]
(6*a^2*b^3*c^3 - 23*a^3*b^2*c^2*d + 28*a^4*b*c*d^2 - 11*a^5*d^3)*arctan(sq rt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^6) + (sqrt(d*x + c)*a^3*b^2*c^2*d - 2*sqrt(d*x + c)*a^4*b*c*d^2 + sqrt(d*x + c)*a^5*d^3) /(((d*x + c)*b - b*c + a*d)*b^6) + 2/315*(35*(d*x + c)^(9/2)*b^16*d^16 - 4 5*(d*x + c)^(7/2)*b^16*c*d^16 - 90*(d*x + c)^(7/2)*a*b^15*d^17 + 189*(d*x + c)^(5/2)*a^2*b^14*d^18 + 315*(d*x + c)^(3/2)*a^2*b^14*c*d^18 + 945*sqrt( d*x + c)*a^2*b^14*c^2*d^18 - 420*(d*x + c)^(3/2)*a^3*b^13*d^19 - 2520*sqrt (d*x + c)*a^3*b^13*c*d^19 + 1575*sqrt(d*x + c)*a^4*b^12*d^20)/(b^18*d^18)
Time = 0.15 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.09 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx={\left (c+d\,x\right )}^{5/2}\,\left (\frac {6\,c^2}{5\,b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{5\,b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{5\,b}\right )-\left (\frac {{\left (a\,d-b\,c\right )}^2\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b^2}-\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {2\,c^3}{b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b}\right )}{b}\right )\,\sqrt {c+d\,x}-{\left (c+d\,x\right )}^{3/2}\,\left (\frac {2\,c^3}{3\,b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{3\,b}\right )-\left (\frac {6\,c}{7\,b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{7\,b^3\,d^2}\right )\,{\left (c+d\,x\right )}^{7/2}+\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,b^2\,d^2}+\frac {\sqrt {c+d\,x}\,\left (a^5\,d^3-2\,a^4\,b\,c\,d^2+a^3\,b^2\,c^2\,d\right )}{b^7\,\left (c+d\,x\right )-b^7\,c+a\,b^6\,d}-\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )\,\sqrt {c+d\,x}}{11\,a^5\,d^3-28\,a^4\,b\,c\,d^2+23\,a^3\,b^2\,c^2\,d-6\,a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )}{b^{13/2}} \]
(c + d*x)^(5/2)*((6*c^2)/(5*b^2*d^2) - (2*(a*d - b*c)^2)/(5*b^4*d^2) + (2* ((6*c)/(b^2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c))/(5*b)) - (((a*d - b*c)^2*((6*c^2)/(b^2*d^2) - (2*(a*d - b*c)^2)/(b^4*d^2) + (2*((6*c)/(b^ 2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c))/b))/b^2 - (2*(a*d - b*c)* ((2*c^3)/(b^2*d^2) - (((6*c)/(b^2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c)^2)/b^2 + (2*(a*d - b*c)*((6*c^2)/(b^2*d^2) - (2*(a*d - b*c)^2)/(b^4* d^2) + (2*((6*c)/(b^2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c))/b))/b ))/b)*(c + d*x)^(1/2) - (c + d*x)^(3/2)*((2*c^3)/(3*b^2*d^2) - (((6*c)/(b^ 2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c)^2)/(3*b^2) + (2*(a*d - b*c )*((6*c^2)/(b^2*d^2) - (2*(a*d - b*c)^2)/(b^4*d^2) + (2*((6*c)/(b^2*d^2) + (4*(a*d - b*c))/(b^3*d^2))*(a*d - b*c))/b))/(3*b)) - ((6*c)/(7*b^2*d^2) + (4*(a*d - b*c))/(7*b^3*d^2))*(c + d*x)^(7/2) + (2*(c + d*x)^(9/2))/(9*b^2 *d^2) + ((c + d*x)^(1/2)*(a^5*d^3 + a^3*b^2*c^2*d - 2*a^4*b*c*d^2))/(b^7*( c + d*x) - b^7*c + a*b^6*d) - (a^2*atan((a^2*b^(1/2)*(a*d - b*c)^(3/2)*(11 *a*d - 6*b*c)*(c + d*x)^(1/2))/(11*a^5*d^3 - 6*a^2*b^3*c^3 + 23*a^3*b^2*c^ 2*d - 28*a^4*b*c*d^2))*(a*d - b*c)^(3/2)*(11*a*d - 6*b*c))/b^(13/2)