Integrand size = 22, antiderivative size = 276 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {11 b^7 (13 b B-20 A c) (b+2 c x) \sqrt {b x+c x^2}}{131072 c^7}-\frac {11 b^5 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac {11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac {(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {11 b^9 (13 b B-20 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{131072 c^{15/2}} \]
-11/49152*b^5*(-20*A*c+13*B*b)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^6+11/15360*b^ 3*(-20*A*c+13*B*b)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^5-11/4480*b^2*(-20*A*c+13 *B*b)*(c*x^2+b*x)^(7/2)/c^4+11/2880*b*(-20*A*c+13*B*b)*x*(c*x^2+b*x)^(7/2) /c^3-1/180*(-20*A*c+13*B*b)*x^2*(c*x^2+b*x)^(7/2)/c^2+1/10*B*x^3*(c*x^2+b* x)^(7/2)/c-11/131072*b^9*(-20*A*c+13*B*b)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1 /2))/c^(15/2)+11/131072*b^7*(-20*A*c+13*B*b)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c ^7
Time = 1.89 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.07 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (45045 b^9 B+5120 b^3 c^6 x^5 (5 A+3 B x)+458752 c^9 x^8 (10 A+9 B x)-1280 b^4 c^5 x^4 (22 A+13 B x)+1848 b^7 c^2 x (25 A+13 B x)-2310 b^8 c (30 A+13 B x)+704 b^5 c^4 x^3 (45 A+26 B x)-528 b^6 c^3 x^2 (70 A+39 B x)+57344 b c^8 x^7 (185 A+164 B x)+2048 b^2 c^7 x^6 (3090 A+2681 B x)\right )+90090 b^{10} B \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+138600 A b^9 c \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{41287680 c^{15/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(45045*b^9*B + 5120* b^3*c^6*x^5*(5*A + 3*B*x) + 458752*c^9*x^8*(10*A + 9*B*x) - 1280*b^4*c^5*x ^4*(22*A + 13*B*x) + 1848*b^7*c^2*x*(25*A + 13*B*x) - 2310*b^8*c*(30*A + 1 3*B*x) + 704*b^5*c^4*x^3*(45*A + 26*B*x) - 528*b^6*c^3*x^2*(70*A + 39*B*x) + 57344*b*c^8*x^7*(185*A + 164*B*x) + 2048*b^2*c^7*x^6*(3090*A + 2681*B*x )) + 90090*b^10*B*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 1 38600*A*b^9*c*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(412 87680*c^(15/2)*Sqrt[x*(b + c*x)])
Time = 0.43 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1221, 1134, 1134, 1160, 1087, 1087, 1087, 1091, 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.
| \(\displaystyle \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \int x^3 \left (c x^2+b x\right )^{5/2}dx}{20 c}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \int x^2 \left (c x^2+b x\right )^{5/2}dx}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \int x \left (c x^2+b x\right )^{5/2}dx}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \int \left (c x^2+b x\right )^{5/2}dx}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{3/2}dx}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac {(13 b B-20 A c) \left (\frac {x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {11 b \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\) |
(B*x^3*(b*x + c*x^2)^(7/2))/(10*c) - ((13*b*B - 20*A*c)*((x^2*(b*x + c*x^2 )^(7/2))/(9*c) - (11*b*((x*(b*x + c*x^2)^(7/2))/(8*c) - (9*b*((b*x + c*x^2 )^(7/2)/(7*c) - (b*(((b + 2*c*x)*(b*x + c*x^2)^(5/2))/(12*c) - (5*b^2*(((b + 2*c*x)*(b*x + c*x^2)^(3/2))/(8*c) - (3*b^2*(((b + 2*c*x)*Sqrt[b*x + c*x ^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(3/2))))/(1 6*c)))/(24*c)))/(2*c)))/(16*c)))/(18*c)))/(20*c)
3.1.93.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {\left (-4128768 B \,c^{9} x^{9}-4587520 A \,c^{9} x^{8}-9404416 B b \,c^{8} x^{8}-10608640 A b \,c^{8} x^{7}-5490688 B \,b^{2} c^{7} x^{7}-6328320 A \,b^{2} c^{7} x^{6}-15360 B \,b^{3} c^{6} x^{6}-25600 A \,b^{3} c^{6} x^{5}+16640 B \,b^{4} c^{5} x^{5}+28160 A \,b^{4} c^{5} x^{4}-18304 B \,b^{5} c^{4} x^{4}-31680 A \,b^{5} c^{4} x^{3}+20592 B \,b^{6} c^{3} x^{3}+36960 A \,b^{6} c^{3} x^{2}-24024 B \,b^{7} c^{2} x^{2}-46200 A \,b^{7} c^{2} x +30030 B \,b^{8} c x +69300 A \,b^{8} c -45045 B \,b^{9}\right ) x \left (c x +b \right )}{41287680 c^{7} \sqrt {x \left (c x +b \right )}}+\frac {11 b^{9} \left (20 A c -13 B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{262144 c^{\frac {15}{2}}}\) | \(265\) |
| default | \(B \left (\frac {x^{3} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{10 c}-\frac {13 b \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\right )+A \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )\) | \(412\) |
-1/41287680/c^7*(-4128768*B*c^9*x^9-4587520*A*c^9*x^8-9404416*B*b*c^8*x^8- 10608640*A*b*c^8*x^7-5490688*B*b^2*c^7*x^7-6328320*A*b^2*c^7*x^6-15360*B*b ^3*c^6*x^6-25600*A*b^3*c^6*x^5+16640*B*b^4*c^5*x^5+28160*A*b^4*c^5*x^4-183 04*B*b^5*c^4*x^4-31680*A*b^5*c^4*x^3+20592*B*b^6*c^3*x^3+36960*A*b^6*c^3*x ^2-24024*B*b^7*c^2*x^2-46200*A*b^7*c^2*x+30030*B*b^8*c*x+69300*A*b^8*c-450 45*B*b^9)*x*(c*x+b)/(x*(c*x+b))^(1/2)+11/262144*b^9*(20*A*c-13*B*b)/c^(15/ 2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))
Time = 0.28 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.96 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\left [-\frac {3465 \, {\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 69300 \, A b^{8} c^{2} + 229376 \, {\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \, {\left (383 \, B b^{2} c^{8} + 740 \, A b c^{9}\right )} x^{7} + 15360 \, {\left (B b^{3} c^{7} + 412 \, A b^{2} c^{8}\right )} x^{6} - 1280 \, {\left (13 \, B b^{4} c^{6} - 20 \, A b^{3} c^{7}\right )} x^{5} + 1408 \, {\left (13 \, B b^{5} c^{5} - 20 \, A b^{4} c^{6}\right )} x^{4} - 1584 \, {\left (13 \, B b^{6} c^{4} - 20 \, A b^{5} c^{5}\right )} x^{3} + 1848 \, {\left (13 \, B b^{7} c^{3} - 20 \, A b^{6} c^{4}\right )} x^{2} - 2310 \, {\left (13 \, B b^{8} c^{2} - 20 \, A b^{7} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{82575360 \, c^{8}}, \frac {3465 \, {\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 69300 \, A b^{8} c^{2} + 229376 \, {\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \, {\left (383 \, B b^{2} c^{8} + 740 \, A b c^{9}\right )} x^{7} + 15360 \, {\left (B b^{3} c^{7} + 412 \, A b^{2} c^{8}\right )} x^{6} - 1280 \, {\left (13 \, B b^{4} c^{6} - 20 \, A b^{3} c^{7}\right )} x^{5} + 1408 \, {\left (13 \, B b^{5} c^{5} - 20 \, A b^{4} c^{6}\right )} x^{4} - 1584 \, {\left (13 \, B b^{6} c^{4} - 20 \, A b^{5} c^{5}\right )} x^{3} + 1848 \, {\left (13 \, B b^{7} c^{3} - 20 \, A b^{6} c^{4}\right )} x^{2} - 2310 \, {\left (13 \, B b^{8} c^{2} - 20 \, A b^{7} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{41287680 \, c^{8}}\right ] \]
[-1/82575360*(3465*(13*B*b^10 - 20*A*b^9*c)*sqrt(c)*log(2*c*x + b + 2*sqrt (c*x^2 + b*x)*sqrt(c)) - 2*(4128768*B*c^10*x^9 + 45045*B*b^9*c - 69300*A*b ^8*c^2 + 229376*(41*B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 740* A*b*c^9)*x^7 + 15360*(B*b^3*c^7 + 412*A*b^2*c^8)*x^6 - 1280*(13*B*b^4*c^6 - 20*A*b^3*c^7)*x^5 + 1408*(13*B*b^5*c^5 - 20*A*b^4*c^6)*x^4 - 1584*(13*B* b^6*c^4 - 20*A*b^5*c^5)*x^3 + 1848*(13*B*b^7*c^3 - 20*A*b^6*c^4)*x^2 - 231 0*(13*B*b^8*c^2 - 20*A*b^7*c^3)*x)*sqrt(c*x^2 + b*x))/c^8, 1/41287680*(346 5*(13*B*b^10 - 20*A*b^9*c)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x )) + (4128768*B*c^10*x^9 + 45045*B*b^9*c - 69300*A*b^8*c^2 + 229376*(41*B* b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 740*A*b*c^9)*x^7 + 15360*( B*b^3*c^7 + 412*A*b^2*c^8)*x^6 - 1280*(13*B*b^4*c^6 - 20*A*b^3*c^7)*x^5 + 1408*(13*B*b^5*c^5 - 20*A*b^4*c^6)*x^4 - 1584*(13*B*b^6*c^4 - 20*A*b^5*c^5 )*x^3 + 1848*(13*B*b^7*c^3 - 20*A*b^6*c^4)*x^2 - 2310*(13*B*b^8*c^2 - 20*A *b^7*c^3)*x)*sqrt(c*x^2 + b*x))/c^8]
Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (274) = 548\).
Time = 0.65 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.99 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\begin {cases} \frac {231 b^{6} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{1024 c^{6}} + \sqrt {b x + c x^{2}} \left (\frac {B c^{2} x^{9}}{10} - \frac {231 b^{5} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{512 c^{6}} + \frac {77 b^{4} x \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{256 c^{5}} - \frac {77 b^{3} x^{2} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{320 c^{4}} + \frac {33 b^{2} x^{3} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{160 c^{3}} - \frac {11 b x^{4} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{60 c^{2}} + \frac {x^{8} \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{9 c} + \frac {x^{7} \cdot \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{8 c} + \frac {x^{6} \cdot \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{7 c} + \frac {x^{5} \left (A b^{3} - \frac {13 b \left (3 A b^{2} c + B b^{3} - \frac {15 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {17 b \left (A c^{3} + \frac {41 B b c^{2}}{20}\right )}{18 c}\right )}{16 c}\right )}{14 c}\right )}{6 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {13}{2}}}{13} + \frac {B \left (b x\right )^{\frac {15}{2}}}{15 b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((231*b**6*(A*b**3 - 13*b*(3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(16*c))/(14*c))*Pie cewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True))/(102 4*c**6) + sqrt(b*x + c*x**2)*(B*c**2*x**9/10 - 231*b**5*(A*b**3 - 13*b*(3* A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b* c**2/20)/(18*c))/(16*c))/(14*c))/(512*c**6) + 77*b**4*x*(A*b**3 - 13*b*(3* A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b* c**2/20)/(18*c))/(16*c))/(14*c))/(256*c**5) - 77*b**3*x**2*(A*b**3 - 13*b* (3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B *b*c**2/20)/(18*c))/(16*c))/(14*c))/(320*c**4) + 33*b**2*x**3*(A*b**3 - 13 *b*(3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 4 1*B*b*c**2/20)/(18*c))/(16*c))/(14*c))/(160*c**3) - 11*b*x**4*(A*b**3 - 13 *b*(3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 4 1*B*b*c**2/20)/(18*c))/(16*c))/(14*c))/(60*c**2) + x**8*(A*c**3 + 41*B*b*c **2/20)/(9*c) + x**7*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2 /20)/(18*c))/(8*c) + x**6*(3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b* *2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(16*c))/(7*c) + x**5*(A*b**3 - 13*b*(3*A*b**2*c + B*b**3 - 15*b*(3*A*b*c**2 + 3*B*b**2*c - 17*b*(A*c** 3 + 41*B*b*c**2/20)/(18*c))/(16*c))/(14*c))/(6*c)), Ne(c, 0)), (2*(A*(b...
Time = 0.18 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.64 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x^{3}}{10 \, c} - \frac {13 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b x^{2}}{180 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A x^{2}}{9 \, c} + \frac {143 \, \sqrt {c x^{2} + b x} B b^{8} x}{65536 \, c^{6}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{6} x}{24576 \, c^{5}} - \frac {55 \, \sqrt {c x^{2} + b x} A b^{7} x}{16384 \, c^{5}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{4} x}{7680 \, c^{4}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{5} x}{6144 \, c^{4}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{2} x}{2880 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{3} x}{384 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b x}{144 \, c^{2}} - \frac {143 \, B b^{10} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{262144 \, c^{\frac {15}{2}}} + \frac {55 \, A b^{9} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{65536 \, c^{\frac {13}{2}}} + \frac {143 \, \sqrt {c x^{2} + b x} B b^{9}}{131072 \, c^{7}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{7}}{49152 \, c^{6}} - \frac {55 \, \sqrt {c x^{2} + b x} A b^{8}}{32768 \, c^{6}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{5}}{15360 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{6}}{12288 \, c^{5}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{3}}{4480 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{4}}{768 \, c^{4}} + \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b^{2}}{224 \, c^{3}} \]
1/10*(c*x^2 + b*x)^(7/2)*B*x^3/c - 13/180*(c*x^2 + b*x)^(7/2)*B*b*x^2/c^2 + 1/9*(c*x^2 + b*x)^(7/2)*A*x^2/c + 143/65536*sqrt(c*x^2 + b*x)*B*b^8*x/c^ 6 - 143/24576*(c*x^2 + b*x)^(3/2)*B*b^6*x/c^5 - 55/16384*sqrt(c*x^2 + b*x) *A*b^7*x/c^5 + 143/7680*(c*x^2 + b*x)^(5/2)*B*b^4*x/c^4 + 55/6144*(c*x^2 + b*x)^(3/2)*A*b^5*x/c^4 + 143/2880*(c*x^2 + b*x)^(7/2)*B*b^2*x/c^3 - 11/38 4*(c*x^2 + b*x)^(5/2)*A*b^3*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*A*b*x/c^2 - 143/262144*B*b^10*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(15/2) + 55/65536*A*b^9*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(13/2) + 14 3/131072*sqrt(c*x^2 + b*x)*B*b^9/c^7 - 143/49152*(c*x^2 + b*x)^(3/2)*B*b^7 /c^6 - 55/32768*sqrt(c*x^2 + b*x)*A*b^8/c^6 + 143/15360*(c*x^2 + b*x)^(5/2 )*B*b^5/c^5 + 55/12288*(c*x^2 + b*x)^(3/2)*A*b^6/c^5 - 143/4480*(c*x^2 + b *x)^(7/2)*B*b^3/c^4 - 11/768*(c*x^2 + b*x)^(5/2)*A*b^4/c^4 + 11/224*(c*x^2 + b*x)^(7/2)*A*b^2/c^3
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.11 \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{41287680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, {\left (18 \, B c^{2} x + \frac {41 \, B b c^{10} + 20 \, A c^{11}}{c^{9}}\right )} x + \frac {383 \, B b^{2} c^{9} + 740 \, A b c^{10}}{c^{9}}\right )} x + \frac {15 \, {\left (B b^{3} c^{8} + 412 \, A b^{2} c^{9}\right )}}{c^{9}}\right )} x - \frac {5 \, {\left (13 \, B b^{4} c^{7} - 20 \, A b^{3} c^{8}\right )}}{c^{9}}\right )} x + \frac {11 \, {\left (13 \, B b^{5} c^{6} - 20 \, A b^{4} c^{7}\right )}}{c^{9}}\right )} x - \frac {99 \, {\left (13 \, B b^{6} c^{5} - 20 \, A b^{5} c^{6}\right )}}{c^{9}}\right )} x + \frac {231 \, {\left (13 \, B b^{7} c^{4} - 20 \, A b^{6} c^{5}\right )}}{c^{9}}\right )} x - \frac {1155 \, {\left (13 \, B b^{8} c^{3} - 20 \, A b^{7} c^{4}\right )}}{c^{9}}\right )} x + \frac {3465 \, {\left (13 \, B b^{9} c^{2} - 20 \, A b^{8} c^{3}\right )}}{c^{9}}\right )} + \frac {11 \, {\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{262144 \, c^{\frac {15}{2}}} \]
1/41287680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + (41*B *b*c^10 + 20*A*c^11)/c^9)*x + (383*B*b^2*c^9 + 740*A*b*c^10)/c^9)*x + 15*( B*b^3*c^8 + 412*A*b^2*c^9)/c^9)*x - 5*(13*B*b^4*c^7 - 20*A*b^3*c^8)/c^9)*x + 11*(13*B*b^5*c^6 - 20*A*b^4*c^7)/c^9)*x - 99*(13*B*b^6*c^5 - 20*A*b^5*c ^6)/c^9)*x + 231*(13*B*b^7*c^4 - 20*A*b^6*c^5)/c^9)*x - 1155*(13*B*b^8*c^3 - 20*A*b^7*c^4)/c^9)*x + 3465*(13*B*b^9*c^2 - 20*A*b^8*c^3)/c^9) + 11/262 144*(13*B*b^10 - 20*A*b^9*c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqr t(c) + b))/c^(15/2)
Timed out. \[ \int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\int x^3\,{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]