Integrand size = 26, antiderivative size = 345 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \]
1/384*(48*A*c^3*d^2-9*b^3*B*e^2+14*b^2*c*e*(A*e+2*B*d)-24*b*c^2*d*(2*A*e+B *d))*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/7*B*(e*x+d)^2*(c*x^2+b*x)^(5/2)/c+1 /840*(14*A*c*e*(-7*b*e+24*c*d)+B*(63*b^2*e^2-196*b*c*d*e+48*c^2*d^2)+10*c* e*(14*A*c*e-9*B*b*e+4*B*c*d)*x)*(c*x^2+b*x)^(5/2)/c^3+1/1024*b^4*(48*A*c^3 *d^2-9*b^3*B*e^2+14*b^2*c*e*(A*e+2*B*d)-24*b*c^2*d*(2*A*e+B*d))*arctanh(x* c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)-1/1024*b^2*(48*A*c^3*d^2-9*b^3*B*e^2+1 4*b^2*c*e*(A*e+2*B*d)-24*b*c^2*d*(2*A*e+B*d))*(2*c*x+b)*(c*x^2+b*x)^(1/2)/ c^5
Time = 2.49 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.37 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (945 b^6 B e^2-210 b^5 c e (14 B d+7 A e+3 B e x)+96 b^2 c^4 x \left (7 A \left (5 d^2+4 d e x+e^2 x^2\right )+2 B x \left (7 d^2+7 d e x+2 e^2 x^2\right )\right )+28 b^4 c^2 \left (5 A e (36 d+7 e x)+2 B \left (45 d^2+35 d e x+9 e^2 x^2\right )\right )+256 c^6 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )-16 b^3 c^3 \left (7 A \left (45 d^2+30 d e x+7 e^2 x^2\right )+B x \left (105 d^2+98 d e x+27 e^2 x^2\right )\right )+128 b c^5 x^2 \left (7 A \left (45 d^2+66 d e x+26 e^2 x^2\right )+B x \left (231 d^2+364 d e x+150 e^2 x^2\right )\right )\right )+630 b^5 \left (8 B c^2 d^2+16 A c^2 d e+3 b^2 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+420 b^4 c \left (24 A c^2 d^2+14 b^2 B d e+7 A b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{107520 c^{11/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(945*b^6*B*e^2 - 210 *b^5*c*e*(14*B*d + 7*A*e + 3*B*e*x) + 96*b^2*c^4*x*(7*A*(5*d^2 + 4*d*e*x + e^2*x^2) + 2*B*x*(7*d^2 + 7*d*e*x + 2*e^2*x^2)) + 28*b^4*c^2*(5*A*e*(36*d + 7*e*x) + 2*B*(45*d^2 + 35*d*e*x + 9*e^2*x^2)) + 256*c^6*x^3*(7*A*(15*d^ 2 + 24*d*e*x + 10*e^2*x^2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2)) - 16* b^3*c^3*(7*A*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + B*x*(105*d^2 + 98*d*e*x + 2 7*e^2*x^2)) + 128*b*c^5*x^2*(7*A*(45*d^2 + 66*d*e*x + 26*e^2*x^2) + B*x*(2 31*d^2 + 364*d*e*x + 150*e^2*x^2))) + 630*b^5*(8*B*c^2*d^2 + 16*A*c^2*d*e + 3*b^2*B*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 420* b^4*c*(24*A*c^2*d^2 + 14*b^2*B*d*e + 7*A*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x] )/(-Sqrt[b] + Sqrt[b + c*x])]))/(107520*c^(11/2)*Sqrt[x*(b + c*x)])
Time = 0.48 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1236, 27, 1225, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^{3/2} (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {1}{2} (d+e x) ((5 b B-14 A c) d-(4 B c d-9 b B e+14 A c e) x) \left (c x^2+b x\right )^{3/2}dx}{7 c}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {\int (d+e x) ((5 b B-14 A c) d-(4 B c d-9 b B e+14 A c e) x) \left (c x^2+b x\right )^{3/2}dx}{14 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {-\frac {7 \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right ) \int \left (c x^2+b x\right )^{3/2}dx}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+2 B \left (\frac {63 b^2 e^2}{2}-98 b c d e+24 c^2 d^2\right )\right )}{60 c^2}}{14 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {-\frac {7 \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+2 B \left (\frac {63 b^2 e^2}{2}-98 b c d e+24 c^2 d^2\right )\right )}{60 c^2}}{14 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {-\frac {7 \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+2 B \left (\frac {63 b^2 e^2}{2}-98 b c d e+24 c^2 d^2\right )\right )}{60 c^2}}{14 c}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {-\frac {7 \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+2 B \left (\frac {63 b^2 e^2}{2}-98 b c d e+24 c^2 d^2\right )\right )}{60 c^2}}{14 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}-\frac {-\frac {7 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+2 B \left (\frac {63 b^2 e^2}{2}-98 b c d e+24 c^2 d^2\right )\right )}{60 c^2}}{14 c}\) |
(B*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) - (-1/60*((14*A*c*e*(24*c*d - 7* b*e) + 2*B*(24*c^2*d^2 - 98*b*c*d*e + (63*b^2*e^2)/2) + 10*c*e*(4*B*c*d - 9*b*B*e + 14*A*c*e)*x)*(b*x + c*x^2)^(5/2))/c^2 - (7*(48*A*c^3*d^2 - 9*b^3 *B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(((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))))/(16*c)))/( 24*c^2))/(14*c)
3.12.72.3.1 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_)^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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[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[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.54 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(-\frac {7 \left (-\left (-\frac {9 b^{3} B \,e^{2}}{14}+b^{2} c e \left (A e +2 B d \right )-\frac {24 c^{2} \left (A e +\frac {B d}{2}\right ) d b}{7}+\frac {24 A \,c^{3} d^{2}}{7}\right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\sqrt {x \left (c x +b \right )}\, \left (-\frac {192 \left (\frac {26 x^{2} \left (\frac {75 B x}{91}+A \right ) e^{2}}{45}+\frac {22 x d \left (\frac {26 B x}{33}+A \right ) e}{15}+d^{2} \left (\frac {11 B x}{15}+A \right )\right ) x^{2} b \,c^{\frac {11}{2}}}{7}-\frac {128 x^{3} \left (\frac {2 x^{2} \left (\frac {6 B x}{7}+A \right ) e^{2}}{3}+\frac {8 x \left (\frac {5 B x}{6}+A \right ) d e}{5}+d^{2} \left (\frac {4 B x}{5}+A \right )\right ) c^{\frac {13}{2}}}{7}+\left (\frac {24 \left (\left (\frac {3}{35} B \,x^{3}+\frac {7}{45} A \,x^{2}\right ) e^{2}+\frac {2 x d \left (\frac {7 B x}{15}+A \right ) e}{3}+d^{2} \left (\frac {B x}{3}+A \right )\right ) b \,c^{\frac {7}{2}}}{7}-\frac {16 \left (\frac {\left (\frac {4 B x}{7}+A \right ) x^{2} e^{2}}{5}+\frac {4 \left (\frac {B x}{2}+A \right ) x d e}{5}+d^{2} \left (\frac {2 B x}{5}+A \right )\right ) x \,c^{\frac {9}{2}}}{7}+\left (\left (-\frac {2 x \left (\frac {18 B x}{35}+A \right ) e^{2}}{3}-\frac {24 \left (\frac {7 B x}{18}+A \right ) d e}{7}-\frac {12 B \,d^{2}}{7}\right ) c^{\frac {5}{2}}+\left (\left (\left (\frac {3 B x}{7}+A \right ) e +2 B d \right ) c^{\frac {3}{2}}-\frac {9 B b e \sqrt {c}}{14}\right ) e b \right ) b^{2}\right ) b^{2}\right )\right )}{512 c^{\frac {11}{2}}}\) | \(336\) |
| default | \(A \,d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+B \,e^{2} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+\left (2 A d e +B \,d^{2}\right ) \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(523\) |
| risch | \(-\frac {\left (-15360 B \,c^{6} e^{2} x^{6}-17920 A \,c^{6} e^{2} x^{5}-19200 B b \,c^{5} e^{2} x^{5}-35840 B \,c^{6} d e \,x^{5}-23296 A b \,c^{5} e^{2} x^{4}-43008 A \,c^{6} d e \,x^{4}-384 B \,b^{2} c^{4} e^{2} x^{4}-46592 B b \,c^{5} d e \,x^{4}-21504 B \,c^{6} d^{2} x^{4}-672 A \,b^{2} c^{4} e^{2} x^{3}-59136 A b \,c^{5} d e \,x^{3}-26880 A \,c^{6} d^{2} x^{3}+432 B \,b^{3} c^{3} e^{2} x^{3}-1344 B \,b^{2} c^{4} d e \,x^{3}-29568 B b \,c^{5} d^{2} x^{3}+784 A \,b^{3} c^{3} e^{2} x^{2}-2688 A \,b^{2} c^{4} d e \,x^{2}-40320 A b \,c^{5} d^{2} x^{2}-504 B \,b^{4} c^{2} e^{2} x^{2}+1568 B \,b^{3} c^{3} d e \,x^{2}-1344 B \,b^{2} c^{4} d^{2} x^{2}-980 A \,b^{4} c^{2} e^{2} x +3360 A \,b^{3} c^{3} d e x -3360 A \,b^{2} c^{4} d^{2} x +630 B \,b^{5} c \,e^{2} x -1960 B \,b^{4} c^{2} d e x +1680 B \,b^{3} c^{3} d^{2} x +1470 A \,b^{5} c \,e^{2}-5040 A \,b^{4} c^{2} d e +5040 A \,b^{3} c^{3} d^{2}-945 B \,b^{6} e^{2}+2940 B \,b^{5} c d e -2520 B \,b^{4} c^{2} d^{2}\right ) x \left (c x +b \right )}{107520 c^{5} \sqrt {x \left (c x +b \right )}}+\frac {b^{4} \left (14 A \,b^{2} c \,e^{2}-48 A b \,c^{2} d e +48 A \,c^{3} d^{2}-9 b^{3} B \,e^{2}+28 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}\) | \(527\) |
-7/512/c^(11/2)*(-(-9/14*b^3*B*e^2+b^2*c*e*(A*e+2*B*d)-24/7*c^2*(A*e+1/2*B *d)*d*b+24/7*A*c^3*d^2)*b^4*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+(x*(c*x+b ))^(1/2)*(-192/7*(26/45*x^2*(75/91*B*x+A)*e^2+22/15*x*d*(26/33*B*x+A)*e+d^ 2*(11/15*B*x+A))*x^2*b*c^(11/2)-128/7*x^3*(2/3*x^2*(6/7*B*x+A)*e^2+8/5*x*( 5/6*B*x+A)*d*e+d^2*(4/5*B*x+A))*c^(13/2)+(24/7*((3/35*B*x^3+7/45*A*x^2)*e^ 2+2/3*x*d*(7/15*B*x+A)*e+d^2*(1/3*B*x+A))*b*c^(7/2)-16/7*(1/5*(4/7*B*x+A)* x^2*e^2+4/5*(1/2*B*x+A)*x*d*e+d^2*(2/5*B*x+A))*x*c^(9/2)+((-2/3*x*(18/35*B *x+A)*e^2-24/7*(7/18*B*x+A)*d*e-12/7*B*d^2)*c^(5/2)+(((3/7*B*x+A)*e+2*B*d) *c^(3/2)-9/14*B*b*e*c^(1/2))*e*b)*b^2)*b^2))
Time = 0.40 (sec) , antiderivative size = 987, normalized size of antiderivative = 2.86 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]
[-1/215040*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6*c - 12*A*b^ 5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c* x^2 + b*x)*sqrt(c)) - 2*(15360*B*c^7*e^2*x^6 + 1280*(28*B*c^7*d*e + (15*B* b*c^6 + 14*A*c^7)*e^2)*x^5 + 128*(168*B*c^7*d^2 + 28*(13*B*b*c^6 + 12*A*c^ 7)*d*e + (3*B*b^2*c^5 + 182*A*b*c^6)*e^2)*x^4 + 48*(56*(11*B*b*c^6 + 10*A* c^7)*d^2 + 28*(B*b^2*c^5 + 44*A*b*c^6)*d*e - (9*B*b^3*c^4 - 14*A*b^2*c^5)* e^2)*x^3 + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^2 - 420*(7*B*b^5*c^2 - 12*A*b^ 4*c^3)*d*e + 105*(9*B*b^6*c - 14*A*b^5*c^2)*e^2 + 56*(24*(B*b^2*c^5 + 30*A *b*c^6)*d^2 - 4*(7*B*b^3*c^4 - 12*A*b^2*c^5)*d*e + (9*B*b^4*c^3 - 14*A*b^3 *c^4)*e^2)*x^2 - 70*(24*(B*b^3*c^4 - 2*A*b^2*c^5)*d^2 - 4*(7*B*b^4*c^3 - 1 2*A*b^3*c^4)*d*e + (9*B*b^5*c^2 - 14*A*b^4*c^3)*e^2)*x)*sqrt(c*x^2 + b*x)) /c^6, 1/107520*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6*c - 12* A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (15360*B*c^7*e^2*x^6 + 1280*(28*B*c^7*d*e + (15*B*b *c^6 + 14*A*c^7)*e^2)*x^5 + 128*(168*B*c^7*d^2 + 28*(13*B*b*c^6 + 12*A*c^7 )*d*e + (3*B*b^2*c^5 + 182*A*b*c^6)*e^2)*x^4 + 48*(56*(11*B*b*c^6 + 10*A*c ^7)*d^2 + 28*(B*b^2*c^5 + 44*A*b*c^6)*d*e - (9*B*b^3*c^4 - 14*A*b^2*c^5)*e ^2)*x^3 + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^2 - 420*(7*B*b^5*c^2 - 12*A*b^4 *c^3)*d*e + 105*(9*B*b^6*c - 14*A*b^5*c^2)*e^2 + 56*(24*(B*b^2*c^5 + 30*A* b*c^6)*d^2 - 4*(7*B*b^3*c^4 - 12*A*b^2*c^5)*d*e + (9*B*b^4*c^3 - 14*A*b...
Leaf count of result is larger than twice the leaf count of optimal. 1224 vs. \(2 (357) = 714\).
Time = 0.76 (sec) , antiderivative size = 1224, normalized size of antiderivative = 3.55 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((3*b**2*(A*b**2*d**2 - 5*b*(2*A*b**2*d*e + 2*A*b*c*d**2 + B*b**2 *d**2 - 7*b*(A*b**2*e**2 + 4*A*b*c*d*e + A*c**2*d**2 + 2*B*b**2*d*e + 2*B* b*c*d**2 - 9*b*(2*A*b*c*e**2 + 2*A*c**2*d*e + B*b**2*e**2 + 4*B*b*c*d*e + B*c**2*d**2 - 11*b*(A*c**2*e**2 + 15*B*b*c*e**2/14 + 2*B*c**2*d*e)/(12*c)) /(10*c))/(8*c))/(6*c))*Piecewise((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))/(8*c**2) + sqrt(b*x + c*x**2)*(B*c*e**2*x**6/7 - 3*b *(A*b**2*d**2 - 5*b*(2*A*b**2*d*e + 2*A*b*c*d**2 + B*b**2*d**2 - 7*b*(A*b* *2*e**2 + 4*A*b*c*d*e + A*c**2*d**2 + 2*B*b**2*d*e + 2*B*b*c*d**2 - 9*b*(2 *A*b*c*e**2 + 2*A*c**2*d*e + B*b**2*e**2 + 4*B*b*c*d*e + B*c**2*d**2 - 11* b*(A*c**2*e**2 + 15*B*b*c*e**2/14 + 2*B*c**2*d*e)/(12*c))/(10*c))/(8*c))/( 6*c))/(4*c**2) + x**5*(A*c**2*e**2 + 15*B*b*c*e**2/14 + 2*B*c**2*d*e)/(6*c ) + x**4*(2*A*b*c*e**2 + 2*A*c**2*d*e + B*b**2*e**2 + 4*B*b*c*d*e + B*c**2 *d**2 - 11*b*(A*c**2*e**2 + 15*B*b*c*e**2/14 + 2*B*c**2*d*e)/(12*c))/(5*c) + x**3*(A*b**2*e**2 + 4*A*b*c*d*e + A*c**2*d**2 + 2*B*b**2*d*e + 2*B*b*c* d**2 - 9*b*(2*A*b*c*e**2 + 2*A*c**2*d*e + B*b**2*e**2 + 4*B*b*c*d*e + B*c* *2*d**2 - 11*b*(A*c**2*e**2 + 15*B*b*c*e**2/14 + 2*B*c**2*d*e)/(12*c))/(10 *c))/(4*c) + x**2*(2*A*b**2*d*e + 2*A*b*c*d**2 + B*b**2*d**2 - 7*b*(A*b**2 *e**2 + 4*A*b*c*d*e + A*c**2*d**2 + 2*B*b**2*d*e + 2*B*b*c*d**2 - 9*b*(2*A *b*c*e**2 + 2*A*c**2*d*e + B*b**2*e**2 + 4*B*b*c*d*e + B*c**2*d**2 - 11...
Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (321) = 642\).
Time = 0.20 (sec) , antiderivative size = 728, normalized size of antiderivative = 2.11 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B e^{2} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d^{2} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{5} e^{2} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e^{2} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b e^{2} x}{28 \, c^{2}} + \frac {3 \, A b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {9 \, B b^{7} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d^{2}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{6} e^{2}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e^{2}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e^{2}}{40 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x}{96 \, c^{2}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} x}{6 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{8 \, c} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}}{192 \, c^{3}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b}{60 \, c^{2}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{16 \, c^{2}} + \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{5 \, c} \]
1/7*(c*x^2 + b*x)^(5/2)*B*e^2*x^2/c + 1/4*(c*x^2 + b*x)^(3/2)*A*d^2*x - 3/ 32*sqrt(c*x^2 + b*x)*A*b^2*d^2*x/c + 9/512*sqrt(c*x^2 + b*x)*B*b^5*e^2*x/c ^4 - 3/64*(c*x^2 + b*x)^(3/2)*B*b^3*e^2*x/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*B *b*e^2*x/c^2 + 3/128*A*b^4*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c) )/c^(5/2) - 9/2048*B*b^7*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/ c^(11/2) - 3/64*sqrt(c*x^2 + b*x)*A*b^3*d^2/c^2 + 1/8*(c*x^2 + b*x)^(3/2)* A*b*d^2/c + 9/1024*sqrt(c*x^2 + b*x)*B*b^6*e^2/c^5 - 3/128*(c*x^2 + b*x)^( 3/2)*B*b^4*e^2/c^4 + 3/40*(c*x^2 + b*x)^(5/2)*B*b^2*e^2/c^3 - 7/256*(2*B*d *e + A*e^2)*sqrt(c*x^2 + b*x)*b^4*x/c^3 + 7/96*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(3/2)*b^2*x/c^2 + 3/64*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^3*x/c^2 + 1/6*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(5/2)*x/c - 1/8*(B*d^2 + 2*A*d*e)*(c *x^2 + b*x)^(3/2)*b*x/c + 7/1024*(2*B*d*e + A*e^2)*b^6*log(2*c*x + b + 2*s qrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 3/256*(B*d^2 + 2*A*d*e)*b^5*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 7/512*(2*B*d*e + A*e^2)*sqrt( c*x^2 + b*x)*b^5/c^4 + 7/192*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(3/2)*b^3/c^3 + 3/128*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^4/c^3 - 7/60*(2*B*d*e + A*e ^2)*(c*x^2 + b*x)^(5/2)*b/c^2 - 1/16*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2) *b^2/c^2 + 1/5*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(5/2)/c
Time = 0.31 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.50 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{107520} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c e^{2} x + \frac {28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac {168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac {3 \, {\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac {35 \, {\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac {{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \]
1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*e^2*x + (28*B*c^7*d*e + 15*B*b*c^6*e^2 + 14*A*c^7*e^2)/c^6)*x + (168*B*c^7*d^2 + 364*B*b*c^6*d*e + 336*A*c^7*d*e + 3*B*b^2*c^5*e^2 + 182*A*b*c^6*e^2)/c^6)*x + 3*(616*B*b*c ^6*d^2 + 560*A*c^7*d^2 + 28*B*b^2*c^5*d*e + 1232*A*b*c^6*d*e - 9*B*b^3*c^4 *e^2 + 14*A*b^2*c^5*e^2)/c^6)*x + 7*(24*B*b^2*c^5*d^2 + 720*A*b*c^6*d^2 - 28*B*b^3*c^4*d*e + 48*A*b^2*c^5*d*e + 9*B*b^4*c^3*e^2 - 14*A*b^3*c^4*e^2)/ c^6)*x - 35*(24*B*b^3*c^4*d^2 - 48*A*b^2*c^5*d^2 - 28*B*b^4*c^3*d*e + 48*A *b^3*c^4*d*e + 9*B*b^5*c^2*e^2 - 14*A*b^4*c^3*e^2)/c^6)*x + 105*(24*B*b^4* c^3*d^2 - 48*A*b^3*c^4*d^2 - 28*B*b^5*c^2*d*e + 48*A*b^4*c^3*d*e + 9*B*b^6 *c*e^2 - 14*A*b^5*c^2*e^2)/c^6) + 1/2048*(24*B*b^5*c^2*d^2 - 48*A*b^4*c^3* d^2 - 28*B*b^6*c*d*e + 48*A*b^5*c^2*d*e + 9*B*b^7*e^2 - 14*A*b^6*c*e^2)*lo g(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)
Timed out. \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2 \,d x \]