Integrand size = 19, antiderivative size = 171 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=-\frac {5 b^4 (b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}+\frac {5 b^2 (b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {5 b^6 (b B-2 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}} \]
5/384*b^2*(-2*A*c+B*b)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^3-1/24*(-2*A*c+B*b)*( 2*c*x+b)*(c*x^2+b*x)^(5/2)/c^2+1/7*B*(c*x^2+b*x)^(7/2)/c+5/1024*b^6*(-2*A* c+B*b)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(9/2)-5/1024*b^4*(-2*A*c+B*b )*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^4
Time = 1.07 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.39 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-105 b^6 B+70 b^5 c (3 A+B x)-28 b^4 c^2 x (5 A+2 B x)+16 b^3 c^3 x^2 (7 A+3 B x)+512 c^6 x^5 (7 A+6 B x)+256 b c^5 x^4 (35 A+29 B x)+32 b^2 c^4 x^3 (189 A+148 B x)\right )+420 A b^6 c \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+210 b^7 B \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{21504 c^{9/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-105*b^6*B + 70*b^5 *c*(3*A + B*x) - 28*b^4*c^2*x*(5*A + 2*B*x) + 16*b^3*c^3*x^2*(7*A + 3*B*x) + 512*c^6*x^5*(7*A + 6*B*x) + 256*b*c^5*x^4*(35*A + 29*B*x) + 32*b^2*c^4* x^3*(189*A + 148*B*x)) + 420*A*b^6*c*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 210*b^7*B*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(21504*c^(9/2)*Sqrt[x*(b + c*x)])
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \int \left (c x^2+b x\right )^{5/2}dx}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{3/2}dx}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
(B*(b*x + c*x^2)^(7/2))/(7*c) - ((b*B - 2*A*c)*(((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))))/(16*c)))/(24*c)))/(2*c)
3.1.96.3.1 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_))*((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]
Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\left (3072 B \,c^{6} x^{6}+3584 A \,c^{6} x^{5}+7424 B b \,c^{5} x^{5}+8960 A b \,c^{5} x^{4}+4736 B \,b^{2} c^{4} x^{4}+6048 A \,b^{2} c^{4} x^{3}+48 B \,b^{3} c^{3} x^{3}+112 A \,b^{3} c^{3} x^{2}-56 B \,b^{4} c^{2} x^{2}-140 A \,b^{4} c^{2} x +70 B \,b^{5} c x +210 A \,b^{5} c -105 B \,b^{6}\right ) x \left (c x +b \right )}{21504 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{6} \left (2 A c -B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}\) | \(193\) |
| default | \(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+B \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(263\) |
1/21504/c^4*(3072*B*c^6*x^6+3584*A*c^6*x^5+7424*B*b*c^5*x^5+8960*A*b*c^5*x ^4+4736*B*b^2*c^4*x^4+6048*A*b^2*c^4*x^3+48*B*b^3*c^3*x^3+112*A*b^3*c^3*x^ 2-56*B*b^4*c^2*x^2-140*A*b^4*c^2*x+70*B*b^5*c*x+210*A*b^5*c-105*B*b^6)*x*( c*x+b)/(x*(c*x+b))^(1/2)-5/2048*b^6*(2*A*c-B*b)/c^(9/2)*ln((1/2*b+c*x)/c^( 1/2)+(c*x^2+b*x)^(1/2))
Time = 0.27 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.29 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\left [-\frac {105 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 210 \, A b^{5} c^{2} + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (37 \, B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{4} + 48 \, {\left (B b^{3} c^{4} + 126 \, A b^{2} c^{5}\right )} x^{3} - 56 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{2} + 70 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{43008 \, c^{5}}, -\frac {105 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 210 \, A b^{5} c^{2} + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (37 \, B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{4} + 48 \, {\left (B b^{3} c^{4} + 126 \, A b^{2} c^{5}\right )} x^{3} - 56 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{2} + 70 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{21504 \, c^{5}}\right ] \]
[-1/43008*(105*(B*b^7 - 2*A*b^6*c)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(3072*B*c^7*x^6 - 105*B*b^6*c + 210*A*b^5*c^2 + 256*(29* B*b*c^6 + 14*A*c^7)*x^5 + 128*(37*B*b^2*c^5 + 70*A*b*c^6)*x^4 + 48*(B*b^3* c^4 + 126*A*b^2*c^5)*x^3 - 56*(B*b^4*c^3 - 2*A*b^3*c^4)*x^2 + 70*(B*b^5*c^ 2 - 2*A*b^4*c^3)*x)*sqrt(c*x^2 + b*x))/c^5, -1/21504*(105*(B*b^7 - 2*A*b^6 *c)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (3072*B*c^7*x^6 - 105*B*b^6*c + 210*A*b^5*c^2 + 256*(29*B*b*c^6 + 14*A*c^7)*x^5 + 128*(37*B* b^2*c^5 + 70*A*b*c^6)*x^4 + 48*(B*b^3*c^4 + 126*A*b^2*c^5)*x^3 - 56*(B*b^4 *c^3 - 2*A*b^3*c^4)*x^2 + 70*(B*b^5*c^2 - 2*A*b^4*c^3)*x)*sqrt(c*x^2 + b*x ))/c^5]
Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (163) = 326\).
Time = 0.55 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.33 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 b^{3} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 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 )}{16 c^{3}} + \sqrt {b x + c x^{2}} \left (\frac {B c^{2} x^{6}}{7} + \frac {5 b^{2} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{8 c^{3}} - \frac {5 b x \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{12 c^{2}} + \frac {x^{5} \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{6 c} + \frac {x^{4} \cdot \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{5 c} + \frac {x^{3} \cdot \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{4 c} + \frac {x^{2} \left (A b^{3} - \frac {7 b \left (3 A b^{2} c + B b^{3} - \frac {9 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {11 b \left (A c^{3} + \frac {29 B b c^{2}}{14}\right )}{12 c}\right )}{10 c}\right )}{8 c}\right )}{3 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {7}{2}}}{7} + \frac {B \left (b x\right )^{\frac {9}{2}}}{9 b}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-5*b**3*(A*b**3 - 7*b*(3*A*b**2*c + B*b**3 - 9*b*(3*A*b*c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(10*c))/(8*c))*Piecewi se((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))/(16*c**3 ) + sqrt(b*x + c*x**2)*(B*c**2*x**6/7 + 5*b**2*(A*b**3 - 7*b*(3*A*b**2*c + B*b**3 - 9*b*(3*A*b*c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(1 2*c))/(10*c))/(8*c))/(8*c**3) - 5*b*x*(A*b**3 - 7*b*(3*A*b**2*c + B*b**3 - 9*b*(3*A*b*c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(10 *c))/(8*c))/(12*c**2) + x**5*(A*c**3 + 29*B*b*c**2/14)/(6*c) + x**4*(3*A*b *c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(5*c) + x**3*( 3*A*b**2*c + B*b**3 - 9*b*(3*A*b*c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b *c**2/14)/(12*c))/(10*c))/(4*c) + x**2*(A*b**3 - 7*b*(3*A*b**2*c + B*b**3 - 9*b*(3*A*b*c**2 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(1 0*c))/(8*c))/(3*c)), Ne(c, 0)), (2*(A*(b*x)**(7/2)/7 + B*(b*x)**(9/2)/(9*b ))/b, Ne(b, 0)), (0, True))
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (147) = 294\).
Time = 0.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.86 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A x - \frac {5 \, \sqrt {c x^{2} + b x} B b^{5} x}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} x}{192 \, c^{2}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{4} x}{256 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b x}{12 \, c} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2} x}{96 \, c} + \frac {5 \, B b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} - \frac {5 \, A b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} - \frac {5 \, \sqrt {c x^{2} + b x} B b^{6}}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4}}{384 \, c^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{5}}{512 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2}}{24 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B}{7 \, c} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b}{12 \, c} \]
1/6*(c*x^2 + b*x)^(5/2)*A*x - 5/512*sqrt(c*x^2 + b*x)*B*b^5*x/c^3 + 5/192* (c*x^2 + b*x)^(3/2)*B*b^3*x/c^2 + 5/256*sqrt(c*x^2 + b*x)*A*b^4*x/c^2 - 1/ 12*(c*x^2 + b*x)^(5/2)*B*b*x/c - 5/96*(c*x^2 + b*x)^(3/2)*A*b^2*x/c + 5/20 48*B*b^7*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 5/1024*A*b ^6*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 5/1024*sqrt(c*x^ 2 + b*x)*B*b^6/c^4 + 5/384*(c*x^2 + b*x)^(3/2)*B*b^4/c^3 + 5/512*sqrt(c*x^ 2 + b*x)*A*b^5/c^3 - 1/24*(c*x^2 + b*x)^(5/2)*B*b^2/c^2 - 5/192*(c*x^2 + b *x)^(3/2)*A*b^3/c^2 + 1/7*(c*x^2 + b*x)^(7/2)*B/c + 1/12*(c*x^2 + b*x)^(5/ 2)*A*b/c
Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.28 \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, B c^{2} x + \frac {29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac {37 \, B b^{2} c^{6} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac {3 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )}}{c^{6}}\right )} x - \frac {7 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )}}{c^{6}}\right )} - \frac {5 \, {\left (B b^{7} - 2 \, A b^{6} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \]
1/21504*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*B*c^2*x + (29*B*b*c^7 + 14*A* c^8)/c^6)*x + (37*B*b^2*c^6 + 70*A*b*c^7)/c^6)*x + 3*(B*b^3*c^5 + 126*A*b^ 2*c^6)/c^6)*x - 7*(B*b^4*c^4 - 2*A*b^3*c^5)/c^6)*x + 35*(B*b^5*c^3 - 2*A*b ^4*c^4)/c^6)*x - 105*(B*b^6*c^2 - 2*A*b^5*c^3)/c^6) - 5/2048*(B*b^7 - 2*A* b^6*c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(9/2)
Timed out. \[ \int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]