Integrand size = 20, antiderivative size = 189 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {5 b^5 (9 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^3 (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 b^7 (9 b B-16 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}} \]
-5/6144*b^3*(-16*A*c+9*B*b)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/384*b*(-16*A *c+9*B*b)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^3-1/112*(-14*B*c*x-16*A*c+9*B*b)*( c*x^2+b*x)^(7/2)/c^2-5/16384*b^7*(-16*A*c+9*B*b)*arctanh(x*c^(1/2)/(c*x^2+ b*x)^(1/2))/c^(11/2)+5/16384*b^5*(-16*A*c+9*B*b)*(2*c*x+b)*(c*x^2+b*x)^(1/ 2)/c^5
Time = 1.38 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.36 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (945 b^7 B+384 b^3 c^4 x^3 (2 A+B x)-210 b^6 c (8 A+3 B x)+6144 c^7 x^6 (8 A+7 B x)+56 b^5 c^2 x (20 A+9 B x)-16 b^4 c^3 x^2 (56 A+27 B x)+1024 b c^6 x^5 (116 A+99 B x)+256 b^2 c^5 x^4 (296 A+243 B x)\right )+1890 b^8 B \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+3360 A b^7 c \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{344064 c^{11/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(945*b^7*B + 384*b^3 *c^4*x^3*(2*A + B*x) - 210*b^6*c*(8*A + 3*B*x) + 6144*c^7*x^6*(8*A + 7*B*x ) + 56*b^5*c^2*x*(20*A + 9*B*x) - 16*b^4*c^3*x^2*(56*A + 27*B*x) + 1024*b* c^6*x^5*(116*A + 99*B*x) + 256*b^2*c^5*x^4*(296*A + 243*B*x)) + 1890*b^8*B *ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 3360*A*b^7*c*ArcTa nh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(344064*c^(11/2)*Sqrt[x *(b + c*x)])
Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1225, 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.
| \(\displaystyle \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {b (9 b B-16 A c) \int \left (c x^2+b x\right )^{5/2}dx}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {b (9 b B-16 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{3/2}dx}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {b (9 b B-16 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {b (9 b B-16 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {b (9 b B-16 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b (9 b B-16 A c) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\) |
-1/112*((9*b*B - 16*A*c - 14*B*c*x)*(b*x + c*x^2)^(7/2))/c^2 + (b*(9*b*B - 16*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)))/(32*c^2)
3.1.95.3.1 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_))*((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]
Time = 0.12 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {\left (-43008 B \,c^{7} x^{7}-49152 A \,c^{7} x^{6}-101376 B b \,c^{6} x^{6}-118784 A b \,c^{6} x^{5}-62208 B \,b^{2} c^{5} x^{5}-75776 A \,b^{2} c^{5} x^{4}-384 B \,b^{3} c^{4} x^{4}-768 A \,b^{3} c^{4} x^{3}+432 B \,b^{4} c^{3} x^{3}+896 A \,b^{4} c^{3} x^{2}-504 B \,b^{5} c^{2} x^{2}-1120 A \,b^{5} c^{2} x +630 B \,b^{6} c x +1680 A \,b^{6} c -945 B \,b^{7}\right ) x \left (c x +b \right )}{344064 c^{5} \sqrt {x \left (c x +b \right )}}+\frac {5 b^{7} \left (16 A c -9 B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}\) | \(217\) |
| default | \(B \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+A \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(310\) |
-1/344064/c^5*(-43008*B*c^7*x^7-49152*A*c^7*x^6-101376*B*b*c^6*x^6-118784* A*b*c^6*x^5-62208*B*b^2*c^5*x^5-75776*A*b^2*c^5*x^4-384*B*b^3*c^4*x^4-768* A*b^3*c^4*x^3+432*B*b^4*c^3*x^3+896*A*b^4*c^3*x^2-504*B*b^5*c^2*x^2-1120*A *b^5*c^2*x+630*B*b^6*c*x+1680*A*b^6*c-945*B*b^7)*x*(c*x+b)/(x*(c*x+b))^(1/ 2)+5/32768*b^7*(16*A*c-9*B*b)/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^ (1/2))
Time = 0.28 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.37 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\left [-\frac {105 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (43008 \, B c^{8} x^{7} + 945 \, B b^{7} c - 1680 \, A b^{6} c^{2} + 3072 \, {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 256 \, {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (43008 \, B c^{8} x^{7} + 945 \, B b^{7} c - 1680 \, A b^{6} c^{2} + 3072 \, {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 256 \, {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \]
[-1/688128*(105*(9*B*b^8 - 16*A*b^7*c)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^ 2 + b*x)*sqrt(c)) - 2*(43008*B*c^8*x^7 + 945*B*b^7*c - 1680*A*b^6*c^2 + 30 72*(33*B*b*c^7 + 16*A*c^8)*x^6 + 256*(243*B*b^2*c^6 + 464*A*b*c^7)*x^5 + 1 28*(3*B*b^3*c^5 + 592*A*b^2*c^6)*x^4 - 48*(9*B*b^4*c^4 - 16*A*b^3*c^5)*x^3 + 56*(9*B*b^5*c^3 - 16*A*b^4*c^4)*x^2 - 70*(9*B*b^6*c^2 - 16*A*b^5*c^3)*x )*sqrt(c*x^2 + b*x))/c^6, 1/344064*(105*(9*B*b^8 - 16*A*b^7*c)*sqrt(-c)*ar ctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (43008*B*c^8*x^7 + 945*B*b^7*c - 1680*A*b^6*c^2 + 3072*(33*B*b*c^7 + 16*A*c^8)*x^6 + 256*(243*B*b^2*c^6 + 4 64*A*b*c^7)*x^5 + 128*(3*B*b^3*c^5 + 592*A*b^2*c^6)*x^4 - 48*(9*B*b^4*c^4 - 16*A*b^3*c^5)*x^3 + 56*(9*B*b^5*c^3 - 16*A*b^4*c^4)*x^2 - 70*(9*B*b^6*c^ 2 - 16*A*b^5*c^3)*x)*sqrt(c*x^2 + b*x))/c^6]
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (189) = 378\).
Time = 0.68 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.47 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\begin {cases} \frac {35 b^{4} \left (A b^{3} - \frac {9 b \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{10 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 )}{128 c^{4}} + \sqrt {b x + c x^{2}} \left (\frac {B c^{2} x^{7}}{8} - \frac {35 b^{3} \left (A b^{3} - \frac {9 b \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{10 c}\right )}{64 c^{4}} + \frac {35 b^{2} x \left (A b^{3} - \frac {9 b \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{10 c}\right )}{96 c^{3}} - \frac {7 b x^{2} \left (A b^{3} - \frac {9 b \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{10 c}\right )}{24 c^{2}} + \frac {x^{6} \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{7 c} + \frac {x^{5} \cdot \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{6 c} + \frac {x^{4} \cdot \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{5 c} + \frac {x^{3} \left (A b^{3} - \frac {9 b \left (3 A b^{2} c + B b^{3} - \frac {11 b \left (3 A b c^{2} + 3 B b^{2} c - \frac {13 b \left (A c^{3} + \frac {33 B b c^{2}}{16}\right )}{14 c}\right )}{12 c}\right )}{10 c}\right )}{4 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {9}{2}}}{9} + \frac {B \left (b x\right )^{\frac {11}{2}}}{11 b}\right )}{b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((35*b**4*(A*b**3 - 9*b*(3*A*b**2*c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16)/(14*c))/(12*c))/(10*c))*Piece wise((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))/(128*c **4) + sqrt(b*x + c*x**2)*(B*c**2*x**7/8 - 35*b**3*(A*b**3 - 9*b*(3*A*b**2 *c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/1 6)/(14*c))/(12*c))/(10*c))/(64*c**4) + 35*b**2*x*(A*b**3 - 9*b*(3*A*b**2*c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16) /(14*c))/(12*c))/(10*c))/(96*c**3) - 7*b*x**2*(A*b**3 - 9*b*(3*A*b**2*c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16)/(1 4*c))/(12*c))/(10*c))/(24*c**2) + x**6*(A*c**3 + 33*B*b*c**2/16)/(7*c) + x **5*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16)/(14*c))/(6*c ) + x**4*(3*A*b**2*c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c* *3 + 33*B*b*c**2/16)/(14*c))/(12*c))/(5*c) + x**3*(A*b**3 - 9*b*(3*A*b**2* c + B*b**3 - 11*b*(3*A*b*c**2 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16 )/(14*c))/(12*c))/(10*c))/(4*c)), Ne(c, 0)), (2*(A*(b*x)**(9/2)/9 + B*(b*x )**(11/2)/(11*b))/b**2, Ne(b, 0)), (0, True))
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (165) = 330\).
Time = 0.19 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.92 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {45 \, \sqrt {c x^{2} + b x} B b^{6} x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} x}{1024 \, c^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{5} x}{512 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} x}{64 \, c^{2}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3} x}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x}{8 \, c} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b x}{12 \, c} - \frac {45 \, B b^{8} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, A b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{7}}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5}}{2048 \, c^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{6}}{1024 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3}}{128 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{4}}{384 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b}{112 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{2}}{24 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A}{7 \, c} \]
45/8192*sqrt(c*x^2 + b*x)*B*b^6*x/c^4 - 15/1024*(c*x^2 + b*x)^(3/2)*B*b^4* x/c^3 - 5/512*sqrt(c*x^2 + b*x)*A*b^5*x/c^3 + 3/64*(c*x^2 + b*x)^(5/2)*B*b ^2*x/c^2 + 5/192*(c*x^2 + b*x)^(3/2)*A*b^3*x/c^2 + 1/8*(c*x^2 + b*x)^(7/2) *B*x/c - 1/12*(c*x^2 + b*x)^(5/2)*A*b*x/c - 45/32768*B*b^8*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 5/2048*A*b^7*log(2*c*x + b + 2*sq rt(c*x^2 + b*x)*sqrt(c))/c^(9/2) + 45/16384*sqrt(c*x^2 + b*x)*B*b^7/c^5 - 15/2048*(c*x^2 + b*x)^(3/2)*B*b^5/c^4 - 5/1024*sqrt(c*x^2 + b*x)*A*b^6/c^4 + 3/128*(c*x^2 + b*x)^(5/2)*B*b^3/c^3 + 5/384*(c*x^2 + b*x)^(3/2)*A*b^4/c ^3 - 9/112*(c*x^2 + b*x)^(7/2)*B*b/c^2 - 1/24*(c*x^2 + b*x)^(5/2)*A*b^2/c^ 2 + 1/7*(c*x^2 + b*x)^(7/2)*A/c
Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.33 \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} x + \frac {33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac {243 \, B b^{2} c^{7} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac {3 \, B b^{3} c^{6} + 592 \, A b^{2} c^{7}}{c^{7}}\right )} x - \frac {3 \, {\left (9 \, B b^{4} c^{5} - 16 \, A b^{3} c^{6}\right )}}{c^{7}}\right )} x + \frac {7 \, {\left (9 \, B b^{5} c^{4} - 16 \, A b^{4} c^{5}\right )}}{c^{7}}\right )} x - \frac {35 \, {\left (9 \, B b^{6} c^{3} - 16 \, A b^{5} c^{4}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (9 \, B b^{7} c^{2} - 16 \, A b^{6} c^{3}\right )}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \]
1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x + (33*B*b*c^8 + 16*A*c^9)/c^7)*x + (243*B*b^2*c^7 + 464*A*b*c^8)/c^7)*x + (3*B*b^3*c^6 + 5 92*A*b^2*c^7)/c^7)*x - 3*(9*B*b^4*c^5 - 16*A*b^3*c^6)/c^7)*x + 7*(9*B*b^5* c^4 - 16*A*b^4*c^5)/c^7)*x - 35*(9*B*b^6*c^3 - 16*A*b^5*c^4)/c^7)*x + 105* (9*B*b^7*c^2 - 16*A*b^6*c^3)/c^7) + 5/32768*(9*B*b^8 - 16*A*b^7*c)*log(abs (2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)
Timed out. \[ \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\int x\,{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]