Integrand size = 24, antiderivative size = 264 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}} \]
-5/6144*b^2*(32*A*c^2*d+9*b^2*B*e-16*b*c*(A*e+B*d))*(2*c*x+b)*(c*x^2+b*x)^ (3/2)/c^4+1/384*(32*A*c^2*d+9*b^2*B*e-16*b*c*(A*e+B*d))*(2*c*x+b)*(c*x^2+b *x)^(5/2)/c^3-1/112*(9*B*b*e-16*c*(A*e+B*d)-14*B*c*e*x)*(c*x^2+b*x)^(7/2)/ c^2-5/16384*b^6*(32*A*c^2*d+9*b^2*B*e-16*b*c*(A*e+B*d))*arctanh(x*c^(1/2)/ (c*x^2+b*x)^(1/2))/c^(11/2)+5/16384*b^4*(32*A*c^2*d+9*b^2*B*e-16*b*c*(A*e+ B*d))*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5
Time = 2.12 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.41 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (945 b^7 B e-210 b^6 c (8 B d+8 A e+3 B e x)+128 b^3 c^4 x^2 (3 B x (2 d+e x)+2 A (7 d+3 e x))+2048 c^7 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+56 b^5 c^2 (20 A (3 d+e x)+B x (20 d+9 e x))-16 b^4 c^3 x (28 A (5 d+2 e x)+B x (56 d+27 e x))+1024 b c^6 x^4 (4 A (35 d+29 e x)+B x (116 d+99 e x))+256 b^2 c^5 x^3 (B x (296 d+243 e x)+A (378 d+296 e x))\right )+210 b^6 \left (32 A c^2 d+9 b^2 B e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+3360 b^7 c (B d+A e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{344064 c^{11/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(945*b^7*B*e - 210*b ^6*c*(8*B*d + 8*A*e + 3*B*e*x) + 128*b^3*c^4*x^2*(3*B*x*(2*d + e*x) + 2*A* (7*d + 3*e*x)) + 2048*c^7*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x)) + 56*b^5*c^2*(20*A*(3*d + e*x) + B*x*(20*d + 9*e*x)) - 16*b^4*c^3*x*(28*A*(5 *d + 2*e*x) + B*x*(56*d + 27*e*x)) + 1024*b*c^6*x^4*(4*A*(35*d + 29*e*x) + B*x*(116*d + 99*e*x)) + 256*b^2*c^5*x^3*(B*x*(296*d + 243*e*x) + A*(378*d + 296*e*x))) + 210*b^6*(32*A*c^2*d + 9*b^2*B*e)*ArcTanh[(Sqrt[c]*Sqrt[x]) /(Sqrt[b] - Sqrt[b + c*x])] + 3360*b^7*c*(B*d + A*e)*ArcTanh[(Sqrt[c]*Sqrt [x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(344064*c^(11/2)*Sqrt[x*(b + c*x)])
Time = 0.39 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^{5/2} (d+e x) \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right ) \int \left (c x^2+b x\right )^{5/2}dx}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{3/2}dx}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right )}{24 c}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{32 c^2}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2}\) |
-1/112*((9*b*B*e - 16*c*(B*d + A*e) - 14*B*c*e*x)*(b*x + c*x^2)^(7/2))/c^2 + ((32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(((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)
3.12.84.3.1 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 = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(-\frac {\left (-43008 B \,c^{7} e \,x^{7}-49152 A \,c^{7} e \,x^{6}-101376 B b \,c^{6} e \,x^{6}-49152 B \,c^{7} d \,x^{6}-118784 A b \,c^{6} e \,x^{5}-57344 A \,c^{7} d \,x^{5}-62208 B \,b^{2} c^{5} e \,x^{5}-118784 B b \,c^{6} d \,x^{5}-75776 A \,b^{2} c^{5} e \,x^{4}-143360 A b \,c^{6} d \,x^{4}-384 B \,b^{3} c^{4} e \,x^{4}-75776 B \,b^{2} c^{5} d \,x^{4}-768 A \,b^{3} c^{4} e \,x^{3}-96768 A \,b^{2} c^{5} d \,x^{3}+432 B \,b^{4} c^{3} e \,x^{3}-768 B \,b^{3} c^{4} d \,x^{3}+896 A \,b^{4} c^{3} e \,x^{2}-1792 A \,b^{3} c^{4} d \,x^{2}-504 B \,b^{5} c^{2} e \,x^{2}+896 B \,b^{4} c^{3} d \,x^{2}-1120 A \,b^{5} c^{2} e x +2240 A \,b^{4} c^{3} d x +630 B \,b^{6} c e x -1120 B \,b^{5} c^{2} d x +1680 A \,b^{6} c e -3360 A \,b^{5} c^{2} d -945 B \,b^{7} e +1680 B \,b^{6} c d \right ) x \left (c x +b \right )}{344064 c^{5} \sqrt {x \left (c x +b \right )}}+\frac {5 b^{6} \left (16 A b c e -32 A \,c^{2} d -9 b^{2} B e +16 B b c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}\) | \(397\) |
| default | \(d A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+B e \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+\left (A e +B d \right ) \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(437\) |
-1/344064/c^5*(-43008*B*c^7*e*x^7-49152*A*c^7*e*x^6-101376*B*b*c^6*e*x^6-4 9152*B*c^7*d*x^6-118784*A*b*c^6*e*x^5-57344*A*c^7*d*x^5-62208*B*b^2*c^5*e* x^5-118784*B*b*c^6*d*x^5-75776*A*b^2*c^5*e*x^4-143360*A*b*c^6*d*x^4-384*B* b^3*c^4*e*x^4-75776*B*b^2*c^5*d*x^4-768*A*b^3*c^4*e*x^3-96768*A*b^2*c^5*d* x^3+432*B*b^4*c^3*e*x^3-768*B*b^3*c^4*d*x^3+896*A*b^4*c^3*e*x^2-1792*A*b^3 *c^4*d*x^2-504*B*b^5*c^2*e*x^2+896*B*b^4*c^3*d*x^2-1120*A*b^5*c^2*e*x+2240 *A*b^4*c^3*d*x+630*B*b^6*c*e*x-1120*B*b^5*c^2*d*x+1680*A*b^6*c*e-3360*A*b^ 5*c^2*d-945*B*b^7*e+1680*B*b^6*c*d)*x*(c*x+b)/(x*(c*x+b))^(1/2)+5/32768*b^ 6*(16*A*b*c*e-32*A*c^2*d-9*B*b^2*e+16*B*b*c*d)/c^(11/2)*ln((1/2*b+c*x)/c^( 1/2)+(c*x^2+b*x)^(1/2))
Time = 0.33 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.04 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (16 \, {\left (B b^{7} c - 2 \, A b^{6} c^{2}\right )} d - {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} e\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} e x^{7} + 3072 \, {\left (16 \, B c^{8} d + {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} e\right )} x^{6} + 256 \, {\left (16 \, {\left (29 \, B b c^{7} + 14 \, A c^{8}\right )} d + {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} e\right )} x^{5} + 128 \, {\left (16 \, {\left (37 \, B b^{2} c^{6} + 70 \, A b c^{7}\right )} d + {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} e\right )} x^{4} + 48 \, {\left (16 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )} d - {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} e\right )} x^{3} - 56 \, {\left (16 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )} d - {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} e\right )} x^{2} - 1680 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )} d + 105 \, {\left (9 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )} e + 70 \, {\left (16 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )} d - {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, -\frac {105 \, {\left (16 \, {\left (B b^{7} c - 2 \, A b^{6} c^{2}\right )} d - {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (43008 \, B c^{8} e x^{7} + 3072 \, {\left (16 \, B c^{8} d + {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} e\right )} x^{6} + 256 \, {\left (16 \, {\left (29 \, B b c^{7} + 14 \, A c^{8}\right )} d + {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} e\right )} x^{5} + 128 \, {\left (16 \, {\left (37 \, B b^{2} c^{6} + 70 \, A b c^{7}\right )} d + {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} e\right )} x^{4} + 48 \, {\left (16 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )} d - {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} e\right )} x^{3} - 56 \, {\left (16 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )} d - {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} e\right )} x^{2} - 1680 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )} d + 105 \, {\left (9 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )} e + 70 \, {\left (16 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )} d - {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \]
[1/688128*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*s qrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(43008*B*c^8*e*x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16*(29*B*b*c^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B*b^2* c^6 + 70*A*b*c^7)*d + (3*B*b^3*c^5 + 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B*b^3 *c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^3 - 56*(16*(B* b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680*(B*b ^6*c^2 - 2*A*b^5*c^3)*d + 105*(9*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B*b^5 *c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)*sqrt(c*x^2 + b* x))/c^6, -1/344064*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^ 7*c)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (43008*B*c^8*e *x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16*(29*B*b *c^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B* b^2*c^6 + 70*A*b*c^7)*d + (3*B*b^3*c^5 + 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B *b^3*c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^3 - 56*(16 *(B*b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680* (B*b^6*c^2 - 2*A*b^5*c^3)*d + 105*(9*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B *b^5*c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)*sqrt(c*x^2 + b*x))/c^6]
Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (270) = 540\).
Time = 0.92 (sec) , antiderivative size = 1188, normalized size of antiderivative = 4.50 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((-5*b**3*(A*b**3*d - 7*b*(A*b**3*e + 3*A*b**2*c*d + B*b**3*d - 9 *b*(3*A*b**2*c*e + 3*A*b*c**2*d + B*b**3*e + 3*B*b**2*c*d - 11*b*(3*A*b*c* *2*e + A*c**3*d + 3*B*b**2*c*e + 3*B*b*c**2*d - 13*b*(A*c**3*e + 33*B*b*c* *2*e/16 + B*c**3*d)/(14*c))/(12*c))/(10*c))/(8*c))*Piecewise((log(b + 2*sq rt(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*e*x**7/8 + 5*b**2*(A*b**3*d - 7*b*(A*b**3*e + 3*A*b**2*c*d + B*b**3*d - 9*b*(3*A*b**2*c*e + 3*A*b*c**2*d + B*b**3*e + 3*B*b**2*c*d - 11*b*(3*A*b*c**2*e + A*c**3*d + 3*B*b**2*c*e + 3*B*b*c**2*d - 13*b*(A*c**3 *e + 33*B*b*c**2*e/16 + B*c**3*d)/(14*c))/(12*c))/(10*c))/(8*c))/(8*c**3) - 5*b*x*(A*b**3*d - 7*b*(A*b**3*e + 3*A*b**2*c*d + B*b**3*d - 9*b*(3*A*b** 2*c*e + 3*A*b*c**2*d + B*b**3*e + 3*B*b**2*c*d - 11*b*(3*A*b*c**2*e + A*c* *3*d + 3*B*b**2*c*e + 3*B*b*c**2*d - 13*b*(A*c**3*e + 33*B*b*c**2*e/16 + B *c**3*d)/(14*c))/(12*c))/(10*c))/(8*c))/(12*c**2) + x**6*(A*c**3*e + 33*B* b*c**2*e/16 + B*c**3*d)/(7*c) + x**5*(3*A*b*c**2*e + A*c**3*d + 3*B*b**2*c *e + 3*B*b*c**2*d - 13*b*(A*c**3*e + 33*B*b*c**2*e/16 + B*c**3*d)/(14*c))/ (6*c) + x**4*(3*A*b**2*c*e + 3*A*b*c**2*d + B*b**3*e + 3*B*b**2*c*d - 11*b *(3*A*b*c**2*e + A*c**3*d + 3*B*b**2*c*e + 3*B*b*c**2*d - 13*b*(A*c**3*e + 33*B*b*c**2*e/16 + B*c**3*d)/(14*c))/(12*c))/(5*c) + x**3*(A*b**3*e + 3*A *b**2*c*d + B*b**3*d - 9*b*(3*A*b**2*c*e + 3*A*b*c**2*d + B*b**3*e + 3*...
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (240) = 480\).
Time = 0.20 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.17 \[ \int (A+B x) (d+e 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 d x + \frac {5 \, \sqrt {c x^{2} + b x} A b^{4} d x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2} d x}{96 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{6} e x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e x}{1024 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B e x}{8 \, c} - \frac {5 \, A b^{6} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} - \frac {45 \, B b^{8} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{5} d}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3} d}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b d}{12 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{7} e}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5} e}{2048 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3} e}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b e}{112 \, c^{2}} - \frac {5 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{5} x}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{3} x}{192 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )} b x}{12 \, c} + \frac {5 \, {\left (B d + A e\right )} 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 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{6}}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{4}}{384 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )} b^{2}}{24 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} {\left (B d + A e\right )}}{7 \, c} \]
1/6*(c*x^2 + b*x)^(5/2)*A*d*x + 5/256*sqrt(c*x^2 + b*x)*A*b^4*d*x/c^2 - 5/ 96*(c*x^2 + b*x)^(3/2)*A*b^2*d*x/c + 45/8192*sqrt(c*x^2 + b*x)*B*b^6*e*x/c ^4 - 15/1024*(c*x^2 + b*x)^(3/2)*B*b^4*e*x/c^3 + 3/64*(c*x^2 + b*x)^(5/2)* B*b^2*e*x/c^2 + 1/8*(c*x^2 + b*x)^(7/2)*B*e*x/c - 5/1024*A*b^6*d*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 45/32768*B*b^8*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 5/512*sqrt(c*x^2 + b*x)*A*b^5 *d/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*A*b^3*d/c^2 + 1/12*(c*x^2 + b*x)^(5/2)* A*b*d/c + 45/16384*sqrt(c*x^2 + b*x)*B*b^7*e/c^5 - 15/2048*(c*x^2 + b*x)^( 3/2)*B*b^5*e/c^4 + 3/128*(c*x^2 + b*x)^(5/2)*B*b^3*e/c^3 - 9/112*(c*x^2 + b*x)^(7/2)*B*b*e/c^2 - 5/512*sqrt(c*x^2 + b*x)*(B*d + A*e)*b^5*x/c^3 + 5/1 92*(c*x^2 + b*x)^(3/2)*(B*d + A*e)*b^3*x/c^2 - 1/12*(c*x^2 + b*x)^(5/2)*(B *d + A*e)*b*x/c + 5/2048*(B*d + A*e)*b^7*log(2*c*x + b + 2*sqrt(c*x^2 + b* x)*sqrt(c))/c^(9/2) - 5/1024*sqrt(c*x^2 + b*x)*(B*d + A*e)*b^6/c^4 + 5/384 *(c*x^2 + b*x)^(3/2)*(B*d + A*e)*b^4/c^3 - 1/24*(c*x^2 + b*x)^(5/2)*(B*d + A*e)*b^2/c^2 + 1/7*(c*x^2 + b*x)^(7/2)*(B*d + A*e)/c
Time = 0.31 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.54 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} e x + \frac {16 \, B c^{9} d + 33 \, B b c^{8} e + 16 \, A c^{9} e}{c^{7}}\right )} x + \frac {464 \, B b c^{8} d + 224 \, A c^{9} d + 243 \, B b^{2} c^{7} e + 464 \, A b c^{8} e}{c^{7}}\right )} x + \frac {592 \, B b^{2} c^{7} d + 1120 \, A b c^{8} d + 3 \, B b^{3} c^{6} e + 592 \, A b^{2} c^{7} e}{c^{7}}\right )} x + \frac {3 \, {\left (16 \, B b^{3} c^{6} d + 2016 \, A b^{2} c^{7} d - 9 \, B b^{4} c^{5} e + 16 \, A b^{3} c^{6} e\right )}}{c^{7}}\right )} x - \frac {7 \, {\left (16 \, B b^{4} c^{5} d - 32 \, A b^{3} c^{6} d - 9 \, B b^{5} c^{4} e + 16 \, A b^{4} c^{5} e\right )}}{c^{7}}\right )} x + \frac {35 \, {\left (16 \, B b^{5} c^{4} d - 32 \, A b^{4} c^{5} d - 9 \, B b^{6} c^{3} e + 16 \, A b^{5} c^{4} e\right )}}{c^{7}}\right )} x - \frac {105 \, {\left (16 \, B b^{6} c^{3} d - 32 \, A b^{5} c^{4} d - 9 \, B b^{7} c^{2} e + 16 \, A b^{6} c^{3} e\right )}}{c^{7}}\right )} - \frac {5 \, {\left (16 \, B b^{7} c d - 32 \, A b^{6} c^{2} d - 9 \, B b^{8} e + 16 \, A b^{7} c e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \]
1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*e*x + (16*B*c^9*d + 33*B*b*c^8*e + 16*A*c^9*e)/c^7)*x + (464*B*b*c^8*d + 224*A*c^9*d + 243*B *b^2*c^7*e + 464*A*b*c^8*e)/c^7)*x + (592*B*b^2*c^7*d + 1120*A*b*c^8*d + 3 *B*b^3*c^6*e + 592*A*b^2*c^7*e)/c^7)*x + 3*(16*B*b^3*c^6*d + 2016*A*b^2*c^ 7*d - 9*B*b^4*c^5*e + 16*A*b^3*c^6*e)/c^7)*x - 7*(16*B*b^4*c^5*d - 32*A*b^ 3*c^6*d - 9*B*b^5*c^4*e + 16*A*b^4*c^5*e)/c^7)*x + 35*(16*B*b^5*c^4*d - 32 *A*b^4*c^5*d - 9*B*b^6*c^3*e + 16*A*b^5*c^4*e)/c^7)*x - 105*(16*B*b^6*c^3* d - 32*A*b^5*c^4*d - 9*B*b^7*c^2*e + 16*A*b^6*c^3*e)/c^7) - 5/32768*(16*B* b^7*c*d - 32*A*b^6*c^2*d - 9*B*b^8*e + 16*A*b^7*c*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)
Timed out. \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )\,\left (d+e\,x\right ) \,d x \]