Integrand size = 24, antiderivative size = 198 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=-\frac {2 (B d-A e) \sqrt {a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 b (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b^2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{105 e (b d-a e)^4 \sqrt {d+e x}} \]
-2/7*(-A*e+B*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)/(e*x+d)^(7/2)+2/35*(6*A*b*e-7*B *a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^2/(e*x+d)^(5/2)+8/105*b*(6*A*b*e-7* B*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^3/(e*x+d)^(3/2)+16/105*b^2*(6*A*b* e-7*B*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^4/(e*x+d)^(1/2)
Time = 0.22 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=\frac {2 \sqrt {a+b x} \left (15 B d e^2 (a+b x)^3-15 A e^3 (a+b x)^3-42 b B d e (a+b x)^2 (d+e x)+63 A b e^2 (a+b x)^2 (d+e x)-21 a B e^2 (a+b x)^2 (d+e x)+35 b^2 B d (a+b x) (d+e x)^2-105 A b^2 e (a+b x) (d+e x)^2+70 a b B e (a+b x) (d+e x)^2+105 A b^3 (d+e x)^3-105 a b^2 B (d+e x)^3\right )}{105 (b d-a e)^4 (d+e x)^{7/2}} \]
(2*Sqrt[a + b*x]*(15*B*d*e^2*(a + b*x)^3 - 15*A*e^3*(a + b*x)^3 - 42*b*B*d *e*(a + b*x)^2*(d + e*x) + 63*A*b*e^2*(a + b*x)^2*(d + e*x) - 21*a*B*e^2*( a + b*x)^2*(d + e*x) + 35*b^2*B*d*(a + b*x)*(d + e*x)^2 - 105*A*b^2*e*(a + b*x)*(d + e*x)^2 + 70*a*b*B*e*(a + b*x)*(d + e*x)^2 + 105*A*b^3*(d + e*x) ^3 - 105*a*b^2*B*(d + e*x)^3))/(105*(b*d - a*e)^4*(d + e*x)^(7/2))
Time = 0.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 55, 55, 48}
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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}}dx}{7 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {4 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right ) (-7 a B e+6 A b e+b B d)}{7 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
(-2*(B*d - A*e)*Sqrt[a + b*x])/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + ((b*B*d + 6*A*b*e - 7*a*B*e)*((2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (4*b*((2*Sqrt[a + b*x])/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (4*b*Sqrt[a + b *x])/(3*(b*d - a*e)^2*Sqrt[d + e*x])))/(5*(b*d - a*e))))/(7*e*(b*d - a*e))
3.23.43.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Time = 1.10 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} e^{3} x^{3}+56 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-168 A \,b^{3} d \,e^{2} x^{2}-28 B \,a^{2} b \,e^{3} x^{2}+200 B a \,b^{2} d \,e^{2} x^{2}-28 B \,b^{3} d^{2} e \,x^{2}-18 A \,a^{2} b \,e^{3} x +84 A a \,b^{2} d \,e^{2} x -210 A \,b^{3} d^{2} e x +21 B \,a^{3} e^{3} x -101 B \,a^{2} b d \,e^{2} x +259 B a \,b^{2} d^{2} e x -35 b^{3} B \,d^{3} x +15 a^{3} A \,e^{3}-63 A \,a^{2} b d \,e^{2}+105 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+6 B \,a^{3} d \,e^{2}-28 B \,a^{2} b \,d^{2} e +70 B a \,b^{2} d^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (a e -b d \right )^{4}}\) | \(281\) |
| gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} e^{3} x^{3}+56 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-168 A \,b^{3} d \,e^{2} x^{2}-28 B \,a^{2} b \,e^{3} x^{2}+200 B a \,b^{2} d \,e^{2} x^{2}-28 B \,b^{3} d^{2} e \,x^{2}-18 A \,a^{2} b \,e^{3} x +84 A a \,b^{2} d \,e^{2} x -210 A \,b^{3} d^{2} e x +21 B \,a^{3} e^{3} x -101 B \,a^{2} b d \,e^{2} x +259 B a \,b^{2} d^{2} e x -35 b^{3} B \,d^{3} x +15 a^{3} A \,e^{3}-63 A \,a^{2} b d \,e^{2}+105 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+6 B \,a^{3} d \,e^{2}-28 B \,a^{2} b \,d^{2} e +70 B a \,b^{2} d^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
-2/105*(b*x+a)^(1/2)*(-48*A*b^3*e^3*x^3+56*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x ^3+24*A*a*b^2*e^3*x^2-168*A*b^3*d*e^2*x^2-28*B*a^2*b*e^3*x^2+200*B*a*b^2*d *e^2*x^2-28*B*b^3*d^2*e*x^2-18*A*a^2*b*e^3*x+84*A*a*b^2*d*e^2*x-210*A*b^3* d^2*e*x+21*B*a^3*e^3*x-101*B*a^2*b*d*e^2*x+259*B*a*b^2*d^2*e*x-35*B*b^3*d^ 3*x+15*A*a^3*e^3-63*A*a^2*b*d*e^2+105*A*a*b^2*d^2*e-105*A*b^3*d^3+6*B*a^3* d*e^2-28*B*a^2*b*d^2*e+70*B*a*b^2*d^3)/(e*x+d)^(7/2)/(a*e-b*d)^4
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (174) = 348\).
Time = 6.01 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.76 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=-\frac {2 \, {\left (15 \, A a^{3} e^{3} + 35 \, {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} d^{3} - 7 \, {\left (4 \, B a^{2} b - 15 \, A a b^{2}\right )} d^{2} e + 3 \, {\left (2 \, B a^{3} - 21 \, A a^{2} b\right )} d e^{2} - 8 \, {\left (B b^{3} d e^{2} - {\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \, {\left (7 \, B b^{3} d^{2} e - 2 \, {\left (25 \, B a b^{2} - 21 \, A b^{3}\right )} d e^{2} + {\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (35 \, B b^{3} d^{3} - 7 \, {\left (37 \, B a b^{2} - 30 \, A b^{3}\right )} d^{2} e + {\left (101 \, B a^{2} b - 84 \, A a b^{2}\right )} d e^{2} - 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \]
-2/105*(15*A*a^3*e^3 + 35*(2*B*a*b^2 - 3*A*b^3)*d^3 - 7*(4*B*a^2*b - 15*A* a*b^2)*d^2*e + 3*(2*B*a^3 - 21*A*a^2*b)*d*e^2 - 8*(B*b^3*d*e^2 - (7*B*a*b^ 2 - 6*A*b^3)*e^3)*x^3 - 4*(7*B*b^3*d^2*e - 2*(25*B*a*b^2 - 21*A*b^3)*d*e^2 + (7*B*a^2*b - 6*A*a*b^2)*e^3)*x^2 - (35*B*b^3*d^3 - 7*(37*B*a*b^2 - 30*A *b^3)*d^2*e + (101*B*a^2*b - 84*A*a*b^2)*d*e^2 - 3*(7*B*a^3 - 6*A*a^2*b)*e ^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^8 - 4*a*b^3*d^7*e + 6*a^2*b^2*d^ 6*e^2 - 4*a^3*b*d^5*e^3 + a^4*d^4*e^4 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6 *a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*x^4 + 4*(b^4*d^5*e^3 - 4*a*b^3 *d^4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^3 + 6*(b^4*d ^6*e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e ^6)*x^2 + 4*(b^4*d^7*e - 4*a*b^3*d^6*e^2 + 6*a^2*b^2*d^5*e^3 - 4*a^3*b*d^4 *e^4 + a^4*d^3*e^5)*x)
\[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=\int \frac {A + B x}{\sqrt {a + b x} \left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (174) = 348\).
Time = 0.41 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.13 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{8} d e^{5} {\left | b \right |} - 7 \, B a b^{7} e^{6} {\left | b \right |} + 6 \, A b^{8} e^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}} + \frac {7 \, {\left (B b^{9} d^{2} e^{4} {\left | b \right |} - 8 \, B a b^{8} d e^{5} {\left | b \right |} + 6 \, A b^{9} d e^{5} {\left | b \right |} + 7 \, B a^{2} b^{7} e^{6} {\left | b \right |} - 6 \, A a b^{8} e^{6} {\left | b \right |}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} + \frac {35 \, {\left (B b^{10} d^{3} e^{3} {\left | b \right |} - 9 \, B a b^{9} d^{2} e^{4} {\left | b \right |} + 6 \, A b^{10} d^{2} e^{4} {\left | b \right |} + 15 \, B a^{2} b^{8} d e^{5} {\left | b \right |} - 12 \, A a b^{9} d e^{5} {\left | b \right |} - 7 \, B a^{3} b^{7} e^{6} {\left | b \right |} + 6 \, A a^{2} b^{8} e^{6} {\left | b \right |}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} {\left (b x + a\right )} - \frac {105 \, {\left (B a b^{10} d^{3} e^{3} {\left | b \right |} - A b^{11} d^{3} e^{3} {\left | b \right |} - 3 \, B a^{2} b^{9} d^{2} e^{4} {\left | b \right |} + 3 \, A a b^{10} d^{2} e^{4} {\left | b \right |} + 3 \, B a^{3} b^{8} d e^{5} {\left | b \right |} - 3 \, A a^{2} b^{9} d e^{5} {\left | b \right |} - B a^{4} b^{7} e^{6} {\left | b \right |} + A a^{3} b^{8} e^{6} {\left | b \right |}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]
2/105*((4*(b*x + a)*(2*(B*b^8*d*e^5*abs(b) - 7*B*a*b^7*e^6*abs(b) + 6*A*b^ 8*e^6*abs(b))*(b*x + a)/(b^6*d^4*e^3 - 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7) + 7*(B*b^9*d^2*e^4*abs(b) - 8*B*a*b^8*d* e^5*abs(b) + 6*A*b^9*d*e^5*abs(b) + 7*B*a^2*b^7*e^6*abs(b) - 6*A*a*b^8*e^6 *abs(b))/(b^6*d^4*e^3 - 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d* e^6 + a^4*b^2*e^7)) + 35*(B*b^10*d^3*e^3*abs(b) - 9*B*a*b^9*d^2*e^4*abs(b) + 6*A*b^10*d^2*e^4*abs(b) + 15*B*a^2*b^8*d*e^5*abs(b) - 12*A*a*b^9*d*e^5* abs(b) - 7*B*a^3*b^7*e^6*abs(b) + 6*A*a^2*b^8*e^6*abs(b))/(b^6*d^4*e^3 - 4 *a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7))*(b*x + a) - 105*(B*a*b^10*d^3*e^3*abs(b) - A*b^11*d^3*e^3*abs(b) - 3*B*a^2*b^9* d^2*e^4*abs(b) + 3*A*a*b^10*d^2*e^4*abs(b) + 3*B*a^3*b^8*d*e^5*abs(b) - 3* A*a^2*b^9*d*e^5*abs(b) - B*a^4*b^7*e^6*abs(b) + A*a^3*b^8*e^6*abs(b))/(b^6 *d^4*e^3 - 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2 *e^7))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)
Time = 2.92 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.12 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x\,\left (-42\,B\,a^4\,e^3+190\,B\,a^3\,b\,d\,e^2+6\,A\,a^3\,b\,e^3-462\,B\,a^2\,b^2\,d^2\,e-42\,A\,a^2\,b^2\,d\,e^2-70\,B\,a\,b^3\,d^3+210\,A\,a\,b^3\,d^2\,e+210\,A\,b^4\,d^3\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}-\frac {12\,B\,a^4\,d\,e^2+30\,A\,a^4\,e^3-56\,B\,a^3\,b\,d^2\,e-126\,A\,a^3\,b\,d\,e^2+140\,B\,a^2\,b^2\,d^3+210\,A\,a^2\,b^2\,d^2\,e-210\,A\,a\,b^3\,d^3}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^3\,x^4\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b^2\,x^3\,\left (a\,e+7\,b\,d\right )\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,x^2\,\left (-a^2\,e^2+14\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}\right )}{x^4\,\sqrt {a+b\,x}+\frac {d^4\,\sqrt {a+b\,x}}{e^4}+\frac {6\,d^2\,x^2\,\sqrt {a+b\,x}}{e^2}+\frac {4\,d\,x^3\,\sqrt {a+b\,x}}{e}+\frac {4\,d^3\,x\,\sqrt {a+b\,x}}{e^3}} \]
((d + e*x)^(1/2)*((x*(210*A*b^4*d^3 - 42*B*a^4*e^3 + 6*A*a^3*b*e^3 - 70*B* a*b^3*d^3 - 42*A*a^2*b^2*d*e^2 - 462*B*a^2*b^2*d^2*e + 210*A*a*b^3*d^2*e + 190*B*a^3*b*d*e^2))/(105*e^4*(a*e - b*d)^4) - (30*A*a^4*e^3 - 210*A*a*b^3 *d^3 + 12*B*a^4*d*e^2 + 140*B*a^2*b^2*d^3 + 210*A*a^2*b^2*d^2*e - 126*A*a^ 3*b*d*e^2 - 56*B*a^3*b*d^2*e)/(105*e^4*(a*e - b*d)^4) + (16*b^3*x^4*(6*A*b *e - 7*B*a*e + B*b*d))/(105*e^2*(a*e - b*d)^4) + (8*b^2*x^3*(a*e + 7*b*d)* (6*A*b*e - 7*B*a*e + B*b*d))/(105*e^3*(a*e - b*d)^4) + (2*b*x^2*(35*b^2*d^ 2 - a^2*e^2 + 14*a*b*d*e)*(6*A*b*e - 7*B*a*e + B*b*d))/(105*e^4*(a*e - b*d )^4)))/(x^4*(a + b*x)^(1/2) + (d^4*(a + b*x)^(1/2))/e^4 + (6*d^2*x^2*(a + b*x)^(1/2))/e^2 + (4*d*x^3*(a + b*x)^(1/2))/e + (4*d^3*x*(a + b*x)^(1/2))/ e^3)