3.23.61 \(\int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx\) [2261]

Optimal. Leaf size=298 \[ -\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {256 b^2 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^6 \sqrt {d+e x}} \]

[Out]

-2/3*(A*b-B*a)/b/(-a*e+b*d)/(b*x+a)^(3/2)/(e*x+d)^(7/2)-2/3*(-10*A*b*e+7*B*a*e+3*B*b*d)/b/(-a*e+b*d)^2/(e*x+d)
^(7/2)/(b*x+a)^(1/2)-16/21*e*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^3/(e*x+d)^(7/2)-32/35*e*(-
10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(5/2)-128/105*b*e*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*
x+a)^(1/2)/(-a*e+b*d)^5/(e*x+d)^(3/2)-256/105*b^2*e*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^6/(e*
x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \begin {gather*} -\frac {256 b^2 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt {d+e x} (b d-a e)^6}-\frac {128 b e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac {32 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac {16 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac {2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d - 10*A*b*e + 7*a*B*e))/(3*b*(
b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d -
a*e)^3*(d + e*x)^(7/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(5/
2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*
(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}+\frac {(3 b B d-10 A b e+7 a B e) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {(8 e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {(16 e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {(64 b e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {\left (128 b^2 e (3 b B d-10 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{105 (b d-a e)^5}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {256 b^2 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^6 \sqrt {d+e x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.38, size = 341, normalized size = 1.14 \begin {gather*} -\frac {2 \left (-15 B d e^4 (a+b x)^5+15 A e^5 (a+b x)^5+84 b B d e^3 (a+b x)^4 (d+e x)-105 A b e^4 (a+b x)^4 (d+e x)+21 a B e^4 (a+b x)^4 (d+e x)-210 b^2 B d e^2 (a+b x)^3 (d+e x)^2+350 A b^2 e^3 (a+b x)^3 (d+e x)^2-140 a b B e^3 (a+b x)^3 (d+e x)^2+420 b^3 B d e (a+b x)^2 (d+e x)^3-1050 A b^3 e^2 (a+b x)^2 (d+e x)^3+630 a b^2 B e^2 (a+b x)^2 (d+e x)^3+105 b^4 B d (a+b x) (d+e x)^4-525 A b^4 e (a+b x) (d+e x)^4+420 a b^3 B e (a+b x) (d+e x)^4+35 A b^5 (d+e x)^5-35 a b^4 B (d+e x)^5\right )}{105 (b d-a e)^6 (a+b x)^{3/2} (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]

[Out]

(-2*(-15*B*d*e^4*(a + b*x)^5 + 15*A*e^5*(a + b*x)^5 + 84*b*B*d*e^3*(a + b*x)^4*(d + e*x) - 105*A*b*e^4*(a + b*
x)^4*(d + e*x) + 21*a*B*e^4*(a + b*x)^4*(d + e*x) - 210*b^2*B*d*e^2*(a + b*x)^3*(d + e*x)^2 + 350*A*b^2*e^3*(a
+ b*x)^3*(d + e*x)^2 - 140*a*b*B*e^3*(a + b*x)^3*(d + e*x)^2 + 420*b^3*B*d*e*(a + b*x)^2*(d + e*x)^3 - 1050*A
*b^3*e^2*(a + b*x)^2*(d + e*x)^3 + 630*a*b^2*B*e^2*(a + b*x)^2*(d + e*x)^3 + 105*b^4*B*d*(a + b*x)*(d + e*x)^4
- 525*A*b^4*e*(a + b*x)*(d + e*x)^4 + 420*a*b^3*B*e*(a + b*x)*(d + e*x)^4 + 35*A*b^5*(d + e*x)^5 - 35*a*b^4*B
*(d + e*x)^5))/(105*(b*d - a*e)^6*(a + b*x)^(3/2)*(d + e*x)^(7/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(652\) vs. \(2(262)=524\).
time = 0.10, size = 653, normalized size = 2.19

method	result	size
default	\(-\frac {2 \left (-1280 A \,b^{5} e^{5} x^{5}+896 B a \,b^{4} e^{5} x^{5}+384 B \,b^{5} d \,e^{4} x^{5}-1920 A a \,b^{4} e^{5} x^{4}-4480 A \,b^{5} d \,e^{4} x^{4}+1344 B \,a^{2} b^{3} e^{5} x^{4}+3712 B a \,b^{4} d \,e^{4} x^{4}+1344 B \,b^{5} d^{2} e^{3} x^{4}-480 A \,a^{2} b^{3} e^{5} x^{3}-6720 A a \,b^{4} d \,e^{4} x^{3}-5600 A \,b^{5} d^{2} e^{3} x^{3}+336 B \,a^{3} b^{2} e^{5} x^{3}+4848 B \,a^{2} b^{3} d \,e^{4} x^{3}+5936 B a \,b^{4} d^{2} e^{3} x^{3}+1680 B \,b^{5} d^{3} e^{2} x^{3}+80 A \,a^{3} b^{2} e^{5} x^{2}-1680 A \,a^{2} b^{3} d \,e^{4} x^{2}-8400 A a \,b^{4} d^{2} e^{3} x^{2}-2800 A \,b^{5} d^{3} e^{2} x^{2}-56 B \,a^{4} b \,e^{5} x^{2}+1152 B \,a^{3} b^{2} d \,e^{4} x^{2}+6384 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+4480 B a \,b^{4} d^{3} e^{2} x^{2}+840 B \,b^{5} d^{4} e \,x^{2}-30 A \,a^{4} b \,e^{5} x +280 A \,a^{3} b^{2} d \,e^{4} x -2100 A \,a^{2} b^{3} d^{2} e^{3} x -4200 A a \,b^{4} d^{3} e^{2} x -350 A \,b^{5} d^{4} e x +21 B \,a^{5} e^{5} x -187 B \,a^{4} b d \,e^{4} x +1386 B \,a^{3} b^{2} d^{2} e^{3} x +3570 B \,a^{2} b^{3} d^{3} e^{2} x +1505 B a \,b^{4} d^{4} e x +105 B \,b^{5} d^{5} x +15 A \,a^{5} e^{5}-105 A \,a^{4} b d \,e^{4}+350 A \,a^{3} b^{2} d^{2} e^{3}-1050 A \,a^{2} b^{3} d^{3} e^{2}-525 A a \,b^{4} d^{4} e +35 A \,b^{5} d^{5}+6 B \,a^{5} d \,e^{4}-56 B \,a^{4} b \,d^{2} e^{3}+420 B \,a^{3} b^{2} d^{3} e^{2}+840 B \,a^{2} b^{3} d^{4} e +70 B a \,b^{4} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a e -b d \right )^{6}}\)	\(653\)
gosper	\(-\frac {2 \left (-1280 A \,b^{5} e^{5} x^{5}+896 B a \,b^{4} e^{5} x^{5}+384 B \,b^{5} d \,e^{4} x^{5}-1920 A a \,b^{4} e^{5} x^{4}-4480 A \,b^{5} d \,e^{4} x^{4}+1344 B \,a^{2} b^{3} e^{5} x^{4}+3712 B a \,b^{4} d \,e^{4} x^{4}+1344 B \,b^{5} d^{2} e^{3} x^{4}-480 A \,a^{2} b^{3} e^{5} x^{3}-6720 A a \,b^{4} d \,e^{4} x^{3}-5600 A \,b^{5} d^{2} e^{3} x^{3}+336 B \,a^{3} b^{2} e^{5} x^{3}+4848 B \,a^{2} b^{3} d \,e^{4} x^{3}+5936 B a \,b^{4} d^{2} e^{3} x^{3}+1680 B \,b^{5} d^{3} e^{2} x^{3}+80 A \,a^{3} b^{2} e^{5} x^{2}-1680 A \,a^{2} b^{3} d \,e^{4} x^{2}-8400 A a \,b^{4} d^{2} e^{3} x^{2}-2800 A \,b^{5} d^{3} e^{2} x^{2}-56 B \,a^{4} b \,e^{5} x^{2}+1152 B \,a^{3} b^{2} d \,e^{4} x^{2}+6384 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+4480 B a \,b^{4} d^{3} e^{2} x^{2}+840 B \,b^{5} d^{4} e \,x^{2}-30 A \,a^{4} b \,e^{5} x +280 A \,a^{3} b^{2} d \,e^{4} x -2100 A \,a^{2} b^{3} d^{2} e^{3} x -4200 A a \,b^{4} d^{3} e^{2} x -350 A \,b^{5} d^{4} e x +21 B \,a^{5} e^{5} x -187 B \,a^{4} b d \,e^{4} x +1386 B \,a^{3} b^{2} d^{2} e^{3} x +3570 B \,a^{2} b^{3} d^{3} e^{2} x +1505 B a \,b^{4} d^{4} e x +105 B \,b^{5} d^{5} x +15 A \,a^{5} e^{5}-105 A \,a^{4} b d \,e^{4}+350 A \,a^{3} b^{2} d^{2} e^{3}-1050 A \,a^{2} b^{3} d^{3} e^{2}-525 A a \,b^{4} d^{4} e +35 A \,b^{5} d^{5}+6 B \,a^{5} d \,e^{4}-56 B \,a^{4} b \,d^{2} e^{3}+420 B \,a^{3} b^{2} d^{3} e^{2}+840 B \,a^{2} b^{3} d^{4} e +70 B a \,b^{4} d^{5}\right )}{105 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {7}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\)	\(722\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*(-1280*A*b^5*e^5*x^5+896*B*a*b^4*e^5*x^5+384*B*b^5*d*e^4*x^5-1920*A*a*b^4*e^5*x^4-4480*A*b^5*d*e^4*x^4+
1344*B*a^2*b^3*e^5*x^4+3712*B*a*b^4*d*e^4*x^4+1344*B*b^5*d^2*e^3*x^4-480*A*a^2*b^3*e^5*x^3-6720*A*a*b^4*d*e^4*
x^3-5600*A*b^5*d^2*e^3*x^3+336*B*a^3*b^2*e^5*x^3+4848*B*a^2*b^3*d*e^4*x^3+5936*B*a*b^4*d^2*e^3*x^3+1680*B*b^5*
d^3*e^2*x^3+80*A*a^3*b^2*e^5*x^2-1680*A*a^2*b^3*d*e^4*x^2-8400*A*a*b^4*d^2*e^3*x^2-2800*A*b^5*d^3*e^2*x^2-56*B
*a^4*b*e^5*x^2+1152*B*a^3*b^2*d*e^4*x^2+6384*B*a^2*b^3*d^2*e^3*x^2+4480*B*a*b^4*d^3*e^2*x^2+840*B*b^5*d^4*e*x^
2-30*A*a^4*b*e^5*x+280*A*a^3*b^2*d*e^4*x-2100*A*a^2*b^3*d^2*e^3*x-4200*A*a*b^4*d^3*e^2*x-350*A*b^5*d^4*e*x+21*
B*a^5*e^5*x-187*B*a^4*b*d*e^4*x+1386*B*a^3*b^2*d^2*e^3*x+3570*B*a^2*b^3*d^3*e^2*x+1505*B*a*b^4*d^4*e*x+105*B*b
^5*d^5*x+15*A*a^5*e^5-105*A*a^4*b*d*e^4+350*A*a^3*b^2*d^2*e^3-1050*A*a^2*b^3*d^3*e^2-525*A*a*b^4*d^4*e+35*A*b^
5*d^5+6*B*a^5*d*e^4-56*B*a^4*b*d^2*e^3+420*B*a^3*b^2*d^3*e^2+840*B*a^2*b^3*d^4*e+70*B*a*b^4*d^5)/(e*x+d)^(7/2)
/(b*x+a)^(3/2)/(a*e-b*d)^6

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (288) = 576\).
time = 76.20, size = 1263, normalized size = 4.24 \begin {gather*} -\frac {2 \, {\left (105 \, B b^{5} d^{5} x + 35 \, {\left (2 \, B a b^{4} + A b^{5}\right )} d^{5} + {\left (15 \, A a^{5} + 128 \, {\left (7 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} + 192 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \, {\left (7 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} e^{5} + {\left (384 \, B b^{5} d x^{5} + 128 \, {\left (29 \, B a b^{4} - 35 \, A b^{5}\right )} d x^{4} + 48 \, {\left (101 \, B a^{2} b^{3} - 140 \, A a b^{4}\right )} d x^{3} + 48 \, {\left (24 \, B a^{3} b^{2} - 35 \, A a^{2} b^{3}\right )} d x^{2} - {\left (187 \, B a^{4} b - 280 \, A a^{3} b^{2}\right )} d x + 3 \, {\left (2 \, B a^{5} - 35 \, A a^{4} b\right )} d\right )} e^{4} + 14 \, {\left (96 \, B b^{5} d^{2} x^{4} + 8 \, {\left (53 \, B a b^{4} - 50 \, A b^{5}\right )} d^{2} x^{3} + 24 \, {\left (19 \, B a^{2} b^{3} - 25 \, A a b^{4}\right )} d^{2} x^{2} + 3 \, {\left (33 \, B a^{3} b^{2} - 50 \, A a^{2} b^{3}\right )} d^{2} x - {\left (4 \, B a^{4} b - 25 \, A a^{3} b^{2}\right )} d^{2}\right )} e^{3} + 70 \, {\left (24 \, B b^{5} d^{3} x^{3} + 8 \, {\left (8 \, B a b^{4} - 5 \, A b^{5}\right )} d^{3} x^{2} + 3 \, {\left (17 \, B a^{2} b^{3} - 20 \, A a b^{4}\right )} d^{3} x + 3 \, {\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} d^{3}\right )} e^{2} + 35 \, {\left (24 \, B b^{5} d^{4} x^{2} + {\left (43 \, B a b^{4} - 10 \, A b^{5}\right )} d^{4} x + 3 \, {\left (8 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{4}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{105 \, {\left (b^{8} d^{10} x^{2} + 2 \, a b^{7} d^{10} x + a^{2} b^{6} d^{10} + {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )} e^{10} - 2 \, {\left (3 \, a^{5} b^{3} d x^{6} + 4 \, a^{6} b^{2} d x^{5} - a^{7} b d x^{4} - 2 \, a^{8} d x^{3}\right )} e^{9} + 3 \, {\left (5 \, a^{4} b^{4} d^{2} x^{6} + 2 \, a^{5} b^{3} d^{2} x^{5} - 9 \, a^{6} b^{2} d^{2} x^{4} - 4 \, a^{7} b d^{2} x^{3} + 2 \, a^{8} d^{2} x^{2}\right )} e^{8} - 4 \, {\left (5 \, a^{3} b^{5} d^{3} x^{6} - 5 \, a^{4} b^{4} d^{3} x^{5} - 16 \, a^{5} b^{3} d^{3} x^{4} + 2 \, a^{6} b^{2} d^{3} x^{3} + 7 \, a^{7} b d^{3} x^{2} - a^{8} d^{3} x\right )} e^{7} + {\left (15 \, a^{2} b^{6} d^{4} x^{6} - 50 \, a^{3} b^{5} d^{4} x^{5} - 55 \, a^{4} b^{4} d^{4} x^{4} + 76 \, a^{5} b^{3} d^{4} x^{3} + 43 \, a^{6} b^{2} d^{4} x^{2} - 22 \, a^{7} b d^{4} x + a^{8} d^{4}\right )} e^{6} - 6 \, {\left (a b^{7} d^{5} x^{6} - 8 \, a^{2} b^{6} d^{5} x^{5} + a^{3} b^{5} d^{5} x^{4} + 20 \, a^{4} b^{4} d^{5} x^{3} + a^{5} b^{3} d^{5} x^{2} - 8 \, a^{6} b^{2} d^{5} x + a^{7} b d^{5}\right )} e^{5} + {\left (b^{8} d^{6} x^{6} - 22 \, a b^{7} d^{6} x^{5} + 43 \, a^{2} b^{6} d^{6} x^{4} + 76 \, a^{3} b^{5} d^{6} x^{3} - 55 \, a^{4} b^{4} d^{6} x^{2} - 50 \, a^{5} b^{3} d^{6} x + 15 \, a^{6} b^{2} d^{6}\right )} e^{4} + 4 \, {\left (b^{8} d^{7} x^{5} - 7 \, a b^{7} d^{7} x^{4} - 2 \, a^{2} b^{6} d^{7} x^{3} + 16 \, a^{3} b^{5} d^{7} x^{2} + 5 \, a^{4} b^{4} d^{7} x - 5 \, a^{5} b^{3} d^{7}\right )} e^{3} + 3 \, {\left (2 \, b^{8} d^{8} x^{4} - 4 \, a b^{7} d^{8} x^{3} - 9 \, a^{2} b^{6} d^{8} x^{2} + 2 \, a^{3} b^{5} d^{8} x + 5 \, a^{4} b^{4} d^{8}\right )} e^{2} + 2 \, {\left (2 \, b^{8} d^{9} x^{3} + a b^{7} d^{9} x^{2} - 4 \, a^{2} b^{6} d^{9} x - 3 \, a^{3} b^{5} d^{9}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

-2/105*(105*B*b^5*d^5*x + 35*(2*B*a*b^4 + A*b^5)*d^5 + (15*A*a^5 + 128*(7*B*a*b^4 - 10*A*b^5)*x^5 + 192*(7*B*a
^2*b^3 - 10*A*a*b^4)*x^4 + 48*(7*B*a^3*b^2 - 10*A*a^2*b^3)*x^3 - 8*(7*B*a^4*b - 10*A*a^3*b^2)*x^2 + 3*(7*B*a^5
- 10*A*a^4*b)*x)*e^5 + (384*B*b^5*d*x^5 + 128*(29*B*a*b^4 - 35*A*b^5)*d*x^4 + 48*(101*B*a^2*b^3 - 140*A*a*b^4
)*d*x^3 + 48*(24*B*a^3*b^2 - 35*A*a^2*b^3)*d*x^2 - (187*B*a^4*b - 280*A*a^3*b^2)*d*x + 3*(2*B*a^5 - 35*A*a^4*b
)*d)*e^4 + 14*(96*B*b^5*d^2*x^4 + 8*(53*B*a*b^4 - 50*A*b^5)*d^2*x^3 + 24*(19*B*a^2*b^3 - 25*A*a*b^4)*d^2*x^2 +
3*(33*B*a^3*b^2 - 50*A*a^2*b^3)*d^2*x - (4*B*a^4*b - 25*A*a^3*b^2)*d^2)*e^3 + 70*(24*B*b^5*d^3*x^3 + 8*(8*B*a
*b^4 - 5*A*b^5)*d^3*x^2 + 3*(17*B*a^2*b^3 - 20*A*a*b^4)*d^3*x + 3*(2*B*a^3*b^2 - 5*A*a^2*b^3)*d^3)*e^2 + 35*(2
4*B*b^5*d^4*x^2 + (43*B*a*b^4 - 10*A*b^5)*d^4*x + 3*(8*B*a^2*b^3 - 5*A*a*b^4)*d^4)*e)*sqrt(b*x + a)*sqrt(x*e +
d)/(b^8*d^10*x^2 + 2*a*b^7*d^10*x + a^2*b^6*d^10 + (a^6*b^2*x^6 + 2*a^7*b*x^5 + a^8*x^4)*e^10 - 2*(3*a^5*b^3*
d*x^6 + 4*a^6*b^2*d*x^5 - a^7*b*d*x^4 - 2*a^8*d*x^3)*e^9 + 3*(5*a^4*b^4*d^2*x^6 + 2*a^5*b^3*d^2*x^5 - 9*a^6*b^
2*d^2*x^4 - 4*a^7*b*d^2*x^3 + 2*a^8*d^2*x^2)*e^8 - 4*(5*a^3*b^5*d^3*x^6 - 5*a^4*b^4*d^3*x^5 - 16*a^5*b^3*d^3*x
^4 + 2*a^6*b^2*d^3*x^3 + 7*a^7*b*d^3*x^2 - a^8*d^3*x)*e^7 + (15*a^2*b^6*d^4*x^6 - 50*a^3*b^5*d^4*x^5 - 55*a^4*
b^4*d^4*x^4 + 76*a^5*b^3*d^4*x^3 + 43*a^6*b^2*d^4*x^2 - 22*a^7*b*d^4*x + a^8*d^4)*e^6 - 6*(a*b^7*d^5*x^6 - 8*a
^2*b^6*d^5*x^5 + a^3*b^5*d^5*x^4 + 20*a^4*b^4*d^5*x^3 + a^5*b^3*d^5*x^2 - 8*a^6*b^2*d^5*x + a^7*b*d^5)*e^5 + (
b^8*d^6*x^6 - 22*a*b^7*d^6*x^5 + 43*a^2*b^6*d^6*x^4 + 76*a^3*b^5*d^6*x^3 - 55*a^4*b^4*d^6*x^2 - 50*a^5*b^3*d^6
*x + 15*a^6*b^2*d^6)*e^4 + 4*(b^8*d^7*x^5 - 7*a*b^7*d^7*x^4 - 2*a^2*b^6*d^7*x^3 + 16*a^3*b^5*d^7*x^2 + 5*a^4*b
^4*d^7*x - 5*a^5*b^3*d^7)*e^3 + 3*(2*b^8*d^8*x^4 - 4*a*b^7*d^8*x^3 - 9*a^2*b^6*d^8*x^2 + 2*a^3*b^5*d^8*x + 5*a
^4*b^4*d^8)*e^2 + 2*(2*b^8*d^9*x^3 + a*b^7*d^9*x^2 - 4*a^2*b^6*d^9*x - 3*a^3*b^5*d^9)*e)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3473 vs. \(2 (288) = 576\).
time = 2.79, size = 3473, normalized size = 11.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-2/105*(((b*x + a)*((279*B*b^22*d^13*abs(b)*e^7 - 2837*B*a*b^21*d^12*abs(b)*e^8 - 790*A*b^22*d^12*abs(b)*e^8 +
12282*B*a^2*b^20*d^11*abs(b)*e^9 + 9480*A*a*b^21*d^11*abs(b)*e^9 - 27654*B*a^3*b^19*d^10*abs(b)*e^10 - 52140*
A*a^2*b^20*d^10*abs(b)*e^10 + 25685*B*a^4*b^18*d^9*abs(b)*e^11 + 173800*A*a^3*b^19*d^9*abs(b)*e^11 + 31977*B*a
^5*b^17*d^8*abs(b)*e^12 - 391050*A*a^4*b^18*d^8*abs(b)*e^12 - 146916*B*a^6*b^16*d^7*abs(b)*e^13 + 625680*A*a^5
*b^17*d^7*abs(b)*e^13 + 251196*B*a^7*b^15*d^6*abs(b)*e^14 - 729960*A*a^6*b^16*d^6*abs(b)*e^14 - 266607*B*a^8*b
^14*d^5*abs(b)*e^15 + 625680*A*a^7*b^15*d^5*abs(b)*e^15 + 191565*B*a^9*b^13*d^4*abs(b)*e^16 - 391050*A*a^8*b^1
4*d^4*abs(b)*e^16 - 94006*B*a^10*b^12*d^3*abs(b)*e^17 + 173800*A*a^9*b^13*d^3*abs(b)*e^17 + 30378*B*a^11*b^11*
d^2*abs(b)*e^18 - 52140*A*a^10*b^12*d^2*abs(b)*e^18 - 5853*B*a^12*b^10*d*abs(b)*e^19 + 9480*A*a^11*b^11*d*abs(
b)*e^19 + 511*B*a^13*b^9*abs(b)*e^20 - 790*A*a^12*b^10*abs(b)*e^20)*(b*x + a)/(b^22*d^18*e^3 - 18*a*b^21*d^17*
e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 18564*
a^6*b^16*d^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9*b^13*d^9*e^12 + 43758*a^10
*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14*b^8*
d^4*e^17 - 816*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b^4*e^21) + 7*(132*B*b^23
*d^14*abs(b)*e^6 - 1483*B*a*b^22*d^13*abs(b)*e^7 - 365*A*b^23*d^13*abs(b)*e^7 + 7267*B*a^2*b^21*d^12*abs(b)*e^
8 + 4745*A*a*b^22*d^12*abs(b)*e^8 - 19578*B*a^3*b^20*d^11*abs(b)*e^9 - 28470*A*a^2*b^21*d^11*abs(b)*e^9 + 2774
2*B*a^4*b^19*d^10*abs(b)*e^10 + 104390*A*a^3*b^20*d^10*abs(b)*e^10 - 3289*B*a^5*b^18*d^9*abs(b)*e^11 - 260975*
A*a^4*b^19*d^9*abs(b)*e^11 - 73359*B*a^6*b^17*d^8*abs(b)*e^12 + 469755*A*a^5*b^18*d^8*abs(b)*e^12 + 173316*B*a
^7*b^16*d^7*abs(b)*e^13 - 626340*A*a^6*b^17*d^7*abs(b)*e^13 - 229944*B*a^8*b^15*d^6*abs(b)*e^14 + 626340*A*a^7
*b^16*d^6*abs(b)*e^14 + 205491*B*a^9*b^14*d^5*abs(b)*e^15 - 469755*A*a^8*b^15*d^5*abs(b)*e^15 - 128843*B*a^10*
b^13*d^4*abs(b)*e^16 + 260975*A*a^9*b^14*d^4*abs(b)*e^16 + 56342*B*a^11*b^12*d^3*abs(b)*e^17 - 104390*A*a^10*b
^13*d^3*abs(b)*e^17 - 16458*B*a^12*b^11*d^2*abs(b)*e^18 + 28470*A*a^11*b^12*d^2*abs(b)*e^18 + 2897*B*a^13*b^10
*d*abs(b)*e^19 - 4745*A*a^12*b^11*d*abs(b)*e^19 - 233*B*a^14*b^9*abs(b)*e^20 + 365*A*a^13*b^10*abs(b)*e^20)/(b
^22*d^18*e^3 - 18*a*b^21*d^17*e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8
568*a^5*b^17*d^13*e^8 + 18564*a^6*b^16*d^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*
a^9*b^13*d^9*e^12 + 43758*a^10*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13
*b^9*d^5*e^16 + 3060*a^14*b^8*d^4*e^17 - 816*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 +
a^18*b^4*e^21)) + 350*(3*B*b^24*d^15*abs(b)*e^5 - 37*B*a*b^23*d^14*abs(b)*e^6 - 8*A*b^24*d^14*abs(b)*e^6 + 203
*B*a^2*b^22*d^13*abs(b)*e^7 + 112*A*a*b^23*d^13*abs(b)*e^7 - 637*B*a^3*b^21*d^12*abs(b)*e^8 - 728*A*a^2*b^22*d
^12*abs(b)*e^8 + 1183*B*a^4*b^20*d^11*abs(b)*e^9 + 2912*A*a^3*b^21*d^11*abs(b)*e^9 - 1001*B*a^5*b^19*d^10*abs(
b)*e^10 - 8008*A*a^4*b^20*d^10*abs(b)*e^10 - 1001*B*a^6*b^18*d^9*abs(b)*e^11 + 16016*A*a^5*b^19*d^9*abs(b)*e^1
1 + 4719*B*a^7*b^17*d^8*abs(b)*e^12 - 24024*A*a^6*b^18*d^8*abs(b)*e^12 - 8151*B*a^8*b^16*d^7*abs(b)*e^13 + 274
56*A*a^7*b^17*d^7*abs(b)*e^13 + 9009*B*a^9*b^15*d^6*abs(b)*e^14 - 24024*A*a^8*b^16*d^6*abs(b)*e^14 - 7007*B*a^
10*b^14*d^5*abs(b)*e^15 + 16016*A*a^9*b^15*d^5*abs(b)*e^15 + 3913*B*a^11*b^13*d^4*abs(b)*e^16 - 8008*A*a^10*b^
14*d^4*abs(b)*e^16 - 1547*B*a^12*b^12*d^3*abs(b)*e^17 + 2912*A*a^11*b^13*d^3*abs(b)*e^17 + 413*B*a^13*b^11*d^2
*abs(b)*e^18 - 728*A*a^12*b^12*d^2*abs(b)*e^18 - 67*B*a^14*b^10*d*abs(b)*e^19 + 112*A*a^13*b^11*d*abs(b)*e^19
+ 5*B*a^15*b^9*abs(b)*e^20 - 8*A*a^14*b^10*abs(b)*e^20)/(b^22*d^18*e^3 - 18*a*b^21*d^17*e^4 + 153*a^2*b^20*d^1
6*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 18564*a^6*b^16*d^12*e^9 - 31
824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9*b^13*d^9*e^12 + 43758*a^10*b^12*d^8*e^13 - 31824
*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14*b^8*d^4*e^17 - 816*a^15*b^
7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b^4*e^21))*(b*x + a) + 210*(2*B*b^25*d^16*abs(b
)*e^4 - 27*B*a*b^24*d^15*abs(b)*e^5 - 5*A*b^25*d^15*abs(b)*e^5 + 165*B*a^2*b^23*d^14*abs(b)*e^6 + 75*A*a*b^24*
d^14*abs(b)*e^6 - 595*B*a^3*b^22*d^13*abs(b)*e^7 - 525*A*a^2*b^23*d^13*abs(b)*e^7 + 1365*B*a^4*b^21*d^12*abs(b
)*e^8 + 2275*A*a^3*b^22*d^12*abs(b)*e^8 - 1911*B*a^5*b^20*d^11*abs(b)*e^9 - 6825*A*a^4*b^21*d^11*abs(b)*e^9 +
1001*B*a^6*b^19*d^10*abs(b)*e^10 + 15015*A*a^5*b^20*d^10*abs(b)*e^10 + 2145*B*a^7*b^18*d^9*abs(b)*e^11 - 25025
*A*a^6*b^19*d^9*abs(b)*e^11 - 6435*B*a^8*b^17*d^8*abs(b)*e^12 + 32175*A*a^7*b^18*d^8*abs(b)*e^12 + 9295*B*a^9*
b^16*d^7*abs(b)*e^13 - 32175*A*a^8*b^17*d^7*abs...

________________________________________________________________________________________

Mupad [B]
time = 3.23, size = 596, normalized size = 2.00 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {12\,B\,a^5\,d\,e^4+30\,A\,a^5\,e^5-112\,B\,a^4\,b\,d^2\,e^3-210\,A\,a^4\,b\,d\,e^4+840\,B\,a^3\,b^2\,d^3\,e^2+700\,A\,a^3\,b^2\,d^2\,e^3+1680\,B\,a^2\,b^3\,d^4\,e-2100\,A\,a^2\,b^3\,d^3\,e^2+140\,B\,a\,b^4\,d^5-1050\,A\,a\,b^4\,d^4\,e+70\,A\,b^5\,d^5}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^3\,x^5\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,{\left (a\,e-b\,d\right )}^6}+\frac {16\,x^2\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+21\,a^2\,b\,d\,e^2+105\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^6}+\frac {128\,b^2\,x^4\,\left (3\,a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b\,x^3\,\left (3\,a^2\,e^2+42\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,x\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (3\,a^4\,e^4-28\,a^3\,b\,d\,e^3+210\,a^2\,b^2\,d^2\,e^2+420\,a\,b^3\,d^3\,e+35\,b^4\,d^4\right )}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a\,d^4\,\sqrt {a+b\,x}}{b\,e^4}+\frac {x^4\,\left (a\,e+4\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x)

[Out]

-((d + e*x)^(1/2)*((30*A*a^5*e^5 + 70*A*b^5*d^5 + 140*B*a*b^4*d^5 + 12*B*a^5*d*e^4 + 1680*B*a^2*b^3*d^4*e - 11
2*B*a^4*b*d^2*e^3 - 2100*A*a^2*b^3*d^3*e^2 + 700*A*a^3*b^2*d^2*e^3 + 840*B*a^3*b^2*d^3*e^2 - 1050*A*a*b^4*d^4*
e - 210*A*a^4*b*d*e^4)/(105*b*e^4*(a*e - b*d)^6) + (256*b^3*x^5*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*(a*e - b*
d)^6) + (16*x^2*(7*B*a*e - 10*A*b*e + 3*B*b*d)*(35*b^3*d^3 - a^3*e^3 + 105*a*b^2*d^2*e + 21*a^2*b*d*e^2))/(105
*e^3*(a*e - b*d)^6) + (128*b^2*x^4*(3*a*e + 7*b*d)*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*e*(a*e - b*d)^6) + (32
*b*x^3*(3*a^2*e^2 + 35*b^2*d^2 + 42*a*b*d*e)*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*e^2*(a*e - b*d)^6) + (2*x*(7
*B*a*e - 10*A*b*e + 3*B*b*d)*(3*a^4*e^4 + 35*b^4*d^4 + 210*a^2*b^2*d^2*e^2 + 420*a*b^3*d^3*e - 28*a^3*b*d*e^3)
)/(105*b*e^4*(a*e - b*d)^6)))/(x^5*(a + b*x)^(1/2) + (a*d^4*(a + b*x)^(1/2))/(b*e^4) + (x^4*(a*e + 4*b*d)*(a +
b*x)^(1/2))/(b*e) + (2*d*x^3*(2*a*e + 3*b*d)*(a + b*x)^(1/2))/(b*e^2) + (d^3*x*(4*a*e + b*d)*(a + b*x)^(1/2))
/(b*e^4) + (2*d^2*x^2*(3*a*e + 2*b*d)*(a + b*x)^(1/2))/(b*e^3))

________________________________________________________________________________________