3.2223 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=412 \[ \frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}-\frac{5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac{(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac{(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

[Out]

(5*(b*d - a*e)^5*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(1024*b^4*e^4) - (5*(b*d - a*e)^4*(2*A
*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(1536*b^4*e^3) + ((b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))
*(a + b*x)^(5/2)*Sqrt[d + e*x])/(384*b^4*e^2) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*Sqrt[
d + e*x])/(64*b^4*e) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(24*b^3*e) + ((
2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(12*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*
b*e) - (5*(b*d - a*e)^6*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(1
024*b^(9/2)*e^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.38341, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ \frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}-\frac{5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac{(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac{(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b*d - a*e)^5*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(1024*b^4*e^4) - (5*(b*d - a*e)^4*(2*A
*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(1536*b^4*e^3) + ((b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))
*(a + b*x)^(5/2)*Sqrt[d + e*x])/(384*b^4*e^2) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*Sqrt[
d + e*x])/(64*b^4*e) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(24*b^3*e) + ((
2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(12*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*
b*e) - (5*(b*d - a*e)^6*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(1
024*b^(9/2)*e^(9/2))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx &=\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac{\left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right ) \int (a+b x)^{5/2} (d+e x)^{5/2} \, dx}{7 b e}\\ &=\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac{\left (5 (b d-a e) \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int (a+b x)^{5/2} (d+e x)^{3/2} \, dx}{84 b^2 e}\\ &=\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac{\left ((b d-a e)^2 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int (a+b x)^{5/2} \sqrt{d+e x} \, dx}{56 b^3 e}\\ &=\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac{\left ((b d-a e)^3 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{d+e x}} \, dx}{448 b^4 e}\\ &=\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{\left (5 (b d-a e)^4 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{2688 b^4 e^2}\\ &=-\frac{5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{1536 b^4 e^3}+\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac{\left (5 (b d-a e)^5 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{3584 b^4 e^3}\\ &=\frac{5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{1024 b^4 e^4}-\frac{5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{1536 b^4 e^3}+\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{7168 b^4 e^4}\\ &=\frac{5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{1024 b^4 e^4}-\frac{5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{1536 b^4 e^3}+\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{3584 b^5 e^4}\\ &=\frac{5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{1024 b^4 e^4}-\frac{5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{1536 b^4 e^3}+\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{\left (5 (b d-a e)^6 \left (7 A b e-B \left (\frac{7 b d}{2}+\frac{7 a e}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{3584 b^5 e^4}\\ &=\frac{5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{1024 b^4 e^4}-\frac{5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{1536 b^4 e^3}+\frac{(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{384 b^4 e^2}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt{d+e x}}{64 b^4 e}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{5 (b d-a e)^6 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{1024 b^{9/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 5.2187, size = 388, normalized size = 0.94 \[ \frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac{\sqrt{b d-a e} \left (\frac{b (d+e x)}{b d-a e}\right )^{3/2} (a B e-2 A b e+b B d) \left (16 e^{7/2} (a+b x)^4 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} \left (3 a^2 e^2-2 a b e (7 d+4 e x)+b^2 \left (27 d^2+40 d e x+16 e^2 x^2\right )\right )-10 e^{3/2} (a+b x)^2 (b d-a e)^{11/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+8 e^{5/2} (a+b x)^3 (b d-a e)^{9/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+15 \sqrt{e} (a+b x) (b d-a e)^{13/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 \sqrt{a+b x} (b d-a e)^7 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{3072 b^6 e^{9/2} \sqrt{a+b x} (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*b*e) - (Sqrt[b*d - a*e]*(b*B*d - 2*A*b*e + a*B*e)*((b*(d + e*x))/(b*d -
a*e))^(3/2)*(15*Sqrt[e]*(b*d - a*e)^(13/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*e^(3/2)*(b*d - a*e)
^(11/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*e^(5/2)*(b*d - a*e)^(9/2)*(a + b*x)^3*Sqrt[(b*(d + e*x
))/(b*d - a*e)] + 16*e^(7/2)*(b*d - a*e)^(3/2)*(a + b*x)^4*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(3*a^2*e^2 - 2*a*b*
e*(7*d + 4*e*x) + b^2*(27*d^2 + 40*d*e*x + 16*e^2*x^2)) - 15*(b*d - a*e)^7*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt
[a + b*x])/Sqrt[b*d - a*e]]))/(3072*b^6*e^(9/2)*Sqrt[a + b*x]*(d + e*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.024, size = 2851, normalized size = 6.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-945*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e
+b*d)/(b*e)^(1/2))*a^5*b^2*d^2*e^5+3150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*
d)/(b*e)^(1/2))*a^2*b^5*d^4*e^3-1260*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/
(b*e)^(1/2))*a*b^6*d^5*e^2+525*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^
(1/2))*a^6*b*d*e^6-6144*B*x^6*b^6*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-7168*A*x^5*b^6*e^6*(b*e)^(1/
2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+525*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d
)/(b*e)^(1/2))*a^4*b^3*d^3*e^4+525*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b
*e)^(1/2))*a^3*b^4*d^4*e^3-945*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^
(1/2))*a^2*b^5*d^5*e^2+525*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2
))*a*b^6*d^6*e-420*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*b*e^6-420*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*b^6*d^5*e-1260*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(
1/2))*a^5*b^2*d*e^6+3150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))
*a^4*b^3*d^2*e^5-105*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^7
*e^7-105*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^7*d^7-1568*A*
(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*b^3*d*e^5+210*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
b^6*d^6-600*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*b^3*d^3*e^3-25504*B*x^3*a^2*b^4*d*e^5*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-25504*B*x^3*a*b^5*d^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-35616*
A*x^2*a^2*b^4*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-35616*A*x^2*a*b^5*d^2*e^4*(b*e)^(1/2)*(b*e*x^2
+a*e*x+b*d*x+a*d)^(1/2)-1016*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*b^3*d^2*e^4-1016*B*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^4*d^3*e^3+644*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*b^5*
d^4*e^2-512*B*x^2*a^3*b^3*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-37376*B*x^4*a*b^5*d*e^5*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-47488*A*x^3*a*b^5*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-4200*A*l
n(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^4*d^3*e^4-33264*A*(b*
e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^4*d^2*e^4-1568*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
x*a*b^5*d^3*e^3+644*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^4*b^2*d*e^5-512*B*x^2*a*b^5*d^3*e^3*(b*e
)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-19680*B*x^2*a^2*b^4*d^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)+280*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^4*b^2*e^6-12096*A*x^3*b^6*d^2*e^4*(b*e)^(1/2)*(b*e*x^2
+a*e*x+b*d*x+a*d)^(1/2)-96*B*x^3*a^3*b^3*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-96*B*x^3*b^6*d^3*e^3*
(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-224*A*x^2*a^3*b^3*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+
280*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^6*d^4*e^2-140*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*x*a^5*b*e^6-140*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^6*d^5*e+2380*A*(b*e)^(1/2)*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*a^4*b^2*d*e^5-5544*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*b^3*d^2*e^4-5544*A*(b*
e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^4*d^3*e^3+2380*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*
b^5*d^4*e^2-224*A*x^2*b^6*d^3*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+112*B*x^2*a^4*b^2*e^6*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+112*B*x^2*b^6*d^4*e^2*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-980*B*(b*e
)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*b*d*e^5-9472*B*x^4*a^2*b^4*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*
d)^(1/2)-14848*B*x^5*a*b^5*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-14848*B*x^5*b^6*d*e^5*(b*e)^(1/2)*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-17920*A*x^4*a*b^5*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-17920*A*x^4*b
^6*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-12096*A*x^3*a^2*b^4*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+
a*d)^(1/2)-9472*B*x^4*b^6*d^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+1582*B*(b*e)^(1/2)*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*a^4*b^2*d^2*e^4+1582*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^4*d^4*e^2-980*B*(b
*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^5*d^5*e+210*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^7*d^6*e+210*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*a^6*b*e^7+210*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^6*e^6)/b^4/e^4/(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)/(b*e)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.25976, size = 3864, normalized size = 9.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/86016*(105*(B*b^7*d^7 - (5*B*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a^3*b^4 +
6*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3)*d^2*e^5 - (5*B*a^6
*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^
2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(3072*B*b^7*e
^7*x^6 - 105*B*b^7*d^6*e + 70*(7*B*a*b^6 + 3*A*b^7)*d^5*e^2 - 7*(113*B*a^2*b^5 + 170*A*a*b^6)*d^4*e^3 + 12*(25
*B*a^3*b^4 + 231*A*a^2*b^5)*d^3*e^4 - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2*e^5 + 70*(7*B*a^5*b^2 - 17*A*a^4*b
^3)*d*e^6 - 105*(B*a^6*b - 2*A*a^5*b^2)*e^7 + 256*(29*B*b^7*d*e^6 + (29*B*a*b^6 + 14*A*b^7)*e^7)*x^5 + 128*(37
*B*b^7*d^2*e^5 + 2*(73*B*a*b^6 + 35*A*b^7)*d*e^6 + (37*B*a^2*b^5 + 70*A*a*b^6)*e^7)*x^4 + 16*(3*B*b^7*d^3*e^4
+ (797*B*a*b^6 + 378*A*b^7)*d^2*e^5 + (797*B*a^2*b^5 + 1484*A*a*b^6)*d*e^6 + 3*(B*a^3*b^4 + 126*A*a^2*b^5)*e^7
)*x^3 - 8*(7*B*b^7*d^4*e^3 - 2*(16*B*a*b^6 + 7*A*b^7)*d^3*e^4 - 6*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*e^5 - 2*(1
6*B*a^3*b^4 + 1113*A*a^2*b^5)*d*e^6 + 7*(B*a^4*b^3 - 2*A*a^3*b^4)*e^7)*x^2 + 2*(35*B*b^7*d^5*e^2 - 7*(23*B*a*b
^6 + 10*A*b^7)*d^4*e^3 + 2*(127*B*a^2*b^5 + 196*A*a*b^6)*d^3*e^4 + 2*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*e^5
- 7*(23*B*a^4*b^3 - 56*A*a^3*b^4)*d*e^6 + 35*(B*a^5*b^2 - 2*A*a^4*b^3)*e^7)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b
^5*e^5), -1/43008*(105*(B*b^7*d^7 - (5*B*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a
^3*b^4 + 6*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3)*d^2*e^5 -
(5*B*a^6*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(
-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(3072*B*b^7*e^7*x^6 - 1
05*B*b^7*d^6*e + 70*(7*B*a*b^6 + 3*A*b^7)*d^5*e^2 - 7*(113*B*a^2*b^5 + 170*A*a*b^6)*d^4*e^3 + 12*(25*B*a^3*b^4
+ 231*A*a^2*b^5)*d^3*e^4 - 7*(113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2*e^5 + 70*(7*B*a^5*b^2 - 17*A*a^4*b^3)*d*e^6
- 105*(B*a^6*b - 2*A*a^5*b^2)*e^7 + 256*(29*B*b^7*d*e^6 + (29*B*a*b^6 + 14*A*b^7)*e^7)*x^5 + 128*(37*B*b^7*d^2
*e^5 + 2*(73*B*a*b^6 + 35*A*b^7)*d*e^6 + (37*B*a^2*b^5 + 70*A*a*b^6)*e^7)*x^4 + 16*(3*B*b^7*d^3*e^4 + (797*B*a
*b^6 + 378*A*b^7)*d^2*e^5 + (797*B*a^2*b^5 + 1484*A*a*b^6)*d*e^6 + 3*(B*a^3*b^4 + 126*A*a^2*b^5)*e^7)*x^3 - 8*
(7*B*b^7*d^4*e^3 - 2*(16*B*a*b^6 + 7*A*b^7)*d^3*e^4 - 6*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*e^5 - 2*(16*B*a^3*b^
4 + 1113*A*a^2*b^5)*d*e^6 + 7*(B*a^4*b^3 - 2*A*a^3*b^4)*e^7)*x^2 + 2*(35*B*b^7*d^5*e^2 - 7*(23*B*a*b^6 + 10*A*
b^7)*d^4*e^3 + 2*(127*B*a^2*b^5 + 196*A*a*b^6)*d^3*e^4 + 2*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*e^5 - 7*(23*B*
a^4*b^3 - 56*A*a^3*b^4)*d*e^6 + 35*(B*a^5*b^2 - 2*A*a^4*b^3)*e^7)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 4.26649, size = 7723, normalized size = 18.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*(560*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*d*e^5 - 1
7*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a
*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2
*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)))/b^(3/2))*A*d^2*abs(b) + 56*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x +
a)*(8*(b*x + a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*
b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^
15)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b
*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*a^5*e^5)*e
^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*d^2*abs(b) +
1120*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - a*e^2)*e^(-4)/b^4)
+ (b^2*d^2 - 2*a*b*d*e + a^2*e^2)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*
e - a*b*e)))/b^(7/2))*A*a^2*d^2*abs(b)/b^2 + 1120*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x +
a)*(6*(b*x + a)/b^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6
)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) +
3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*
sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*a*d^2*abs(b)/b + 112*(sqrt(b^2*d + (b*x + a
)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*
b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2
*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*
e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b
^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)))/b^(5/2))*A*d*abs(b)*e + 28*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*(8*(b*x +
a)*(10*(b*x + a)/b^4 + (b^21*d*e^9 - 49*a*b^20*e^10)*e^(-10)/b^24) - 3*(3*b^22*d^2*e^8 + 10*a*b^21*d*e^9 - 25
3*a^2*b^20*e^10)*e^(-10)/b^24) + (21*b^23*d^3*e^7 + 49*a*b^22*d^2*e^8 + 79*a^2*b^21*d*e^9 - 1429*a^3*b^20*e^10
)*e^(-10)/b^24)*(b*x + a) - 5*(21*b^24*d^4*e^6 + 28*a*b^23*d^3*e^7 + 30*a^2*b^22*d^2*e^8 + 28*a^3*b^21*d*e^9 -
491*a^4*b^20*e^10)*e^(-10)/b^24)*(b*x + a) + 15*(21*b^25*d^5*e^5 + 7*a*b^24*d^4*e^6 + 2*a^2*b^23*d^3*e^7 - 2*
a^3*b^22*d^2*e^8 - 7*a^4*b^21*d*e^9 - 21*a^5*b^20*e^10)*e^(-10)/b^24)*sqrt(b*x + a) + 15*(21*b^6*d^6 - 14*a*b^
5*d^5*e - 5*a^2*b^4*d^4*e^2 - 4*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 14*a^5*b*d*e^5 + 21*a^6*e^6)*e^(-11/2)*l
og(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*d*abs(b)*e + 1120*(sq
rt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-
6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^
2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*
a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)
))/b^(3/2))*B*a^2*d*abs(b)*e/b^2 + 2240*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x
+ a)/b^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b
^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d
^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^
(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*A*a*d*abs(b)*e/b + 224*(sqrt(b^2*d + (b*x + a)*b*e - a*
b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e
^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^
7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^
4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3
- 5*a^4*b*d*e^4 + 7*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a
*b*e)))/b^(5/2))*B*a*d*abs(b)*e/b + 56*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x
+ a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^
2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)))/b^(11/2))*B*a^2*d^2*abs(b)/b^3 + 112*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a
)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d
^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)))/b^(11/2))*A*a*d^2*abs(b)/b^2 + 14*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*
(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*d*e^9 - 49*a*b^20*e^10)*e^(-10)/b^24) - 3*(3*b^22*d^2*e^8 + 10*a*b^21*d
*e^9 - 253*a^2*b^20*e^10)*e^(-10)/b^24) + (21*b^23*d^3*e^7 + 49*a*b^22*d^2*e^8 + 79*a^2*b^21*d*e^9 - 1429*a^3*
b^20*e^10)*e^(-10)/b^24)*(b*x + a) - 5*(21*b^24*d^4*e^6 + 28*a*b^23*d^3*e^7 + 30*a^2*b^22*d^2*e^8 + 28*a^3*b^2
1*d*e^9 - 491*a^4*b^20*e^10)*e^(-10)/b^24)*(b*x + a) + 15*(21*b^25*d^5*e^5 + 7*a*b^24*d^4*e^6 + 2*a^2*b^23*d^3
*e^7 - 2*a^3*b^22*d^2*e^8 - 7*a^4*b^21*d*e^9 - 21*a^5*b^20*e^10)*e^(-10)/b^24)*sqrt(b*x + a) + 15*(21*b^6*d^6
- 14*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 4*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 14*a^5*b*d*e^5 + 21*a^6*e^6)*e^
(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*A*abs(b)*e^2 +
(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(8*(b*x + a)*(10*(b*x + a)*(12*(b*x + a)/b^5 + (b^31*d*e^11 - 7
1*a*b^30*e^12)*e^(-12)/b^35) - (11*b^32*d^2*e^10 + 48*a*b^31*d*e^11 - 1739*a^2*b^30*e^12)*e^(-12)/b^35) + 3*(3
3*b^33*d^3*e^9 + 111*a*b^32*d^2*e^10 + 239*a^2*b^31*d*e^11 - 5983*a^3*b^30*e^12)*e^(-12)/b^35)*(b*x + a) - 7*(
33*b^34*d^4*e^8 + 78*a*b^33*d^3*e^9 + 128*a^2*b^32*d^2*e^10 + 178*a^3*b^31*d*e^11 - 3617*a^4*b^30*e^12)*e^(-12
)/b^35)*(b*x + a) + 35*(33*b^35*d^5*e^7 + 45*a*b^34*d^4*e^8 + 50*a^2*b^33*d^3*e^9 + 50*a^3*b^32*d^2*e^10 + 45*
a^4*b^31*d*e^11 - 991*a^5*b^30*e^12)*e^(-12)/b^35)*(b*x + a) - 105*(33*b^36*d^6*e^6 + 12*a*b^35*d^5*e^7 + 5*a^
2*b^34*d^4*e^8 - 5*a^4*b^32*d^2*e^10 - 12*a^5*b^31*d*e^11 - 33*a^6*b^30*e^12)*e^(-12)/b^35)*sqrt(b*x + a) - 10
5*(33*b^7*d^7 - 21*a*b^6*d^6*e - 7*a^2*b^5*d^5*e^2 - 5*a^3*b^4*d^4*e^3 - 5*a^4*b^3*d^3*e^4 - 7*a^5*b^2*d^2*e^5
- 21*a^6*b*d*e^6 + 33*a^7*e^7)*e^(-13/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)))/b^(9/2))*B*abs(b)*e^2 + 560*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x
+ a)/b^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^
8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^
4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(
1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*A*a^2*abs(b)*e^2/b^2 + 56*(sqrt(b^2*d + (b*x + a)*b*e -
a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2
*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*
e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*
a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e
^3 - 5*a^4*b*d*e^4 + 7*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e -
a*b*e)))/b^(5/2))*B*a^2*abs(b)*e^2/b^2 + 112*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a
)*(8*(b*x + a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b
^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^1
5)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*
x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*a^5*e^5)*e^
(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*A*a*abs(b)*e^2/
b + 28*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*d*e^9 -
49*a*b^20*e^10)*e^(-10)/b^24) - 3*(3*b^22*d^2*e^8 + 10*a*b^21*d*e^9 - 253*a^2*b^20*e^10)*e^(-10)/b^24) + (21*b
^23*d^3*e^7 + 49*a*b^22*d^2*e^8 + 79*a^2*b^21*d*e^9 - 1429*a^3*b^20*e^10)*e^(-10)/b^24)*(b*x + a) - 5*(21*b^24
*d^4*e^6 + 28*a*b^23*d^3*e^7 + 30*a^2*b^22*d^2*e^8 + 28*a^3*b^21*d*e^9 - 491*a^4*b^20*e^10)*e^(-10)/b^24)*(b*x
+ a) + 15*(21*b^25*d^5*e^5 + 7*a*b^24*d^4*e^6 + 2*a^2*b^23*d^3*e^7 - 2*a^3*b^22*d^2*e^8 - 7*a^4*b^21*d*e^9 -
21*a^5*b^20*e^10)*e^(-10)/b^24)*sqrt(b*x + a) + 15*(21*b^6*d^6 - 14*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 4*a^3*b^
3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 14*a^5*b*d*e^5 + 21*a^6*e^6)*e^(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2)
+ sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*a*abs(b)*e^2/b + 112*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*s
qrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4
)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1
/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*A*a^2*d*abs(b)*e/b^3)/b