\(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1834]

Optimal result

Integrand size = 33, antiderivative size = 435 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \]

[Out]

3003/640*e^4*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^6/(e*x+d)^(5/2)+1/5*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^5/(e*x+
d)^(5/2)+1/8*(3*A*b*e-B*a*e-2*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^4/(e*x+d)^(5/2)+13/48*e*(-3*A*b*e+B*a*e+2*B*b*d)/b
/(-a*e+b*d)^3/(b*x+a)^3/(e*x+d)^(5/2)-143/192*e^2*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(5
/2)+429/128*e^3*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(5/2)+1001/128*e^4*(-3*A*b*e+B*a*e+2*B
*b*d)/(-a*e+b*d)^7/(e*x+d)^(3/2)-3003/128*b^(3/2)*e^4*(-3*A*b*e+B*a*e+2*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(
-a*e+b*d)^(1/2))/(-a*e+b*d)^(17/2)+3003/128*b*e^4*(-3*A*b*e+B*a*e+2*B*b*d)/(-a*e+b*d)^8/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 44, 53, 65, 214} \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac {3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt {d+e x} (b d-a e)^8}+\frac {1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac {3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac {429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac {143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac {13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*(a
+ b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13
*e*(2*b*B*d - 3*A*b*e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d - 3*A*b*e
+ a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b
*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) + (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(
3/2)) + (3003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003*b^(3/2)*e^4*(2*b*B*
d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

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*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 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]))))

Rule 214

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{(a+b x)^6 (d+e x)^{7/2}} \, dx \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 b (b d-a e)} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {(13 e (2 b B d-3 A b e+a B e)) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^2} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {\left (143 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{96 b (b d-a e)^3} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {\left (429 e^3 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 b (b d-a e)^4} \\ & = -\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 b (b d-a e)^5} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {\left (3003 b e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^8} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^3 (2 b B d-3 A b e+a B e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 (b d-a e)^8} \\ & = \frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(884\) vs. \(2(435)=870\).

Time = 5.46 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.03 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-3 A \left (256 a^7 e^7-256 a^6 b e^6 (12 d+5 e x)+256 a^5 b^2 e^5 \left (116 d^2+160 d e x+65 e^2 x^2\right )+5 a^4 b^3 e^4 \left (5327 d^3+45677 d^2 e x+66157 d e^2 x^2+27599 e^3 x^3\right )+10 a^3 b^4 e^3 \left (-1211 d^4+5810 d^3 e x+54392 d^2 e^2 x^2+80366 d e^3 x^3+33891 e^4 x^4\right )+2 a^2 b^5 e^2 \left (2324 d^5-6545 d^4 e x+30485 d^3 e^2 x^2+302445 d^2 e^3 x^3+452595 d e^4 x^4+192192 e^5 x^5\right )+2 a b^6 e \left (-568 d^6+1240 d^5 e x-3445 d^4 e^2 x^2+15730 d^3 e^3 x^3+163020 d^2 e^4 x^4+246246 d e^5 x^5+105105 e^6 x^6\right )+b^7 \left (128 d^7-240 d^6 e x+520 d^5 e^2 x^2-1430 d^4 e^3 x^3+6435 d^3 e^4 x^4+69069 d^2 e^5 x^5+105105 d e^6 x^6+45045 e^7 x^7\right )\right )+B \left (-256 a^7 e^6 (2 d+5 e x)+256 a^6 b e^5 \left (64 d^2+150 d e x+65 e^2 x^2\right )+a^5 b^2 e^4 \left (100363 d^3+310305 d^2 e x+364065 d e^2 x^2+137995 e^3 x^3\right )+10 a^4 b^3 e^3 \left (2324 d^4+51487 d^3 e x+120549 d^2 e^2 x^2+107965 d e^3 x^3+33891 e^4 x^4\right )+2 a^3 b^4 e^2 \left (-2618 d^5+51555 d^4 e x+574405 d^3 e^2 x^2+1106105 d^2 e^3 x^3+791505 d e^4 x^4+192192 e^5 x^5\right )+2 b^7 d x \left (-240 d^6+520 d^5 e x-1430 d^4 e^2 x^2+6435 d^3 e^3 x^3+69069 d^2 e^4 x^4+105105 d e^5 x^5+45045 e^6 x^6\right )+2 a^2 b^5 e \left (496 d^6-11850 d^5 e x+57525 d^4 e^2 x^2+620620 d^3 e^3 x^3+1068210 d^2 e^4 x^4+630630 d e^5 x^5+105105 e^6 x^6\right )+a b^6 \left (-96 d^7+4720 d^6 e x-13260 d^5 e^2 x^2+61490 d^4 e^3 x^3+658515 d^3 e^4 x^4+1054053 d^2 e^5 x^5+525525 d e^6 x^6+45045 e^7 x^7\right )\right )}{(b d-a e)^8 (a+b x)^5 (d+e x)^{5/2}}+\frac {45045 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{17/2}}}{1920} \]

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((-3*A*(256*a^7*e^7 - 256*a^6*b*e^6*(12*d + 5*e*x) + 256*a^5*b^2*e^5*(116*d^2 + 160*d*e*x + 65*e^2*x^2) + 5*a^
4*b^3*e^4*(5327*d^3 + 45677*d^2*e*x + 66157*d*e^2*x^2 + 27599*e^3*x^3) + 10*a^3*b^4*e^3*(-1211*d^4 + 5810*d^3*
e*x + 54392*d^2*e^2*x^2 + 80366*d*e^3*x^3 + 33891*e^4*x^4) + 2*a^2*b^5*e^2*(2324*d^5 - 6545*d^4*e*x + 30485*d^
3*e^2*x^2 + 302445*d^2*e^3*x^3 + 452595*d*e^4*x^4 + 192192*e^5*x^5) + 2*a*b^6*e*(-568*d^6 + 1240*d^5*e*x - 344
5*d^4*e^2*x^2 + 15730*d^3*e^3*x^3 + 163020*d^2*e^4*x^4 + 246246*d*e^5*x^5 + 105105*e^6*x^6) + b^7*(128*d^7 - 2
40*d^6*e*x + 520*d^5*e^2*x^2 - 1430*d^4*e^3*x^3 + 6435*d^3*e^4*x^4 + 69069*d^2*e^5*x^5 + 105105*d*e^6*x^6 + 45
045*e^7*x^7)) + B*(-256*a^7*e^6*(2*d + 5*e*x) + 256*a^6*b*e^5*(64*d^2 + 150*d*e*x + 65*e^2*x^2) + a^5*b^2*e^4*
(100363*d^3 + 310305*d^2*e*x + 364065*d*e^2*x^2 + 137995*e^3*x^3) + 10*a^4*b^3*e^3*(2324*d^4 + 51487*d^3*e*x +
120549*d^2*e^2*x^2 + 107965*d*e^3*x^3 + 33891*e^4*x^4) + 2*a^3*b^4*e^2*(-2618*d^5 + 51555*d^4*e*x + 574405*d^
3*e^2*x^2 + 1106105*d^2*e^3*x^3 + 791505*d*e^4*x^4 + 192192*e^5*x^5) + 2*b^7*d*x*(-240*d^6 + 520*d^5*e*x - 143
0*d^4*e^2*x^2 + 6435*d^3*e^3*x^3 + 69069*d^2*e^4*x^4 + 105105*d*e^5*x^5 + 45045*e^6*x^6) + 2*a^2*b^5*e*(496*d^
6 - 11850*d^5*e*x + 57525*d^4*e^2*x^2 + 620620*d^3*e^3*x^3 + 1068210*d^2*e^4*x^4 + 630630*d*e^5*x^5 + 105105*e
^6*x^6) + a*b^6*(-96*d^7 + 4720*d^6*e*x - 13260*d^5*e^2*x^2 + 61490*d^4*e^3*x^3 + 658515*d^3*e^4*x^4 + 1054053
*d^2*e^5*x^5 + 525525*d*e^6*x^6 + 45045*e^7*x^7)))/((b*d - a*e)^8*(a + b*x)^5*(d + e*x)^(5/2)) + (45045*b^(3/2
)*e^4*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(17/2))/1
920

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(2 e^{4} \left (-\frac {A e -B d}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {-6 A b e +B a e +5 B b d}{3 \left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b \left (7 A b e -2 B a e -5 B b d \right )}{\left (a e -b d \right )^{8} \sqrt {e x +d}}-\frac {b^{2} \left (\frac {\left (\frac {3633}{256} A e \,b^{5}-\frac {1467}{256} B e \,b^{4} a -\frac {1083}{128} B d \,b^{5}\right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (23511 A a b \,e^{2}-23511 A \,b^{2} d e -9629 a^{2} B \,e^{2}-4253 B a b d e +13882 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1001}{10} A \,a^{2} b^{3} e^{3}-\frac {1001}{5} A a \,b^{4} d \,e^{2}+\frac {1001}{10} A \,b^{5} d^{2} e -\frac {1253}{30} B \,e^{3} a^{3} b^{2}+\frac {749}{10} B a \,b^{4} d^{2} e -\frac {175}{3} B \,b^{5} d^{3}+\frac {126}{5} B \,a^{2} b^{3} d \,e^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} A \,a^{3} b^{2} e^{4}-\frac {28329}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {28329}{128} A a \,b^{4} d^{2} e^{2}-\frac {9443}{128} A \,b^{5} d^{3} e -\frac {12131}{384} B \,e^{4} a^{4} b +\frac {20195}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4067}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {36463}{384} B a \,b^{4} d^{3} e +\frac {8099}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} A \,a^{4} b \,e^{5}-\frac {5327}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {15981}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {5327}{64} A a \,b^{4} d^{3} e^{2}+\frac {5327}{256} A \,b^{5} d^{4} e -\frac {2373}{256} B \,a^{5} e^{5}+\frac {3269}{128} B \,a^{4} b d \,e^{4}-\frac {1211}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {1029}{32} B \,a^{2} b^{3} d^{3} e^{2}+\frac {9443}{256} B a \,b^{4} d^{4} e -\frac {1477}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (3 A b e -B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{8}}\right )\)	\(618\)
default	\(2 e^{4} \left (-\frac {A e -B d}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {-6 A b e +B a e +5 B b d}{3 \left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b \left (7 A b e -2 B a e -5 B b d \right )}{\left (a e -b d \right )^{8} \sqrt {e x +d}}-\frac {b^{2} \left (\frac {\left (\frac {3633}{256} A e \,b^{5}-\frac {1467}{256} B e \,b^{4} a -\frac {1083}{128} B d \,b^{5}\right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (23511 A a b \,e^{2}-23511 A \,b^{2} d e -9629 a^{2} B \,e^{2}-4253 B a b d e +13882 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1001}{10} A \,a^{2} b^{3} e^{3}-\frac {1001}{5} A a \,b^{4} d \,e^{2}+\frac {1001}{10} A \,b^{5} d^{2} e -\frac {1253}{30} B \,e^{3} a^{3} b^{2}+\frac {749}{10} B a \,b^{4} d^{2} e -\frac {175}{3} B \,b^{5} d^{3}+\frac {126}{5} B \,a^{2} b^{3} d \,e^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} A \,a^{3} b^{2} e^{4}-\frac {28329}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {28329}{128} A a \,b^{4} d^{2} e^{2}-\frac {9443}{128} A \,b^{5} d^{3} e -\frac {12131}{384} B \,e^{4} a^{4} b +\frac {20195}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4067}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {36463}{384} B a \,b^{4} d^{3} e +\frac {8099}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} A \,a^{4} b \,e^{5}-\frac {5327}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {15981}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {5327}{64} A a \,b^{4} d^{3} e^{2}+\frac {5327}{256} A \,b^{5} d^{4} e -\frac {2373}{256} B \,a^{5} e^{5}+\frac {3269}{128} B \,a^{4} b d \,e^{4}-\frac {1211}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {1029}{32} B \,a^{2} b^{3} d^{3} e^{2}+\frac {9443}{256} B a \,b^{4} d^{4} e -\frac {1477}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (3 A b e -B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{8}}\right )\)	\(618\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(735\)

[In]

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^4*(-1/5*(A*e-B*d)/(a*e-b*d)^6/(e*x+d)^(5/2)-1/3*(-6*A*b*e+B*a*e+5*B*b*d)/(a*e-b*d)^7/(e*x+d)^(3/2)-3*b*(7*
A*b*e-2*B*a*e-5*B*b*d)/(a*e-b*d)^8/(e*x+d)^(1/2)-1/(a*e-b*d)^8*b^2*(((3633/256*A*e*b^5-1467/256*B*e*b^4*a-1083
/128*B*d*b^5)*(e*x+d)^(9/2)+1/384*b^3*(23511*A*a*b*e^2-23511*A*b^2*d*e-9629*B*a^2*e^2-4253*B*a*b*d*e+13882*B*b
^2*d^2)*(e*x+d)^(7/2)+(1001/10*A*a^2*b^3*e^3-1001/5*A*a*b^4*d*e^2+1001/10*A*b^5*d^2*e-1253/30*B*e^3*a^3*b^2+74
9/10*B*a*b^4*d^2*e-175/3*B*b^5*d^3+126/5*B*a^2*b^3*d*e^2)*(e*x+d)^(5/2)+(9443/128*A*a^3*b^2*e^4-28329/128*A*a^
2*b^3*d*e^3+28329/128*A*a*b^4*d^2*e^2-9443/128*A*b^5*d^3*e-12131/384*B*e^4*a^4*b+20195/384*B*a^3*b^2*d*e^3+406
7/128*B*a^2*b^3*d^2*e^2-36463/384*B*a*b^4*d^3*e+8099/192*B*b^5*d^4)*(e*x+d)^(3/2)+(5327/256*A*a^4*b*e^5-5327/6
4*A*a^3*b^2*d*e^4+15981/128*A*a^2*b^3*d^2*e^3-5327/64*A*a*b^4*d^3*e^2+5327/256*A*b^5*d^4*e-2373/256*B*a^5*e^5+
3269/128*B*a^4*b*d*e^4-1211/128*B*a^3*b^2*d^2*e^3-1029/32*B*a^2*b^3*d^3*e^2+9443/256*B*a*b^4*d^4*e-1477/128*B*
b^5*d^5)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+3003/256*(3*A*b*e-B*a*e-2*B*b*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e
*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3011 vs. \(2 (391) = 782\).

Time = 5.64 (sec) , antiderivative size = 6033, normalized size of antiderivative = 13.87 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,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(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1700 vs. \(2 (391) = 782\).

Time = 0.31 (sec) , antiderivative size = 1700, normalized size of antiderivative = 3.91 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3003/128*(2*B*b^3*d*e^4 + B*a*b^2*e^5 - 3*A*b^3*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^8*d^8 -
8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2
*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*sqrt(-b^2*d + a*b*e)) + 1/1920*(90090*(e*x + d)^7*B*b^7*d*e^4 - 420420*(e*
x + d)^6*B*b^7*d^2*e^4 + 768768*(e*x + d)^5*B*b^7*d^3*e^4 - 677820*(e*x + d)^4*B*b^7*d^4*e^4 + 275990*(e*x + d
)^3*B*b^7*d^5*e^4 - 33280*(e*x + d)^2*B*b^7*d^6*e^4 - 2560*(e*x + d)*B*b^7*d^7*e^4 - 768*B*b^7*d^8*e^4 + 45045
*(e*x + d)^7*B*a*b^6*e^5 - 135135*(e*x + d)^7*A*b^7*e^5 + 210210*(e*x + d)^6*B*a*b^6*d*e^5 + 630630*(e*x + d)^
6*A*b^7*d*e^5 - 1153152*(e*x + d)^5*B*a*b^6*d^2*e^5 - 1153152*(e*x + d)^5*A*b^7*d^2*e^5 + 1694550*(e*x + d)^4*
B*a*b^6*d^3*e^5 + 1016730*(e*x + d)^4*A*b^7*d^3*e^5 - 965965*(e*x + d)^3*B*a*b^6*d^4*e^5 - 413985*(e*x + d)^3*
A*b^7*d^4*e^5 + 149760*(e*x + d)^2*B*a*b^6*d^5*e^5 + 49920*(e*x + d)^2*A*b^7*d^5*e^5 + 14080*(e*x + d)*B*a*b^6
*d^6*e^5 + 3840*(e*x + d)*A*b^7*d^6*e^5 + 5376*B*a*b^6*d^7*e^5 + 768*A*b^7*d^7*e^5 + 210210*(e*x + d)^6*B*a^2*
b^5*e^6 - 630630*(e*x + d)^6*A*a*b^6*e^6 + 2306304*(e*x + d)^5*A*a*b^6*d*e^6 - 1016730*(e*x + d)^4*B*a^2*b^5*d
^2*e^6 - 3050190*(e*x + d)^4*A*a*b^6*d^2*e^6 + 1103960*(e*x + d)^3*B*a^2*b^5*d^3*e^6 + 1655940*(e*x + d)^3*A*a
*b^6*d^3*e^6 - 249600*(e*x + d)^2*B*a^2*b^5*d^4*e^6 - 249600*(e*x + d)^2*A*a*b^6*d^4*e^6 - 30720*(e*x + d)*B*a
^2*b^5*d^5*e^6 - 23040*(e*x + d)*A*a*b^6*d^5*e^6 - 16128*B*a^2*b^5*d^6*e^6 - 5376*A*a*b^6*d^6*e^6 + 384384*(e*
x + d)^5*B*a^3*b^4*e^7 - 1153152*(e*x + d)^5*A*a^2*b^5*e^7 - 338910*(e*x + d)^4*B*a^3*b^4*d*e^7 + 3050190*(e*x
+ d)^4*A*a^2*b^5*d*e^7 - 275990*(e*x + d)^3*B*a^3*b^4*d^2*e^7 - 2483910*(e*x + d)^3*A*a^2*b^5*d^2*e^7 + 16640
0*(e*x + d)^2*B*a^3*b^4*d^3*e^7 + 499200*(e*x + d)^2*A*a^2*b^5*d^3*e^7 + 32000*(e*x + d)*B*a^3*b^4*d^4*e^7 + 5
7600*(e*x + d)*A*a^2*b^5*d^4*e^7 + 26880*B*a^3*b^4*d^5*e^7 + 16128*A*a^2*b^5*d^5*e^7 + 338910*(e*x + d)^4*B*a^
4*b^3*e^8 - 1016730*(e*x + d)^4*A*a^3*b^4*e^8 - 275990*(e*x + d)^3*B*a^4*b^3*d*e^8 + 1655940*(e*x + d)^3*A*a^3
*b^4*d*e^8 - 499200*(e*x + d)^2*A*a^3*b^4*d^2*e^8 - 12800*(e*x + d)*B*a^4*b^3*d^3*e^8 - 76800*(e*x + d)*A*a^3*
b^4*d^3*e^8 - 26880*B*a^4*b^3*d^4*e^8 - 26880*A*a^3*b^4*d^4*e^8 + 137995*(e*x + d)^3*B*a^5*b^2*e^9 - 413985*(e
*x + d)^3*A*a^4*b^3*e^9 - 49920*(e*x + d)^2*B*a^5*b^2*d*e^9 + 249600*(e*x + d)^2*A*a^4*b^3*d*e^9 - 3840*(e*x +
d)*B*a^5*b^2*d^2*e^9 + 57600*(e*x + d)*A*a^4*b^3*d^2*e^9 + 16128*B*a^5*b^2*d^3*e^9 + 26880*A*a^4*b^3*d^3*e^9
+ 16640*(e*x + d)^2*B*a^6*b*e^10 - 49920*(e*x + d)^2*A*a^5*b^2*e^10 + 5120*(e*x + d)*B*a^6*b*d*e^10 - 23040*(e
*x + d)*A*a^5*b^2*d*e^10 - 5376*B*a^6*b*d^2*e^10 - 16128*A*a^5*b^2*d^2*e^10 - 1280*(e*x + d)*B*a^7*e^11 + 3840
*(e*x + d)*A*a^6*b*e^11 + 768*B*a^7*d*e^11 + 5376*A*a^6*b*d*e^11 - 768*A*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e +
28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^
7*b*d*e^7 + a^8*e^8)*((e*x + d)^(3/2)*b - sqrt(e*x + d)*b*d + sqrt(e*x + d)*a*e)^5)

Mupad [B] (verification not implemented)

Time = 12.61 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {27599\,{\left (d+e\,x\right )}^3\,\left (-3\,A\,b^3\,e^5+2\,B\,d\,b^3\,e^4+B\,a\,b^2\,e^5\right )}{384\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{5\,\left (a\,e-b\,d\right )}+\frac {11297\,{\left (d+e\,x\right )}^4\,\left (-3\,A\,b^4\,e^5+2\,B\,d\,b^4\,e^4+B\,a\,b^3\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,{\left (d+e\,x\right )}^6\,\left (-3\,A\,b^6\,e^5+2\,B\,d\,b^6\,e^4+B\,a\,b^5\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^7}-\frac {2\,\left (d+e\,x\right )\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {26\,b\,{\left (d+e\,x\right )}^2\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {1001\,b^4\,{\left (d+e\,x\right )}^5\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,{\left (d+e\,x\right )}^7\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^8}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{9/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{7/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{15/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}+{\left (d+e\,x\right )}^{11/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )\,\left (a^8\,e^8-8\,a^7\,b\,d\,e^7+28\,a^6\,b^2\,d^2\,e^6-56\,a^5\,b^3\,d^3\,e^5+70\,a^4\,b^4\,d^4\,e^4-56\,a^3\,b^5\,d^5\,e^3+28\,a^2\,b^6\,d^6\,e^2-8\,a\,b^7\,d^7\,e+b^8\,d^8\right )}{{\left (a\,e-b\,d\right )}^{17/2}\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{128\,{\left (a\,e-b\,d\right )}^{17/2}} \]

[In]

int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((27599*(d + e*x)^3*(B*a*b^2*e^5 - 3*A*b^3*e^5 + 2*B*b^3*d*e^4))/(384*(a*e - b*d)^4) - (2*(A*e^5 - B*d*e^4))/(
5*(a*e - b*d)) + (11297*(d + e*x)^4*(B*a*b^3*e^5 - 3*A*b^4*e^5 + 2*B*b^4*d*e^4))/(64*(a*e - b*d)^5) + (7007*(d
+ e*x)^6*(B*a*b^5*e^5 - 3*A*b^6*e^5 + 2*B*b^6*d*e^4))/(64*(a*e - b*d)^7) - (2*(d + e*x)*(B*a*e^5 - 3*A*b*e^5
+ 2*B*b*d*e^4))/(3*(a*e - b*d)^2) + (26*b*(d + e*x)^2*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(3*(a*e - b*d)^3) +
(1001*b^4*(d + e*x)^5*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(5*(a*e - b*d)^6) + (3003*b^6*(d + e*x)^7*(B*a*e^5
- 3*A*b*e^5 + 2*B*b*d*e^4))/(128*(a*e - b*d)^8))/((d + e*x)^(5/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 1
0*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(9/2)*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3
*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(7/2)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 -
20*a*b^4*d^3*e) + b^5*(d + e*x)^(15/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(13/2) + (d + e*x)^(11/2)*(10*b^5*d^2
+ 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (3003*b^(3/2)*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(B*a*e - 3*A*b*e + 2*
B*b*d)*(a^8*e^8 + b^8*d^8 + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5
+ 28*a^6*b^2*d^2*e^6 - 8*a*b^7*d^7*e - 8*a^7*b*d*e^7))/((a*e - b*d)^(17/2)*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4)
))*(B*a*e - 3*A*b*e + 2*B*b*d))/(128*(a*e - b*d)^(17/2))