3.1824 \(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=339 \[ -\frac {21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac {21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt {d+e x} (b d-a e)^6}+\frac {7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac {21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac {3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac {5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

[Out]

21/40*e^2*(-11*A*b*e+5*B*a*e+6*B*b*d)/b/(-a*e+b*d)^4/(e*x+d)^(5/2)+1/3*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^3/(e*x+
d)^(5/2)+1/12*(11*A*b*e-5*B*a*e-6*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)^(5/2)+3/8*e*(-11*A*b*e+5*B*a*e+6*B*b
*d)/b/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(5/2)+7/8*e^2*(-11*A*b*e+5*B*a*e+6*B*b*d)/(-a*e+b*d)^5/(e*x+d)^(3/2)-21/8*b
^(3/2)*e^2*(-11*A*b*e+5*B*a*e+6*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(13/2)+21/8*
b*e^2*(-11*A*b*e+5*B*a*e+6*B*b*d)/(-a*e+b*d)^6/(e*x+d)^(1/2)

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Rubi [A]  time = 0.38, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ -\frac {21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac {21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt {d+e x} (b d-a e)^6}+\frac {7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac {21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac {3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac {5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(40*b*(b*d - a*e)^4*(d + e*x)^(5/2)) - (A*b - a*B)/(3*b*(b*d - a*e)*(a
+ b*x)^3*(d + e*x)^(5/2)) - (6*b*B*d - 11*A*b*e + 5*a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) +
(3*e*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) + (7*e^2*(6*b*B*d - 11*A*b
*e + 5*a*B*e))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)) + (21*b*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^6*
Sqrt[d + e*x]) - (21*b^(3/2)*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(8*(b*d - a*e)^(13/2))

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 51

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 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 78

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] || LtQ
[p, n]))))

Rule 208

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

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {A+B x}{(a+b x)^4 (d+e x)^{7/2}} \, dx\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {(6 b B d-11 A b e+5 a B e) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 b (b d-a e)}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {(3 e (6 b B d-11 A b e+5 a B e)) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {\left (21 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac {21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {\left (21 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4}\\ &=\frac {21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac {\left (21 b e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5}\\ &=\frac {21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac {21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt {d+e x}}+\frac {\left (21 b^2 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^6}\\ &=\frac {21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac {21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt {d+e x}}+\frac {\left (21 b^2 e (6 b B d-11 A b e+5 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^6}\\ &=\frac {21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac {7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac {21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt {d+e x}}-\frac {21 b^{3/2} e^2 (6 b B d-11 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 100, normalized size = 0.29 \[ \frac {\frac {5 a B-5 A b}{(a+b x)^3}-\frac {e^2 (-5 a B e+11 A b e-6 b B d) \, _2F_1\left (-\frac {5}{2},3;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{15 b (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*A*b + 5*a*B)/(a + b*x)^3 - (e^2*(-6*b*B*d + 11*A*b*e - 5*a*B*e)*Hypergeometric2F1[-5/2, 3, -3/2, (b*(d +
e*x))/(b*d - a*e)])/(b*d - a*e)^3)/(15*b*(b*d - a*e)*(d + e*x)^(5/2))

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fricas [B]  time = 1.46, size = 3685, normalized size = 10.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/240*(315*(6*B*a^3*b^2*d^4*e^2 + (5*B*a^4*b - 11*A*a^3*b^2)*d^3*e^3 + (6*B*b^5*d*e^5 + (5*B*a*b^4 - 11*A*b^
5)*e^6)*x^6 + 3*(6*B*b^5*d^2*e^4 + 11*(B*a*b^4 - A*b^5)*d*e^5 + (5*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + 3*(6*B*b
^5*d^3*e^3 + (23*B*a*b^4 - 11*A*b^5)*d^2*e^4 + 3*(7*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + (5*B*a^3*b^2 - 11*A*a^2*b^
3)*e^6)*x^4 + (6*B*b^5*d^4*e^2 + (59*B*a*b^4 - 11*A*b^5)*d^3*e^3 + 99*(B*a^2*b^3 - A*a*b^4)*d^2*e^4 + 3*(17*B*
a^3*b^2 - 33*A*a^2*b^3)*d*e^5 + (5*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 + 3*(6*B*a*b^4*d^4*e^2 + (23*B*a^2*b^3 - 1
1*A*a*b^4)*d^3*e^3 + 3*(7*B*a^3*b^2 - 11*A*a^2*b^3)*d^2*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*d*e^5)*x^2 + 3*(6*B*a
^2*b^3*d^4*e^2 + 11*(B*a^3*b^2 - A*a^2*b^3)*d^3*e^3 + (5*B*a^4*b - 11*A*a^3*b^2)*d^2*e^4)*x)*sqrt(b/(b*d - a*e
))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(48*A*a^5*e^5 +
20*(B*a*b^4 + 2*A*b^5)*d^5 - 10*(26*B*a^2*b^3 + 31*A*a*b^4)*d^4*e - 3*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2
- 16*(44*B*a^4*b - 173*A*a^3*b^2)*d^2*e^3 + 32*(B*a^5 - 13*A*a^4*b)*d*e^4 - 315*(6*B*b^5*d*e^4 + (5*B*a*b^4 -
11*A*b^5)*e^5)*x^5 - 105*(42*B*b^5*d^2*e^3 + (83*B*a*b^4 - 77*A*b^5)*d*e^4 + 8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)
*x^4 - 21*(138*B*b^5*d^3*e^2 + (679*B*a*b^4 - 253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4 + 33*
(5*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 - 9*(30*B*b^5*d^4*e + (901*B*a*b^4 - 55*A*b^5)*d^3*e^2 + 2*(914*B*a^2*b^
3 - 803*A*a*b^4)*d^2*e^3 + 3*(337*B*a^3*b^2 - 671*A*a^2*b^3)*d*e^4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 +
(60*B*b^5*d^5 - 10*(73*B*a*b^4 + 11*A*b^5)*d^4*e - 2*(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^
2 - 4103*A*a^2*b^3)*d^2*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*e^5)*x)*sqrt(e
*x + d))/(a^3*b^6*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8
*b*d^4*e^5 + a^9*d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b
^5*d^2*e^7 - 6*a^5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3
*b^6*d^4*e^5 - 5*a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a
*b^8*d^7*e^2 - 2*a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^
2*e^7 - 3*a^7*b^2*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3
- 36*a^4*b^5*d^5*e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)
*x^3 + 3*(a*b^8*d^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b
^3*d^4*e^5 - 2*a^7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b
^5*d^7*e^2 - 5*a^5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x), -1
/120*(315*(6*B*a^3*b^2*d^4*e^2 + (5*B*a^4*b - 11*A*a^3*b^2)*d^3*e^3 + (6*B*b^5*d*e^5 + (5*B*a*b^4 - 11*A*b^5)*
e^6)*x^6 + 3*(6*B*b^5*d^2*e^4 + 11*(B*a*b^4 - A*b^5)*d*e^5 + (5*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + 3*(6*B*b^5*
d^3*e^3 + (23*B*a*b^4 - 11*A*b^5)*d^2*e^4 + 3*(7*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + (5*B*a^3*b^2 - 11*A*a^2*b^3)*
e^6)*x^4 + (6*B*b^5*d^4*e^2 + (59*B*a*b^4 - 11*A*b^5)*d^3*e^3 + 99*(B*a^2*b^3 - A*a*b^4)*d^2*e^4 + 3*(17*B*a^3
*b^2 - 33*A*a^2*b^3)*d*e^5 + (5*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 + 3*(6*B*a*b^4*d^4*e^2 + (23*B*a^2*b^3 - 11*A
*a*b^4)*d^3*e^3 + 3*(7*B*a^3*b^2 - 11*A*a^2*b^3)*d^2*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*d*e^5)*x^2 + 3*(6*B*a^2*
b^3*d^4*e^2 + 11*(B*a^3*b^2 - A*a^2*b^3)*d^3*e^3 + (5*B*a^4*b - 11*A*a^3*b^2)*d^2*e^4)*x)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) + (48*A*a^5*e^5 + 20*(B*a*b^4 + 2*A*b^5
)*d^5 - 10*(26*B*a^2*b^3 + 31*A*a*b^4)*d^4*e - 3*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2 - 16*(44*B*a^4*b - 17
3*A*a^3*b^2)*d^2*e^3 + 32*(B*a^5 - 13*A*a^4*b)*d*e^4 - 315*(6*B*b^5*d*e^4 + (5*B*a*b^4 - 11*A*b^5)*e^5)*x^5 -
105*(42*B*b^5*d^2*e^3 + (83*B*a*b^4 - 77*A*b^5)*d*e^4 + 8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 - 21*(138*B*b^5*
d^3*e^2 + (679*B*a*b^4 - 253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4 + 33*(5*B*a^3*b^2 - 11*A*a
^2*b^3)*e^5)*x^3 - 9*(30*B*b^5*d^4*e + (901*B*a*b^4 - 55*A*b^5)*d^3*e^2 + 2*(914*B*a^2*b^3 - 803*A*a*b^4)*d^2*
e^3 + 3*(337*B*a^3*b^2 - 671*A*a^2*b^3)*d*e^4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (60*B*b^5*d^5 - 10*(7
3*B*a*b^4 + 11*A*b^5)*d^4*e - 2*(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^2 - 4103*A*a^2*b^3)*d
^2*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*e^5)*x)*sqrt(e*x + d))/(a^3*b^6*d^9
- 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*d^3*
e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^5*b^
4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*a^4*
b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*a^2*
b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2*d*e
^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*e^4
- 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d^9 -
3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^7*b^
2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^5*b^
4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x)]

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giac [B]  time = 0.34, size = 779, normalized size = 2.30 \[ \frac {21 \, {\left (6 \, B b^{3} d e^{2} + 5 \, B a b^{2} e^{3} - 11 \, A b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} B b^{2} d e^{2} + 15 \, {\left (x e + d\right )} B b^{2} d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 60 \, {\left (x e + d\right )}^{2} B a b e^{3} - 150 \, {\left (x e + d\right )}^{2} A b^{2} e^{3} - 10 \, {\left (x e + d\right )} B a b d e^{3} - 20 \, {\left (x e + d\right )} A b^{2} d e^{3} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 5 \, {\left (x e + d\right )} B a^{2} e^{4} + 20 \, {\left (x e + d\right )} A a b e^{4} + 3 \, B a^{2} d e^{4} + 6 \, A a b d e^{4} - 3 \, A a^{2} e^{5}\right )}}{15 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} + \frac {90 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d e^{2} - 192 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{2} e^{2} + 102 \, \sqrt {x e + d} B b^{5} d^{3} e^{2} + 123 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} e^{3} - 213 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} e^{3} - 88 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d e^{3} + 472 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d e^{3} - 39 \, \sqrt {x e + d} B a b^{4} d^{2} e^{3} - 267 \, \sqrt {x e + d} A b^{5} d^{2} e^{3} + 280 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} e^{4} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} e^{4} - 228 \, \sqrt {x e + d} B a^{2} b^{3} d e^{4} + 534 \, \sqrt {x e + d} A a b^{4} d e^{4} + 165 \, \sqrt {x e + d} B a^{3} b^{2} e^{5} - 267 \, \sqrt {x e + d} A a^{2} b^{3} e^{5}}{24 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/8*(6*B*b^3*d*e^2 + 5*B*a*b^2*e^3 - 11*A*b^3*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6
*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b
^2*d + a*b*e)) + 2/15*(90*(x*e + d)^2*B*b^2*d*e^2 + 15*(x*e + d)*B*b^2*d^2*e^2 + 3*B*b^2*d^3*e^2 + 60*(x*e + d
)^2*B*a*b*e^3 - 150*(x*e + d)^2*A*b^2*e^3 - 10*(x*e + d)*B*a*b*d*e^3 - 20*(x*e + d)*A*b^2*d*e^3 - 6*B*a*b*d^2*
e^3 - 3*A*b^2*d^2*e^3 - 5*(x*e + d)*B*a^2*e^4 + 20*(x*e + d)*A*a*b*e^4 + 3*B*a^2*d*e^4 + 6*A*a*b*d*e^4 - 3*A*a
^2*e^5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e
^5 + a^6*e^6)*(x*e + d)^(5/2)) + 1/24*(90*(x*e + d)^(5/2)*B*b^5*d*e^2 - 192*(x*e + d)^(3/2)*B*b^5*d^2*e^2 + 10
2*sqrt(x*e + d)*B*b^5*d^3*e^2 + 123*(x*e + d)^(5/2)*B*a*b^4*e^3 - 213*(x*e + d)^(5/2)*A*b^5*e^3 - 88*(x*e + d)
^(3/2)*B*a*b^4*d*e^3 + 472*(x*e + d)^(3/2)*A*b^5*d*e^3 - 39*sqrt(x*e + d)*B*a*b^4*d^2*e^3 - 267*sqrt(x*e + d)*
A*b^5*d^2*e^3 + 280*(x*e + d)^(3/2)*B*a^2*b^3*e^4 - 472*(x*e + d)^(3/2)*A*a*b^4*e^4 - 228*sqrt(x*e + d)*B*a^2*
b^3*d*e^4 + 534*sqrt(x*e + d)*A*a*b^4*d*e^4 + 165*sqrt(x*e + d)*B*a^3*b^2*e^5 - 267*sqrt(x*e + d)*A*a^2*b^3*e^
5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 +
a^6*e^6)*((x*e + d)*b - b*d + a*e)^3)

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maple [B]  time = 0.08, size = 935, normalized size = 2.76 \[ -\frac {89 \sqrt {e x +d}\, A \,a^{2} b^{3} e^{5}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {89 \sqrt {e x +d}\, A a \,b^{4} d \,e^{4}}{4 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {89 \sqrt {e x +d}\, A \,b^{5} d^{2} e^{3}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {55 \sqrt {e x +d}\, B \,a^{3} b^{2} e^{5}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {19 \sqrt {e x +d}\, B \,a^{2} b^{3} d \,e^{4}}{2 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {13 \sqrt {e x +d}\, B a \,b^{4} d^{2} e^{3}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {17 \sqrt {e x +d}\, B \,b^{5} d^{3} e^{2}}{4 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {59 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{4} e^{4}}{3 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {59 \left (e x +d \right )^{\frac {3}{2}} A \,b^{5} d \,e^{3}}{3 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {35 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{3} e^{4}}{3 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {11 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{4} d \,e^{3}}{3 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {8 \left (e x +d \right )^{\frac {3}{2}} B \,b^{5} d^{2} e^{2}}{\left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {71 \left (e x +d \right )^{\frac {5}{2}} A \,b^{5} e^{3}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {41 \left (e x +d \right )^{\frac {5}{2}} B a \,b^{4} e^{3}}{8 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}+\frac {15 \left (e x +d \right )^{\frac {5}{2}} B \,b^{5} d \,e^{2}}{4 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{3}}-\frac {231 A \,b^{3} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{6} \sqrt {\left (a e -b d \right ) b}}+\frac {105 B a \,b^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{6} \sqrt {\left (a e -b d \right ) b}}+\frac {63 B \,b^{3} d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{6} \sqrt {\left (a e -b d \right ) b}}-\frac {20 A \,b^{2} e^{3}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}+\frac {8 B a b \,e^{3}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}+\frac {12 B \,b^{2} d \,e^{2}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}+\frac {8 A b \,e^{3}}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B a \,e^{3}}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B b d \,e^{2}}{\left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A \,e^{3}}{5 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 B d \,e^{2}}{5 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*e^3/(a*e-b*d)^4/(e*x+d)^(5/2)*A+2/5*e^2/(a*e-b*d)^4/(e*x+d)^(5/2)*B*d-13/8*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e
)^3*(e*x+d)^(1/2)*B*a*d^2-11/3*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d+89/4*e^4/(a*e-b*d)^6*b^4/
(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d-19/2*e^4/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d-2/3*e^3/(a*e-b*
d)^5/(e*x+d)^(3/2)*a*B-20*e^3*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*A-89/8*e^5/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(
1/2)*A*a^2-89/8*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2+55/8*e^5/(a*e-b*d)^6*b^2/(b*e*x+a*e)^3*(
e*x+d)^(1/2)*B*a^3+63/4*e^2/(a*e-b*d)^6*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*
d+17/4*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^3-8*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*B*(e*x+d)^(3/
2)*d^2+15/4*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d+8/3*e^3/(a*e-b*d)^5/(e*x+d)^(3/2)*A*b-2*e^2/(a
*e-b*d)^5/(e*x+d)^(3/2)*B*b*d+12*e^2*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*B*d+8*e^3*b/(a*e-b*d)^6/(e*x+d)^(1/2)*a*B-2
31/8*e^3/(a*e-b*d)^6*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A-71/8*e^3/(a*e-b*d)^
6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A+105/8*e^3/(a*e-b*d)^6*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)*b)*a*B+41/8*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a-59/3*e^4/(a*e-b*d)^6*b^4/(b*e*x+a
*e)^3*A*(e*x+d)^(3/2)*a+59/3*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d+35/3*e^4/(a*e-b*d)^6*b^3/(b*e
*x+a*e)^3*B*(e*x+d)^(3/2)*a^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 2.65, size = 547, normalized size = 1.61 \[ \frac {\frac {231\,{\left (d+e\,x\right )}^3\,\left (-11\,A\,b^3\,e^3+6\,B\,d\,b^3\,e^2+5\,B\,a\,b^2\,e^3\right )}{40\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,\left (A\,e^3-B\,d\,e^2\right )}{5\,\left (a\,e-b\,d\right )}+\frac {7\,{\left (d+e\,x\right )}^4\,\left (-11\,A\,b^4\,e^3+6\,B\,d\,b^4\,e^2+5\,B\,a\,b^3\,e^3\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {2\,\left (d+e\,x\right )\,\left (5\,B\,a\,e^3-11\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {6\,b\,{\left (d+e\,x\right )}^2\,\left (5\,B\,a\,e^3-11\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{5\,{\left (a\,e-b\,d\right )}^3}+\frac {21\,b^4\,{\left (d+e\,x\right )}^5\,\left (5\,B\,a\,e^3-11\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^6}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{11/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}+\frac {21\,b^{3/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (5\,B\,a\,e-11\,A\,b\,e+6\,B\,b\,d\right )\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}\,\left (5\,B\,a\,e^3-11\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}\right )\,\left (5\,B\,a\,e-11\,A\,b\,e+6\,B\,b\,d\right )}{8\,{\left (a\,e-b\,d\right )}^{13/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((231*(d + e*x)^3*(5*B*a*b^2*e^3 - 11*A*b^3*e^3 + 6*B*b^3*d*e^2))/(40*(a*e - b*d)^4) - (2*(A*e^3 - B*d*e^2))/(
5*(a*e - b*d)) + (7*(d + e*x)^4*(5*B*a*b^3*e^3 - 11*A*b^4*e^3 + 6*B*b^4*d*e^2))/(a*e - b*d)^5 - (2*(d + e*x)*(
5*B*a*e^3 - 11*A*b*e^3 + 6*B*b*d*e^2))/(15*(a*e - b*d)^2) + (6*b*(d + e*x)^2*(5*B*a*e^3 - 11*A*b*e^3 + 6*B*b*d
*e^2))/(5*(a*e - b*d)^3) + (21*b^4*(d + e*x)^5*(5*B*a*e^3 - 11*A*b*e^3 + 6*B*b*d*e^2))/(8*(a*e - b*d)^6))/((d
+ e*x)^(5/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + b^3*(d + e*x)^(11/2) - (3*b^3*d - 3*a*b^2*e
)*(d + e*x)^(9/2) + (d + e*x)^(7/2)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e)) + (21*b^(3/2)*e^2*atan((b^(1/2)*e
^2*(d + e*x)^(1/2)*(5*B*a*e - 11*A*b*e + 6*B*b*d)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3
+ 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/((a*e - b*d)^(13/2)*(5*B*a*e^3 - 11*A*b*e^3 + 6*B*b*d*
e^2)))*(5*B*a*e - 11*A*b*e + 6*B*b*d))/(8*(a*e - b*d)^(13/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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